Projections of modular forms on Eisenstein series and its application to Siegel's formula
aa r X i v : . [ m a t h . N T ] F e b PROJECTIONS OF MODULAR FORMS ON EISENSTEINSERIES AND ITS APPLICATION TO SIEGEL’S FORMULA
ZAFER SELCUK AYGIN
Dedicated to Professor Emeritus Kenneth S. Williams on the occasion of his th birthday. Abstract.
Let k ≥ N be positive integers and let χ be a Dirichletcharacter modulo N . Let f ( z ) be a modular form in M k (Γ ( N ) , χ ). Then wehave a unique decomposition f ( z ) = E f ( z )+ S f ( z ), where E f ( z ) ∈ E k (Γ ( N ) , χ )and S f ( z ) ∈ S k (Γ ( N ) , χ ). In this paper we give an explicit formula for E f ( z )in terms of Eisenstein series. Then we apply our result to certain families of etaquotients and to representations of positive integers by 2 k –ary positive definitequadratic forms in order to give an alternative version of Siegel’s formula for theweighted average number of representations of an integer by quadratic forms inthe same genus. Our formula for the latter is in terms of generalized divisorfunctions and does not involve computation of local densities. Introduction and notation
Let N , N , Z , Q , C and H denote the sets of positive integers, non-negative inte-gers, integers, rational numbers, complex numbers and upper half plane of complexnumbers, respectively. Throughout the paper z denotes a complex number in H , p always denotes a prime number, all divisors considered are positive divisors, q stands for e πiz and χ d ( n ) denotes the Kronecker symbol (cid:16) dn (cid:17) , we use the subscript K to avoid confusion with fractions. Let N ∈ N and χ be a Dirichlet charactermodulo N . The space of modular forms of weight k for Γ ( N ) with character χ isdenoted by M k (Γ ( N ) , χ ); and E k (Γ ( N ) , χ ), S k (Γ ( N ) , χ ) denote its Eisensteinand cusp form subspaces, respectively. Then we have M k (Γ ( N ) , χ ) = E k (Γ ( N ) , χ ) ⊕ S k (Γ ( N ) , χ ) . That is, given f ( z ) ∈ M k (Γ ( N ) , χ ), we can write f ( z ) = E f ( z ) + S f ( z ) , (1.1)where E f ( z ) ∈ E k (Γ ( N ) , χ ) and S f ( z ) ∈ S k (Γ ( N ) , χ ) are uniquely determinedby f . Let ǫ, ψ be primitive Dirichlet characters such that ǫψ = χ (i.e., ǫ ( n ) ψ ( n ) = χ ( n ) for all n ∈ Z coprime to N ) with conductors say L and M , respectively and Mathematics Subject Classification.
Key words and phrases.
Dedekind eta function; theta functions; Eisenstein series; modularforms; cusp forms; Fourier coefficients. uppose LM | N . Let d be a positive divisor of N/LM and 2 ≤ k ∈ N be suchthat χ ( −
1) = ( − k . Let ω be the primitive Dirichlet character corresponding to ǫψ and M ω be its conductor. We define the Eisenstein series associated with ǫ and ψ by E k ( ǫ, ψ ; dz ) := ǫ (0) + (cid:18) M ω M (cid:19) k (cid:18) W ( ψ ) W ( ω ) (cid:19) (cid:18) − kB k,ω (cid:19) Y p | lcm( L,M ) p k p k − ω ( p ) (1.2) × ∞ X n =1 σ k − ( ǫ, ψ ; n ) e πindz , (1.3)where ω is the complex conjugate of ω , σ k − ( ǫ, ψ ; n ) := X ≤ d | n ǫ ( n/d ) ψ ( d ) d k − is the generalized sum of divisors function associated with ǫ and ψ , W ( ψ ) := M − X a =0 ψ ( a ) e πia/M is the Gauss sum of ψ and B k,ω is the k -th generalized Bernoulli number associatedwith ω defined by ∞ X k =0 B k,ω k ! t k = M ω X a =1 ω ( a ) te at e M ω t − , see [13, end of pg. 94]. By [5, Corollary 8.5.5] (alternatively [13, Theorem 4.7.1,(7.1.13) and Lemma 7.2.19]), we have E k ( ǫ, ψ ; dz ) ∈ M k (Γ ( N ) , χ ) if ( k, ǫ, ψ ) = (2 , χ , χ ),and E ( χ , χ ; z ) − N E ( χ , χ ; N z ) ∈ M (Γ ( N ) , χ ) . Remark 1.1.
Let L ( ǫψ, k ) be the Dirichlet L-function defined by L ( ǫψ, k ) := X n ≥ ǫ ( n ) ψ ( n ) n k . The Eisenstein series we define in (1.3) is equal to E k ( M dz ; ǫ, ψ ) / (2 L ( ǫψ, k )) inthe notation of Theorem 7.1.3 and Theorem 7.2.12 of [13] and to G k ( ψ, ǫ )( dz ) /L ( ǫψ, k ) in the notation of Corollary 8.5.5 and Definition 8.5.10 of [5] . Further treatmentto obtain the form in (1.3) is done by using the formula for L ( ǫψ, k ) given in The-orem 3.3.4 and (3.3.14) of [13] . This normalization is chosen so that the constantterms given in (5.1) and (5.2) are simpler. This in return simplifies the notationin Sections 4 and 5. etting D ( N, C ) to denote the group of Dirichlet characters modulo N , we define E ( k, N, χ ) := { ( ǫ, ψ ) ∈ D ( L, C ) × D ( M, C ) : ǫ , ψ primitive, ǫ ( − ψ ( −
1) = ( − k , ǫψ = χ and LM | N } . The set { E k ( ǫ, ψ ; dz ) : ( ǫ, ψ ) ∈ E ( k, N, χ ) , d | N/LM } constitutes a basis for E k (Γ ( N ) , χ ) whenever k ≥ k, χ ) = (2 , χ ); the set { E ( χ , χ ; z ) − dE ( χ , χ ; dz ) : 1 < d | N/LM }∪ { E ( M dz ; ǫ, ψ ) : ( ǫ, ψ ) ∈ E (2 , N, χ ) , ( ǫ, ψ ) = ( χ , χ ) , d | N/LM } constitutes a basis for E (Γ ( N ) , χ ), see [5, Theorems 8.5.17 and 8.5.22], or [22,Proposition 5]. Then we have E f ( z ) = X ( ǫ,ψ ) ∈E ( k,N,χ ) X d | N/LM a f ( ǫ, ψ, d ) E k ( ǫ, ψ ; dz ) , (1.4)for some a f ( ǫ, ψ, d ) ∈ C . When S f ( z ) = 0, it is easy to determine a f ( ǫ, ψ, d ) bycomparing the first few Fourier coefficients of f ( z ) expanded at i ∞ and the firstfew Fourier coefficients of the right hand side of (1.4) expanded at i ∞ . However,if S f ( z ) = 0 and an explicit basis for S k (Γ ( N ) , χ ) is not known then this methodfails. In this paper we solve this problem, in other words we obtain a f ( ǫ, ψ, d )explicitly in an accessible form for all f ∈ M k (Γ ( N ) , χ ) where k ≥
2, see The-orem 1.1. Our treatment is general and its special cases agree with previouslyknown formulas. Additionally, we give a new treatment of Siegel’s formula forrepresentation numbers of quadratic forms, see Theorem 1.2.Let a ∈ Z and c ∈ N be coprime. For an f ( z ) ∈ M k (Γ ( N ) , χ ) we denote theconstant term of f ( z ) in the Fourier expansion of f ( z ) at the cusp a/c by[0] a/c f = lim z → i ∞ ( cz + d ) − k f (cid:18) az + bcz + d (cid:19) , where b, d ∈ Z such that (cid:20) a bc d (cid:21) ∈ SL ( Z ). The value of [0] a/c f does not dependon the choice of b, d . We denote the n th Fourier coefficient of f ( z ) in the expansionat the cusp i ∞ by [ n ] f . Letting φ ( n ) denote the Euler totient function, we definean average associated with ψ for the constant terms of Fourier series expansionsof modular forms at cusps as follows:[0] c,ψ f := 1 φ ( c ) c X a =1 , gcd( a,c )=1 ψ ( a )[0] a/c f. (1.5)We note that working with this average of constant terms at cusps is a new ideawhich helps studying modular form spaces with nontrivial character, see Section5 for details. etting v p ( n ) to denote the highest power of p dividing n and µ ( n ) to be theM¨obius function we are ready to state the main theorem. Theorem 1.1 (Main Theorem) . Let f ( z ) ∈ M k (Γ ( N ) , χ ) , where N, k ∈ N , k ≥ , χ is a Dirichlet character modulo N that satisfies χ ( −
1) = ( − k . Let E f ( z ) bedefined by (1.1) , then E f ( z ) = X ( ǫ,ψ ) ∈E ( k,N,χ ) X d | N/LM a f ( ǫ, ψ, d ) E k ( ǫ, ψ ; dz ) , where a f ( ǫ, ψ, d ) = Y p | N p k p k − ǫ ( p ) ψ ( p ) X c ∈ C N ( ǫ,ψ ) R k,ǫ,ψ ( d, c/M ) S k,N/LM,ǫ,ψ ( d, c/M )[0] c,ψ f, with C N ( ǫ, ψ ) := { c M : c | N/LM } , (1.6) R k,ǫ,ψ ( d, c ) := ǫ (cid:18) − d gcd( d, c ) (cid:19) ψ (cid:18) c gcd( d, c ) (cid:19) (cid:18) gcd( d, c ) c (cid:19) k , (1.7) and S k,N,ǫ,ψ ( d, c ) := µ (cid:18) dc gcd( d, c ) (cid:19) Y p | gcd( d,c ) , Let c | N , by Lemma 6.1, if a/c and a ′ /c are equivalent cusps of Γ ( N ) and ( ǫ, ψ ) ∈ E ( k, N, χ ) with M | c then ψ ( a )[0] a/c f = ψ ( a ′ )[0] a ′ /c f . There-fore in applications of Theorem 1.1 computing ψ ( a )[0] a/c f at a set of inequivalentcusps will be sufficient, see [5, Corollary 6.3.23] for a description of such a set. Theorem 1.1 agrees with and extends previously known formulas. For example,if we let k ∈ N even, N squarefree, χ = χ in Theorem 1.1, we obtain [3, Theorem1.1] and if we let k ∈ N odd N ∈ { , , , } , χ = χ − N in Theorem 1.1, we obtain[6, (11.20)]. Theorem 1.1 additionally extends the latter to hold for all primes N that are congruent to 3 modulo 4.Before we apply Theorem 1.1 to representation numbers of quadratic forms wegive a snapshot of interesting applications. Since f ( z ) − E f ( z ) is a cusp form, onecan use our main theorem to produce cusp forms. At the end of Section 3 we usethis idea combined with the Modularity Theorem ([9, Theorem 8.8.1]) and considerthe elliptic curve E A : y + y = x − 7. Then we use arithmetic properties ofEisenstein series to obtain E A ( F p ) ≡ p ≡ E A ( F p ) = p + 1 if p ≡ here E A ( F p ) := {∞} ∪ { ( x, y ) ∈ F p × F p : y + y = x − } , with F p denoting the finite field of p elements, see Corollary 3.2.The Fourier coefficients of special functions (expanded at i ∞ ) have been of hugeinterest. A very well studied special function is the Dedekind eta function whichis defined by η ( z ) := e πiz/ Y n ≥ (1 − e πinz ) = q / Y n ≥ (1 − q n ) . Quotients of Dedekind eta functions are often referred to as eta quotients. NathanFine in his book [10] has given several formulas for Fourier coefficients of etaquotients (expanded at i ∞ ). In his work when the weight of the eta quotient isinteger, the formulas are linear combinations of Eisenstein series defined above.For instance he shows that F ( z ) := η (2 z ) η (3 z ) η (8 z ) η (12 z ) η ( z ) η (24 z ) = 1 + X n ≥ σ ( χ , χ − ; n ) q n , see [10, (32.5)], and acknowledges this equation as being very beautiful . We considerthe (2 k + 1)th power of F ( z ), that is we consider F k +1 ( z ) = η k +1 (2 z ) η k +1 (3 z ) η k +1 (8 z ) η k +1 (12 z ) η k +1 ( z ) η k +1 (24 z ) . Using our main theorem (Theorem 1.1) we obtain the following analogous formulafor F k +1 ( z ) when k > E F k +1 ( z ) = 1 − k + 1 B k +1 ,χ − X n ≥ (cid:0) σ k ( χ , χ − ; n ) + ( − k σ k ( χ − , χ ; n ) (cid:1) q n , see Corollary 3.1. Using Theorem 1.1 one can obtain formulas in this fashion forall holomorphic eta quotients of integral weight k ≥ F ( x , . . . , x k ) be a positive definite quadratic form with integer coefficientsand B ( F ) be the matrix associated with F whose entries are given by B ( F ) i,j = (cid:18) ∂ F ∂x i ∂x j (cid:19) . Then the generating function of the number of representations of a positive integerby the quadratic form F is θ F ( z ) = X x ∈ Z k e πiz F ( x ) = X x ∈ Z k e πizxB ( F ) x T / . In [20] Siegel gave a formula for the weighted average for representation numbers ofpositive definite quadratic forms in the same genus. Siegel’s formula is in terms oflocal densities, for other treatments of Siegel’s formula see [23] and [15, Chapter 3]. n the realm of modular forms, Siegel’s formula corresponds to the Eisenstein partof θ F ( z ), see [1], [18], [19, Remark on pg 110] and [20]. For clarity we note that if F and F are in the same genus then E θ F ( z ) = E θ F ( z ). Below we use Theorem1.1 to give an explicit formula for E θ F ( z ), where F is a 2 k –ary positive definitequadratic form with integer coefficients. In Section 2 we give several applicationsof our formula including a comparison of our output for the form P kj =1 x j withthat of Arenas [1, Proposition 1], which uses Siegel’s formula.By [13, Corollary 4.9.5], we have θ F ( z ) ∈ M k (Γ ( N ) , χ ) , (1.9)where χ = (cid:16) ( − k det( B ( F )) ∗ (cid:17) and N is the smallest positive integer such thatthe matrix N B ( F ) − has even diagonal entries. By [21, (10.2)] we have[0] a/c θ F ( z ) = (cid:18) − ic (cid:19) k p det( B ( F )) X x ∈ Z k ,x (mod c ) e πi ( F ( x ) a/c ) . (1.10)Putting (1.3), (1.9) and (1.10) in Theorem 1.1, we obtain the following assertionconcerning the representation numbers of 2 k –ary quadratic forms. Theorem 1.2. Let F ( x , . . . , x k ) be a positive definite quadratic form with k ≥ ;let χ and N be as above and ω be as in (1.3) . Then [ n ] E θ F ( z ) = X ( ǫ,ψ ) ∈E ( k,N,χ ) (cid:18) M ω M (cid:19) k (cid:18) W ( ψ ) W ( ω ) (cid:19) (cid:18) − kB k,ω (cid:19) Y p | lcm( L,M ) p k p k − ω ( p ) (1.11) × X d | N/LM a θ F ( ǫ, ψ, d ) σ k − ( ǫ, ψ ; n/d ) , where a θ F ( ǫ, ψ, d ) = ( − i ) k p det( B ( F )) Y p | N p k p k − ǫ ( p ) ψ ( p ) × X c ∈ C N ( ǫ,ψ ) R k,ǫ,ψ ( d, c/M ) S k,N/LM,ǫ,ψ ( d, c/M ) c k φ ( c ) c X a =1 , gcd( a,c )=1 ψ ( a ) X x ∈ Z k ,x (mod c ) e πi ( F ( x ) a/c ) . The organization of the rest of the paper is as follows. In Section 2, we applyTheorem 1.2 to the representation numbers of diagonal quadratic forms and certainnon-diagonal level 2 quadratic forms. A special case of the latter leads to anequation for Ramanujan’s tau function. In Section 3, we apply our Main Theoremto certain families of eta quotients, these applications give extensions of some wellknown formulas to higher weight eta quotients. In Sections 4–6 we prove the maintheorem. . Applications to representation numbers of certain quadraticforms To apply (1.11) to specific quadratic forms we need to compute the quadraticGauss sum. If F is a diagonal form, say F = P kj =1 α j x j , then we have X x ∈ Z k ,x (mod c ) e πi F ( x ) a/c = k Y j =1 gcd( α j a, c ) g (cid:18) α j a gcd( α j a, c ) , c gcd( α j a, c ) (cid:19) , (2.1)where, if gcd( α, β ) = 1, g ( α, β ) = β ≡ (cid:16) αβ (cid:17)p β if β ≡ i (cid:16) αβ (cid:17)p β if β ≡ i ) (cid:16) βα (cid:17)p β if β ≡ α ≡ − i ) (cid:16) βα (cid:17)p β if β ≡ α ≡ F = P kj =1 x j , that is, α j = 1 for all 1 ≤ j ≤ k . Then we have X x ∈ Z k ,x (mod c ) e πi F ( x ) /c = k Y i =1 g (1 , c ) = c = 1,0 if c = 2,(8 i ) k if c = 4.Thus by Theorem 1.2 when k is even we have E θ F ( z ) = 1 − k (2 k − B k,χ X n ≥ (cid:0) ( − i ) k σ ( χ , χ ; n ) − ( i k + 1) σ ( χ , χ ; n/ k σ ( χ , χ ; n/ (cid:1) q n (2.2)and when k is odd we have E θ F ( z ) = 1 − kB k,χ − X n ≥ (cid:0) σ ( χ , χ − ; n ) + (2 i ) k − σ ( χ − , χ ; n ) (cid:1) q n . (2.3)(2.2) and (2.3) agrees with Ramanujan’s statements [16, (131)–(134)], which wasfirst proven by Mordell in [14]. In [1, Proposition 1] Arenas uses Siegel’s formulato compute E θ F ( z ) and obtains (2.2) and (2.3) in the same form. Now we turnour attention to another diagonal form. Let F ( a, b ; p ) = a X i =1 x i + b X i =1 py i . n [7] Cooper, Kane and Ye found formulas for the representation numbers of F ( k, k ; p ), where p = 3 , , 11 or 23. Their result relies on the existence of aHauptmodul in the levels considered. Inspired by their results, in [2], we derivedformulas for the representation numbers of F (2 a, b ; p ) where a, b ∈ N and p is anodd prime. These results are considered as analogues of the Ramanujan-Mordellformula and specialized version of Theorem 1.2 agrees with these results. Below wegive formulas in all the remaining cases, that is, we find formulas for representationnumbers of F ( a, b ; p ) where a, b ≡ p an odd prime. Corollary 2.1. Let a, b ≥ be odd integers such that a + b ≥ . Set k = ( a + b ) / and p = χ − ( p ) p . Then for any odd prime p , whenever ( − k = χ − ( p ) we have E θ F ( a,b ; p ) ( z ) =1 + ∞ X n =1 k (cid:0) a σ k − ( χ , χ p ; n ) + a σ k − ( χ , χ p ; n/ 2) + a k σ k − ( χ , χ p ; n/ (cid:1) (2 k − χ p (2)) B k,χ p q n + ∞ X n =1 p ( a − / k (cid:0) a σ k − ( χ p , χ ; n ) + a σ k − ( χ p , χ ; n/ 2) + a k σ k − ( χ p , χ ; n/ (cid:1) (2 k − χ p (2)) B k,χ p q n , and whenever ( − k = − χ − ( p ) we have E θ F ( a,b ; p ) ( z ) = 1 − ∞ X n =1 k (cid:0) b σ k − ( χ , χ − p ; n ) + b k σ k − ( χ − , χ p ; n ) (cid:1) B k,χ − p q n − ∞ X n =1 p ( a − / k (cid:0) b σ k − ( χ p , χ − ; n ) + b k σ k − ( χ − p , χ ; n ) (cid:1) B k,χ − p q n where a = ( ( − k/ if p ≡ , ( − ( k + a +2) / if p ≡ , a = ( ( − k/ − χ p (2) if p ≡ , ( − ( k + a ) / − χ p (2) if p ≡ , a = 1 ,a = ( ( − k/ if p ≡ , ( − ( k − / if p ≡ , a = ( ( − k/ χ p (2) − if p ≡ , ( − ( b +1) / + ( − ( k +1) / χ p (2) if p ≡ , a = ( if p ≡ , ( − ( b − / if p ≡ , b = 1 , = ( ( − ( k − / / if p ≡ , ( − ( k + a − / / if p ≡ , b = ( if p ≡ , ( − ( b +1) / if p ≡ , b = ( ( − ( k − / / if p ≡ , ( − k/ / if p ≡ . Using (2.1) and Theorem 1.2 one can obtain results similar to Corollary 2.1 forany diagonal form. Next we consider the non-diagonal form F k = k X m =1 X ≤ i ≤ j ≤ x i,m x j,m ! − x ,m x ,m . We obtain N = 2 , det( B ( F k )) = 2 k and X x ∈ Z k ,x (mod c ) e πi F k ( x ) /c = ( c = 1,( − k if c = 2 . Thus θ F k ∈ M k (Γ (2) , χ ), hence by Theorem 1.2 we have[ n ] E θ F k ( z ) = − k (( − k + 1) B k,χ (cid:0) σ k − ( χ , χ ; n ) + ( − k σ k − ( χ , χ ; n/ (cid:1) . (2.4)When k = 6 we compute the first few coefficients of the cusp part of θ F : θ F ( z ) − E θ F ( z ) = 2 q + 2 − q + 2 q + O ( q ) ∈ S (Γ (2) , χ ) . The Fourier coefficients of η ( z ) are called the Ramanujan’s τ function and firstfew terms are given as follows η ( z ) = X n ≥ τ ( n ) q n = q − q + 252 q + O ( q ) . (2.5)It is well known that η ( z ) and η (2 z ) ∈ S (Γ (2) , χ ) thus by Sturm Theorem[5, Corollary 5.6.14] we obtain θ F ( z ) − E θ F ( z ) = 2 η ( z ) + 2 η (2 z )) . (2.6)If we compare n th coefficient of both sides of (2.6) we get[ n ] θ F ( z ) − σ ( χ , χ ; n ) + 2 σ ( χ , χ ; n/ τ ( n ) + 2 τ ( n/ . ince [ n ] θ F ( z ) ∈ N for all n ∈ N and 2 ≡ − 19 (mod 691)it is not hard to deduce the well known congruence relation τ ( n ) ≡ σ ( χ , χ ; n ) (mod 691) . Applications to eta quotients In this section we give further applications of Theorem 1.1. Recall that theDedekind eta function is defined by η ( z ) = e πiz/ Y n ≥ (1 − e πinz ) . Let k ∈ N . We define f k ( z ) := η k +1 (2 z ) η k +1 (3 z ) η k +1 (8 z ) η k +1 (12 z ) η k +1 ( z ) η k +1 (24 z ) , (3.1) g k ( z ) := η k − (3 z ) η k − (4 z ) η k − ( z ) η k − (2 z ) η k − (6 z ) η k (12 z ) , (3.2) h k ( z ) := η k − (9 z ) η (27 z ) η k − (3 z ) . (3.3)In Corollary 3.1 below we obtain formulas concerning f k ( z ) , g k ( z ) and h k ( z ). Sim-ilar formulas can be obtained via Theorem 1.1 for all integer weight holomorphiceta quotients. Corollary 3.1. Let k ≥ and let f k ( z ) , g k ( z ) and h k ( z ) be defined by (3.1) , (3.2) ,and (3.3) , respectively. Then we have E f k ( z ) = 1 − k + 2 B k +1 ,χ − X n ≥ (cid:0) σ k ( χ , χ − ; n ) + ( − k σ k ( χ − , χ ; n ) (cid:1) q n ,E g k ( z ) = 1 − kB k,χ X n ≥ σ k − ( χ , χ ; n ) q n , and E h k ( z ) = − kB k,χ X n ≥ X d | a d σ k − ( χ , χ ; n/d ) + b σ k − ( χ − , χ − ; n ) q n , (3.4) where a = ( − k − cos (( k + 4) π/ k +1 (3 k − ,a = ( − k / + (3 k + 1) cos (( k + 4) π/ k +1 (3 k − , = − cos (( k + 4) π/ k +1 (3 k − ,b = √ k + 4) π/ k +1 (3 k − . Proof. We use [5, Proposition 5.9.2] to determine f k ( z ) ∈ M k +1 (Γ (24) , χ − ) ,g k ( z ) ∈ M k (Γ (12) , χ ) ,h k ( z ) ∈ M k (Γ (27) , χ ) . We evaluate the constant terms of f k ( z ) , g k ( z ) , h k ( z ) at the relevant cusps using[12, Proposition 2.1]. We do this with the help of some SAGE functions we havewritten, the code is provided in the Appendix A. From these we compute[0] / f k = − i k +1 √ k +1 k +2 , [0] a/c f k = 0 , for a/c = 1 / , / , / , / , / , / , [0] / f k = 1 , see Appendix A for details. We determine the set of tuples of characters as E (2 k + 1 , , χ − ) = { ( χ , χ − ) , ( χ − , χ ) , ( χ , χ − ) , ( χ − , χ ) } . Thus we have E f k ( z ) = X ( ǫ,ψ ) ∈E (2 k +1 , ,χ − ) a f k ( ǫ, ψ, E k +1 ( ǫ, ψ ; z ) . Now we compute a f k ( χ , χ − , 1) = Y p | p k p k − χ ( p ) χ − ( p ) R k,χ ,χ − (1 , S k, ,χ ,χ − (1 , ,χ − f k = R k,χ ,χ − (1 , S k, ,χ ,χ − (1 , ,χ − f k = [0] ,χ − f k . (3.5)We further have[0] ,χ − f k = 1 φ (24) X a =1 , gcd( a, χ − ( a )[0] a/ f k = χ − (1)[0] / f k = 1 . (3.6)Combining (3.5) and (3.6) we have a f k ( χ , χ − , 1) = 1.The rest of the coefficients are obtained similarly. (cid:3) ow we turn our attention to special cases of these formulas. The dimensionof S (Γ (12) , χ ) is 0, so we obtain an exact formula for g , i.e. we have g ( z ) = E g ( z ). When k = 1, (3.4) specializes to E h ( z ) = X n ≥ (cid:18) σ ( χ , χ ; n ) − σ ( χ , χ ; n/ σ ( χ , χ ; n/ 9) + 118 σ ( χ − , χ − ; n ) (cid:19) q n . Clearly h ( z ) − E h ( z ) is a cusp form, and if we normalize h ( z ) − E h ( z ) so thatthe coefficient of q is 1, we obtain the newform N ( z ) in S (Γ (27) , χ ), that is,we have N ( z ) = − η (9 z ) η (27 z ) η (3 z ) + X n ≥ (cid:18) σ ( χ , χ ; n ) − σ ( χ , χ ; n/ σ ( χ , χ ; n/ 9) + 12 σ ( χ − , χ − ; n ) (cid:19) q n . By [8, Table 1] this newform is associated to the elliptic curve E A : y + y = x − . Recall that in Section 1 we defined E A ( F p ) = {∞} ∪ { ( x, y ) ∈ F p × F p : y + y = x − } , where F p is the finite field of p elements. Then by the Modularity Theorem, see[9, Theorem 8.8.1], we have E A ( F p ) = ( p + 1) − [ p ] N ( z ) for all p = 3 . Thus for all p = 3 we have E A ( F p ) = 9[ p ] η (9 z ) η (27 z ) η (3 z ) + ( p + 1) (cid:18) − χ − ( p )2 (cid:19) . Since [ p ] η (9 z ) η (27 z ) η (3 z ) ∈ Z for all p ∈ N and [ p ] η (9 z ) η (27 z ) η (3 z ) = 0 when p ≡ Corollary 3.2. We have E A ( F p ) ≡ if p ≡ , E A ( F p ) = p + 1 if p ≡ . . Orthogonal relations In this section we prove some orthogonal relations involving the functions R k,ǫ,ψ ( d, c )and S k,N,ǫ,ψ ( d, c ) defined in (1.7) and (1.8), respectively. These orthogonal rela-tions concern the constant terms of the Eisenstein series and give the means todetermine a f ( ǫ, ψ, d ) of Theorem 1.1. Throughout the section we assume k, N ∈ N and ǫ, ψ are primitive Dirichlet characters with conductors L, M , respectively, suchthat LM | N . Lemma 4.1. Let p | N be a prime and let t | N/p v , where v = v p ( N ) , then for ≤ i ≤ v we have S k,N,ǫ,ψ ( t · p i , d ) = S k,p v ,ǫ,ψ ( p i , p v p ( d ) ) S k,N/p v ,ǫ,ψ ( t, d/p v p ( d ) ) . Proof. Since t | N/p v we have gcd( t, p ) = 1. Using the multiplicative properties ofthe M¨obius function we obtain S k,N,ǫ,ψ ( t · p i , d ) = µ (cid:18) t · p i · d gcd( t · p i , d ) (cid:19) Y p | gcd( t · p i ,d ) , Let gcd( t, p i ) = 1 , then we have R k,ǫ,ψ ( c, t · p i ) = ǫ ( − R k,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( c/p v p ( c ) , t ) , R k,ǫ,ψ ( t · p i , d ) = ǫ ( − R k,ǫ,ψ ( p i , p v p ( d ) ) R k,ǫ,ψ ( t, d/p v p ( d ) ) . Proof. By elementary manipulations we obtain R k,ǫ,ψ ( c, t · p i ) = ǫ (cid:18) − c gcd( c, t · p i ) (cid:19) ψ (cid:18) t · p i gcd( c, t · p i ) (cid:19) (cid:18) gcd( c, t · p i ) t · p i (cid:19) k = ǫ (cid:18) − c/p v c gcd( c/p v c , t ) (cid:19) ψ (cid:18) t gcd( c/p v p ( c ) , t ) (cid:19) (cid:18) gcd( c/p v p ( c ) , t ) t (cid:19) k × ǫ (cid:18) p v p ( c ) gcd( p v p ( c ) , p i ) (cid:19) ψ (cid:18) p i gcd( p v p ( c ) , p i ) (cid:19) (cid:18) gcd( p v c , p i ) p i (cid:19) k ǫ ( − R k,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( c/p v p ( c ) , t ) . Proof of the second equation is similar. (cid:3) Theorem 4.1. Let c, d | N , then X t | N S k,N,ǫ,ψ ( c, t ) R k,ǫ,ψ ( c, t ) R k,ǫ,ψ ( t, d ) = if c = d , Y p | N p k − ǫ ( p ) ψ ( p ) p k if c = d .Proof. Let p | N be prime and v p ( N ) = v . Then we use Lemmas 4.1 and 4.2 toobtain X t | N S k,N,ǫ,ψ ( c, t ) R k,ǫ,ψ ( c, t ) R k,ǫ,ψ ( t, d )= X ≤ i ≤ v X t | N/p v S k,N,ǫ,ψ ( c, t · p i ) R k,ǫ,ψ ( c, t · p i ) R k,ǫ,ψ ( t · p i , d )= X ≤ i ≤ v X t | N/p v S k,p v ,ǫ,ψ ( p v p ( c ) , p i ) S k,N/p v ,ǫ,ψ ( c/p v p ( c ) , t ) × ǫ ( − R k,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( c/p v p ( c ) , t ) × ǫ ( − R k,ǫ,ψ ( p i , p v p ( d ) ) R k,ǫ,ψ ( t, d/p v p ( d ) )= X ≤ i ≤ v S k,p v ,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p i , p v p ( d ) ) × X t | N/p v S k,N/p v ,ǫ,ψ ( c/p v p ( c ) , t ) R k,ǫ,ψ ( c/p v p ( c ) , t ) R k,ǫ,ψ ( t, c/p v p ( d ) ) . Using this recursively we obtain X t | N S k,N,ǫ,ψ ( c, t ) R k,ǫ,ψ ( c, t ) R k,ǫ,ψ ( t, d )= Y p | N X ≤ i ≤ v S k,p v ,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p i , p v p ( d ) ) . (4.1)Now we prove for all p | N we have X ≤ i ≤ v S k,p v ,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p i , p v p ( d ) )= v p ( c ) = v p ( d ), p k − ǫ ( p ) ψ ( p ) p k if v p ( c ) = v p ( d ). (4.2) e first note that S k,p v ,ǫ,ψ ( p v p ( c ) , p i ) = | v p ( c ) − i | > p k + ǫ ( p ) ψ ( p ) p k if i = v p ( c ) and v > v p ( c ) > i = v p ( c ) and v p ( c ) = v ,1 if i = v p ( c ) and v p ( c ) = 0, − i = v p ( c ) − v p ( c ) > − i = v p ( c ) + 1 and v p ( c ) < v , (4.3)and R k,ǫ,ψ ( p i , p j ) = ǫ ( − 1) if i = j , ǫ ( − ψ ( p j − i ) (cid:18) p j − i (cid:19) k if i < j , ǫ ( − p i − j ) if i > j . (4.4)The cases(Case 1) 0 < v p ( c ) < v p ( d ) ≤ v ,(Case 2) 0 = v p ( c ) < v p ( d ) ≤ v ,(Case 3) v > v p ( c ) > v p ( d ) ≥ v = v p ( c ) > v p ( d ) ≥ < v p ( c ) = v p ( d ) < v ,(Case 6) 0 = v p ( c ) = v p ( d ),(Case 7) v = v p ( c ) = v p ( d ),needs to be handled separately, which is done below. Case 1: Let 0 < v p ( c ) < v p ( d ) ≤ v , then by employing (4.3) for all i such that | v p ( c ) − i | > S k,p v ,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p i , p v p ( d ) ) = 0 . Therefore we have X ≤ i ≤ v S k,p v ,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p i , p v p ( d ) )= S k,p v ,ǫ,ψ ( p v p ( c ) , p v p ( c ) − ) R k,ǫ,ψ ( p v p ( c ) , p v p ( c ) − ) R k,ǫ,ψ ( p v p ( c ) − , p v p ( d ) )+ S k,p v ,ǫ,ψ ( p v p ( c ) , p v p ( c ) ) R k,ǫ,ψ ( p v p ( c ) , p v p ( c ) ) R k,ǫ,ψ ( p v p ( c ) , p v p ( d ) )+ S k,p v ,ǫ,ψ ( p v p ( c ) , p v p ( c )+1 ) R k,ǫ,ψ ( p v p ( c ) , p v p ( c )+1 ) R k,ǫ,ψ ( p v p ( c )+1 , p v p ( d ) ) , which, by (4.3) and (4.4), equals to= − · ǫ ( − p ) · ǫ ( − ψ ( p v p ( d ) − v p ( c )+1 ) (cid:18) p v p ( d ) − v p ( c )+1 (cid:19) k p k + ǫ ( p ) ψ ( p ) p k · ǫ ( − · ǫ ( − ψ ( p v p ( d ) − v p ( c ) ) (cid:18) p v p ( d ) − v p ( c ) (cid:19) k + ( − · ǫ ( − ψ ( p ) (cid:18) p (cid:19) k · ǫ ( − ψ ( p v p ( d ) − v p ( c ) − ) (cid:18) p v p ( d ) − v p ( c ) − (cid:19) k . By using multiplicative properties of Dirichlet characters we conclude that thisexpression is equal to 0. Case 2: Let 0 = v p ( c ) < v p ( d ) ≤ v , then by employing (4.3) and (4.4) we have X ≤ i ≤ v S k,p v ,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p i , p v p ( d ) )= S k,p v ,ǫ,ψ (1 , R k,ǫ,ψ (1 , R k,ǫ,ψ (1 , p v p ( d ) )+ S k,p v ,ǫ,ψ (1 , p ) R k,ǫ,ψ (1 , p ) R k,ǫ,ψ ( p, p v p ( d ) )= ψ ( p v p ( d ) ) (cid:18) p v p ( d ) (cid:19) k − ψ ( p v p ( d ) ) (cid:18) p v p ( d ) (cid:19) k , which equals to 0. Case 3: Let v > v p ( c ) > v p ( d ) ≥ 0, then by employing (4.3) and (4.4) we have X ≤ i ≤ v S k,p v ,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p i , p v p ( d ) )= S k,p v ,ǫ,ψ ( p v p ( c ) , p v p ( c ) − ) R k,ǫ,ψ ( p v p ( c ) , p v p ( c ) − ) R k,ǫ,ψ ( p v p ( c ) − , p v p ( d ) )+ S k,p v ,ǫ,ψ ( p v p ( c ) , p v p ( c ) ) R k,ǫ,ψ ( p v p ( c ) , p v p ( c ) ) R k,ǫ,ψ ( p v p ( c ) , p v p ( d ) )+ S k,p v ,ǫ,ψ ( p v p ( c ) , p v p ( c )+1 ) R k,ǫ,ψ ( p v p ( c ) , p v p ( c )+1 ) R k,ǫ,ψ ( p v p ( c )+1 , p v p ( d ) )= − ǫ ( − p ) ǫ ( − p v p ( c ) − − v p ( d ) ) + p k + ǫ ( p ) ψ ( p ) p k · ǫ ( − · ǫ ( − p v p ( c ) − v p ( d ) ) − ǫ ( − ψ ( p ) 1 p k · ǫ ( − p v p ( c )+1 − v p ( d ) ) . By using multiplicative properties of Dirichlet characters we conclude that thisexpression is equal to 0. Case 4: Let v = v p ( c ) > v p ( d ) ≥ 0, then by employing (4.3) and (4.4) we have X ≤ i ≤ v S k,p v ,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p i , p v p ( d ) )= S k,p v ,ǫ,ψ ( p v p ( c ) , p v p ( c ) − ) R k,ǫ,ψ ( p v p ( c ) , p v p ( c ) − ) R k,ǫ,ψ ( p v p ( c ) − , p v p ( d ) )+ S k,p v ,ǫ,ψ ( p v p ( c ) , p v p ( c ) ) R k,ǫ,ψ ( p v p ( c ) , p v p ( c ) ) R k,ǫ,ψ ( p v p ( c ) , p v p ( d ) )= − ǫ ( p v p ( c ) − v p ( d ) ) + ǫ ( p v p ( c ) − v p ( d ) )= 0 . ase 5: Let 0 < v p ( c ) = v p ( d ) < v , then by employing (4.3), (4.4) and multi-plicative properties of Dirichlet characters we have X ≤ i ≤ v S k,p v ,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p i , p v p ( d ) )= S k,p v ,ǫ,ψ ( p v p ( c ) , p v p ( c ) − ) R k,ǫ,ψ ( p v p ( c ) , p v p ( c ) − ) R k,ǫ,ψ ( p v p ( c ) − , p v p ( c ) )+ S k,p v ,ǫ,ψ ( p v p ( c ) , p v p ( c ) ) R k,ǫ,ψ ( p v p ( c ) , p v p ( c ) ) R k,ǫ,ψ ( p v p ( c ) , p v p ( c ) )+ S k,p v ,ǫ,ψ ( p v p ( c ) , p v p ( c )+1 ) R k,ǫ,ψ ( p v p ( c ) , p v p ( c )+1 ) R k,ǫ,ψ ( p v p ( c )+1 , p v p ( c ) )= − ǫ ( − p ) · ǫ ( − ψ ( p ) 1 p k + p k + ǫ ( p ) ψ ( p ) p k · ǫ ( − · ǫ ( − − ǫ ( − ψ ( p ) 1 p k · ǫ ( − p )= p k − ǫ ( p ) ψ ( p ) p k . Case 6: Let 0 = v p ( c ) = v p ( d ), then by employing (4.3), (4.4) and multiplicativeproperties of Dirichlet characters we have X ≤ i ≤ v S k,p v ,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p i , p v p ( d ) )= S k,p v ,ǫ,ψ (1 , R k,ǫ,ψ (1 , R k,ǫ,ψ (1 , 1) + S k,p v ,ǫ,ψ (1 , p ) R k,ǫ,ψ (1 , p ) R k,ǫ,ψ ( p, · ǫ ( − · ǫ ( − − ǫ ( − ψ ( p ) 1 p k · ǫ ( − p )= p k − ǫ ( p ) ψ ( p ) p k . Case 7: Let v = v p ( c ) = v p ( d ), then by employing (4.3), (4.4) and multiplicativeproperties of Dirichlet characters we have X ≤ i ≤ v S k,p v ,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p v p ( c ) , p i ) R k,ǫ,ψ ( p i , p v p ( d ) )= S k,p v ,ǫ,ψ ( p v , p v − ) R k,ǫ,ψ ( p v , p v − ) R k,ǫ,ψ ( p v − , p v )+ S k,p v ,ǫ,ψ ( p v , p v ) R k,ǫ,ψ ( p v , p v ) R k,ǫ,ψ ( p v , p v )= − · ǫ ( − p ) · ǫ ( − ψ ( p ) 1 p k + 1 · ǫ ( − · ǫ ( − p k − ǫ ( p ) ψ ( p ) p k . Finally, if c = d , then there exists a prime p | N such that v p ( c ) = v p ( d ). Henceby (4.2) the product in (4.1) is 0. If c = d then for all prime divisors p of N wehave v p ( c ) = v p ( d ). Therefore by (4.1) and (4.2) we have the desired result. (cid:3) . Constant terms of expansions of Eisenstein series at the cusps Recall that E k ( ǫ, ψ ; dz ) is defined by (1.3) and we have E k ( ǫ, ψ ; dz ) ∈ E k (Γ ( N ) , χ ) when ( k, ǫ, ψ ) = (2 , χ , χ ) , and L d ( z ) := E ( χ , χ ; z ) − dE ( χ , χ ; dz ) ∈ E (Γ ( N ) , χ ) . The constant terms of Eisenstein series in the expansion at the cusp a/c withgcd( a, c ) = 1 are given by[0] a/c E k ( ǫ, ψ ; dz ) = ψ ( a ) R k,ǫ,ψ ( c, M d ) when ( k, ǫ, ψ ) = (2 , χ , χ ) and (5.1)[0] a/c L d ( z ) = R ,χ ,χ ( c, − d R ,χ ,χ ( c, d ) , (5.2)where R k,ǫ,ψ ( c, t ) is defined by (1.7). For (5.1) see [3, (6.2)], [5, Proposition 8.5.6and Ex. 8.7 (i) on pg. 308]. The formula (5.2) is proved later in this section.The structure of the terms [0] a/c E k ( ǫ, ψ ; dz ) is complicated and difficult to workwith. We observe that taking the average [0] c,ψ E k ( ǫ, ψ ; dz ) gives constant terms avery nice structure which is easier to work with, see (6.5). Throughout the sectionwe assume k, N ∈ N , ǫ and ψ are primitive Dirichlet characters with conductors L and M , respectively, such that LM | N and ( k, ǫ, ψ ) = (2 , χ , χ ). Lemma 5.1. Let c | N and LM d | N . If M ∤ c , or M | c and L ∤ N/c , then [0] a/c E k ( ǫ, ψ ; dz ) = 0 . Proof. First we let M ∤ c . Then M ∤ gcd( M d, c ). Thus gcd (cid:18) M d gcd( M d, c ) , M (cid:19) | M ,which implies ψ (cid:18) M d gcd( M d, c ) (cid:19) = 0 since the conductor of ψ is M . Therefore theresult follows from (5.1).Second, we let M | c , c = M/c , thus c | N/M . Assume L ∤ N/c , thus c ∤ N/LM . Since ( N/M ) L ∈ Z , ( N/M ) c ∈ Z and ( N/M ) /c L Z , we have gcd( c , L ) = 1.Additionally, there exists a prime p dividing c such that v p ( c ) > v p ( N ) − v p ( M ) − v p ( L ) . (5.3)Since c | N/M , for all p | c we have v p ( c ) ≤ v p ( N ) − v p ( M ) . (5.4)By (5.3) and (5.4) we have v p ( N ) − v p ( M ) > v p ( N ) − v p ( M ) − v p ( L ) . Therefore v p ( L ) > . (5.5) ince d | N/LM we have v p ( N ) − v p ( M ) − v p ( L ) ≥ v p ( d ) ≥ . (5.6)Inequalities (5.3) and (5.6) together implies v p ( c ) > v p ( d ). Therefore by employing(5.5) we have p | gcd (cid:18) p v p ( c ) gcd( p v p ( c ) , p v p ( d ) ) , p v p ( L ) (cid:19) . That is, p | gcd (cid:18) c gcd( c , d ) , L (cid:19) . This implies ǫ (cid:18) c gcd( M d, c ) (cid:19) = 0 since conductor of ǫ is L . Therefore the resultfollows from (5.1). (cid:3) Lemma 5.2. Let c ∈ N , and let ψ , ψ be two primitive Dirichlet characters withconductors M and M , respectively. Let both M and M divide c . Then we have c X a =1 , gcd( a,c )=1 ψ ( a ) ψ ( a ) = ( if ψ = ψ , φ ( c ) if ψ = ψ .Proof. The modulus of the character ψ ψ is lcm( M , M ), see [13, pg. 80]. Welet A = lcm( M , M ), then A | c . We have c X a =1 , gcd( a,c )=1 ψ ( a ) ψ ( a ) = c X a =1 ψ ( a ) ψ ( a ) X s | gcd( a,c ) µ ( s )= X s | c µ ( s ) c X a =1 ,s | a ψ ( a ) ψ ( a )= X s | c µ ( s ) ψ ( s ) ψ ( s ) c/s X t =1 ψ ( t ) ψ ( t ) . We have ψ ( s ) ψ ( s ) = 0 whenever gcd( A, s ) > 1, therefore we obtain c X a =1 , gcd( a,c )=1 ψ ( a ) ψ ( a ) = X s | c, gcd( A,s )=1 µ ( s ) ψ ( s ) ψ ( s ) c/s X t =1 ψ ( t ) ψ ( t ) . (5.7) ow if A | c , s | c and gcd( A, s ) = 1, then A | c/s . First we consider ψ = ψ .Then c/s X t =1 ψ ( t ) ψ ( t ) = 0 . (5.8)Therefore if ψ = ψ by (5.7) and (5.8) we have c X a =1 , gcd( a,c )=1 ψ ( a ) ψ ( a ) = 0 . Second we consider the case when ψ = ψ . Noting that in this case A = M , wehave c/s X t =1 ψ ( t ) ψ ( t ) = csM φ ( M ) . (5.9)Therefore if ψ = ψ by by (5.7) and (5.9) we have c X a =1 , gcd( a,c )=1 ψ ( a ) ψ ( a ) = X s | c, gcd( M ,s )=1 µ ( s ) csM φ ( M ) = c · φ ( M ) M X s | c, gcd( M ,s )=1 µ ( s ) s . (5.10)Noting that X s | c µ ( s ) s = φ ( c ) c = Y p | c p − p , we have X s | c, gcd( M ,s )=1 µ ( s ) s = Y p | c, gcd( p,M )=1 p − p = Q p | c p − p Q p | M p − p = φ ( c ) /cφ ( M ) /M . Putting this in (5.10) completes the proof. (cid:3) Before we prove the main result of this section we prove (5.2). Lemma 5.3. Let gcd( a, c ) = 1 , then we have [0] a/c L d ( z ) = R ,χ ,χ ( c, − d R ,χ ,χ ( c, d ) . Proof. Since gcd( a, c ) = 1, there exist β, γ ∈ Z such that A = (cid:20) a βc γ (cid:21) ∈ SL ( Z ).Then by [12, (1.21)] we have E ( χ , χ ; A ( z )) = ( cz + γ ) E ( χ , χ ; z ) − icπ ( cz + γ ) , (5.11) here A ( z ) is the usual linear fractional transformation. Let e = ad gcd( c,ad ) and g = c gcd( c,ad ) . Then since gcd( e, g ) = 1 there exist f, h such that (cid:20) e fg h (cid:21) ∈ SL ( Z ).Hence we have E ( χ , χ ; dA ( z ); ) = E (cid:18) χ , χ ; (cid:20) e fg h (cid:21) (cid:20) ahd − cf βhd − γf − agd + ce − βgd + γe (cid:21) ( z ) (cid:19) = E χ , χ ; " ad gcd( c,ad ) f c gcd( c,ad ) h ahd − cf βhd − γf d gcd( c,ad ) (cid:21) ( z ) ! = (cid:18) gcd( c, d ) d (cid:19) ( cz + γ ) E (cid:18) χ , χ ; (cid:20) ahd − cf βhd − γf d gcd( c,ad ) (cid:21) ( z ) (cid:19) − icπd ( cz + γ ) , where in the last line we used (5.11). Thus we obtain[0] a/c L d ( z ) = [0] a/c ( E ( χ , χ ; z ) − dE ( χ , χ ; dz )) = d − gcd( c, d ) d = R ,χ ,χ ( c, − d R ,χ ,χ ( c, d ) . (cid:3) Theorem 5.1. Let c | N and let ( ǫ , ψ ) , ( ǫ , ψ ) ∈ { ( ǫ, ψ ) ∈ E ( k, N, χ ) : M | c } .Then we have [0] c,ψ E k ( ǫ , ψ ; dz ) = ( [0] /c E k ( ǫ , ψ ; dz ) if ψ = ψ , otherwise.Let c | N and let ( ǫ , ψ ) ∈ { ( ǫ, ψ ) ∈ E (2 , N, χ ) : M | c } , then we have [0] c,ψ L d ( z ) = ( [0] /c L d ( z ) if ψ = χ , otherwise.Proof. If ( k, ǫ, ψ ) = (2 , χ , χ ) by (5.1) we have[0] c,ψ E ( ǫ , ψ ; dz ) = 1 φ ( c ) c X a =1 , gcd( a,c )=1 ψ ( a ) ψ ( a ) R k,ǫ ,ψ ( c, M d )= [0] /c E k ( ǫ , ψ ; dz ) 1 φ ( c ) c X a =1 , gcd( a,c )=1 ψ ( a ) ψ ( a ) . Therefore by Lemma 5.2 we obtain the the first part of the statement. Proof ofthe second part is similar. (cid:3) . Proof of the Main Theorem Recall that E k ( ǫ, ψ ; dz ) is defined by (1.3) and the set { E k ( ǫ, ψ ; dz ) : ( ǫ, ψ ) ∈ E ( k, N, χ ) , d | N/LM } (6.1)constitutes a basis for E k (Γ ( N ) , χ ) whenever ( k, χ ) = (2 , χ ) and the set { E ( χ , χ ; z ) − dE ( χ , χ ; dz ) : 1 < d | N/LM } (6.2) ∪ { E ( ǫ, ψ ; dz ) : ( ǫ, ψ ) ∈ E (2 , N, χ ) , ( ǫ, ψ ) = ( χ , χ ) , d | N/LM } constitutes a basis for E (Γ ( N ) , χ ), see [5, Theorems 8.5.17 and 8.5.22], or [22,Proposition 5].Now we prove the main theorem whenever ( k, χ ) = (2 , χ ). Let f ( z ) ∈ M k (Γ ( N ) , χ )where N, k ∈ N , k ≥ k, χ ) = (2 , χ ). Then by (6.1) we have E f ( z ) = X ( ǫ,ψ ) ∈E ( k,N,χ ) X d | N/LM a f ( ǫ, ψ, d ) E k ( ǫ, ψ ; dz ) , (6.3)for some a f ( ǫ, ψ, d ) ∈ C . Our strategy for the proof is, using the interplay betweenthe constant terms of Eisenstein series, to create sets of linear equations (see (6.5))and to solve those sets of linear equations for a f ( ǫ, ψ, d ) using Theorem 4.1.By (1.1) we have f ( z ) = E f ( z ) + S f ( z ), where E f ( z ) ∈ E k (Γ ( N ) , χ ) and S f ( z ) ∈ S k (Γ ( N ) , χ ) are unique. Since by definition S f ( z ) vanishes at all cusps,we have [0] a/c f ( z ) = [0] a/c E f ( z ). Therefore by (6.3) for each c | N and a ∈ Z suchthat gcd( a, c ) = 1, we obtain[0] a/c f ( z ) = X ( ǫ,ψ ) ∈E ( k,N,χ ) X d | N/LM a f ( ǫ, ψ, d )[0] a/c E k ( ǫ, ψ ; dz ) . Let ( ǫ , ψ ) ∈ E ( k, N, χ ), and let the conductors of ǫ and ψ be L and M , respec-tively. Note that for each ψ there is a unique ǫ such that ( ǫ , ψ ) ∈ E ( k, N, χ ).If we average the constant terms with ψ using (1.5), then for all c | N we obtain[0] c,ψ f ( z ) = X ( ǫ,ψ ) ∈E ( k,N,χ ) X d | N/LM a f ( ǫ, ψ, d )[0] c,ψ E k ( ǫ, ψ ; dz ) . Our goal here is to isolate a set of linear equations from which we can determine a f ( ǫ , ψ , d ) for all d | N/L M . By Lemma 5.1 we have [0] c,ψ E k ( ǫ , ψ ; dz ) = 0if c | N is such that M | c , or M ∤ c and L | N/c . Therefore from now on werestrict c to be in C N ( ǫ , ψ ), see (1.6) for definition. By applying Lemma 5.1 onemore time we have [0] c,ψ E k ( ǫ, ψ ; dz ) = 0 if M ∤ c . Therefore for all c ∈ C N ( ǫ , ψ )we have [0] c,ψ f ( z ) = X ( ǫ,ψ ) ∈E ( k,N,χ ) ,M | c X d | N/LM a f ( ǫ, ψ, d )[0] c,ψ E k ( ǫ, ψ ; dz ) . (6.4) ecall that for each ψ there is a unique ( ǫ , ψ ) ∈ E ( k, N, χ ). Additionally, for all c ∈ C N ( ǫ , ψ ) we have ( ǫ , ψ ) ∈ { ( ǫ, ψ ) ∈ E ( k, N, χ ) : M | c } . Therefore for all c ∈ C N ( ǫ , ψ ) we have[0] c,ψ f ( z ) = X ( ǫ,ψ ) ∈E ( k,N,χ ) , ( ǫ,ψ ) =( ǫ ,ψ ) M | c X d | N/LM a f ( ǫ, ψ, d )[0] c,ψ E k ( ǫ, ψ ; dz )+ X d | N/L M a f ( ǫ , ψ , d )[0] c,ψ E k ( ǫ , ψ ; dz ) . From this, using Theorem 5.1, we obtain[0] c,ψ f ( z ) = X d | N/L M a f ( ǫ , ψ , d )[0] /c E k ( ǫ , ψ ; dz ) . Since M | c we have[0] /c E k ( ǫ , ψ ; dz ) = R k,ǫ ,ψ ( c, M d ) = R k,ǫ ,ψ ( c/M , d ) . Hence for all c ∈ C N ( ǫ , ψ ) we have[0] c,ψ f ( z ) = X d | N/L M a f ( ǫ , ψ , d ) R k,ǫ ,ψ ( c/M , d ) . (6.5)Below we solve the equations coming from (6.5) for a f ( ǫ , ψ , d ) using Theorem4.1. For d | N/L M we consider the sum X c ∈ C N ( ǫ ,ψ ) R k,ǫ ,ψ ( d , c/M ) S k,N/L M ,ǫ ,ψ ( d , c/M )[0] c,ψ f ( z ) , (6.6)which, by (6.5), equals to = X c ∈ C N ( ǫ ,ψ ) R k,ǫ ,ψ ( d , c/M ) S k,N/L M ,ǫ ,ψ ( d , c/M ) X d | N/L M a f ( ǫ , ψ , d ) R k,ǫ ,ψ ( c/M , d ) . (6.7) Rearranging the terms of (6.7) we obtain X c ∈ C N ( ǫ ,ψ ) R k,ǫ ,ψ ( d , c/M ) S k,N/L M ,ǫ ,ψ ( d , c/M )[0] c,ψ f ( z )= X d | N/L M a f ( ǫ , ψ , d ) X c ∈ C N ( ǫ ,ψ ) R k,ǫ ,ψ ( d , c/M ) S k,N/L M ,ǫ ,ψ ( d , c/M ) R k,ǫ ,ψ ( c/M , d ) . (6.8) Recall that C N ( ǫ , ψ ) is defined by (1.6) and is a set equivalent to the set { c : M | c, c/M | N/L M } , .e., c/M runs through all the divisors of N/L M as c runs through all theelements of C N ( ǫ , ψ ). In Theorem 4.1 we use this and we replace N by N/L M , t by c/M , c by d and d by d to obtain X c ∈ C N ( ǫ ,ψ ) R k,ǫ ,ψ ( d , c/M ) S k,N/L M ,ǫ ,ψ ( d , c/M ) R k,ǫ ,ψ ( c/M , d )= Y p | N/L M p k − ǫ ( p ) ψ ( p ) p k if d = d ,0 if d = d . (6.9)Therefore from (6.8) and (6.9) we obtain X c ∈ C N ( ǫ ,ψ ) R k,ǫ ,ψ ( d , c/M ) S k,N/L M ,ǫ ,ψ ( d , c/M )[0] c,ψ f ( z )= a f ( ǫ , ψ , d ) Y p | N/L M p k − ǫ ( p ) ψ ( p ) p k . Since p | L M implies ǫ ( p ) ψ ( p ) = 0 we have a f ( ǫ , ψ , d ) = Y p | N p k p k − ǫ ( p ) ψ ( p ) X c ∈ C N ( ǫ ,ψ ) R k,ǫ ,ψ ( d , c/M ) S k,N/L M ,ǫ ,ψ ( d , c/M )[0] c,ψ f. This completes the proof of Theorem 1.1 when ( k, χ ) = (2 , χ ).Now let ( k, χ ) = (2 , χ ), then a basis of E (Γ ( N ) , χ ) is given by (6.2). UsingLemma 5.3, Theorem 5.1 and arguments similar to the first part of this proof weobtain E f ( z ) = X Let f ( z ) ∈ M k (Γ ( N ) , χ ) and c | N . Let a/c and a ′ /c be equivalentcusps of Γ ( N ) . If ( ǫ, ψ ) ∈ E ( k, N, χ ) with M | c then we have ψ ( a )[0] a/c f = ψ ( a ′ )[0] a ′ /c f. Proof. Let a/c and a ′ /c be equivalent cusps of Γ ( N ), then there exists a matrix (cid:20) α βγ δ (cid:21) ∈ Γ ( N ) such that (cid:20) α βγ δ (cid:21) (cid:20) a bc d (cid:21) = (cid:20) a ′ b ′ c d ′ (cid:21) . (6.10)Then using transformation properties of modular forms we have ψ ( a ′ )[0] a ′ /c f = ψ ( a ′ ) lim z → i ∞ ( cz + d ′ ) − k f (cid:18) a ′ z + b ′ cz + d ′ (cid:19) = ψ ( a ′ ) lim z → i ∞ ( cz + d ′ ) − k χ ( δ ) (cid:18) γ az + bcz + d + δ (cid:19) k f (cid:18) az + bcz + d (cid:19) = ψ ( a ′ ) χ ( δ ) lim z → i ∞ ( cz + d ) − k f (cid:18) az + bcz + d (cid:19) = ψ ( a ′ ) χ ( δ )[0] a/c f. We have M | c and by (6.10) we have a ′ = αa + βc , thus ψ ( a ′ ) = ψ ( α ) ψ ( a ). Since M | c , c | N and N | γ we have M | γ , therefore we have 1 = ψ (1) = ψ ( αδ − γβ ) hich implies ψ ( α ) = ψ ( δ ). Putting these together we obtain ψ ( a ′ ) χ ( δ ) = ψ ( a ) ψ ( δ ) χ ( δ ) . Since gcd( δ, N ) = 1 we have ψ ( δ ) χ ( δ ) = ǫ ( δ ). Now we prove ǫ ( δ ) = 1 whichfinishes the proof. Recall that LM | N , therefore c | M implies L | N/c , i.e., L | γ/c . From (6.10) we have δ = 1 − aγ/c , thus, since gcd( a, c ) = 1 and L | γ/c ,we have ǫ ( δ ) = ǫ (1 − aγ/c ) = ǫ (1) = 1. (cid:3) Acknowledgements I would like to thank Professor Amir Akbary for helpful discussions throughoutthe course of this research. I am also grateful to Professor Shaun Cooper, whogave the vision which initiated this research. Words cannot adequately expressmy gratitude towards Professor Emeritus Kenneth S. Williams, who has givenmany useful suggestions on an earlier version of this manuscript. I would like tothank the referee for pointing out the problems in an earlier version of the proofof Theorem 4.1. References [1] A. Arenas, Quantitative aspects of the representations of integers by quadratic forms, in:Number Theory, Alemania, ISBN 3-11-011791-6, pp. 7–14 (1989).[2] Z.S. Aygin, Extensions of Ramanujan–Mordell formula with coefficients 1 and p , J. Math.Anal. Appl., 465, 690–702 (2018).[3] Z.S. Aygin, On Eisenstein series in M k (Γ ( N )) and their applications, J. Number Theory,195, 358–375 (2019).[4] B.C. Berndt, R.J. Evans and K.S. Williams, Gauss and Jacobi sums, Wiley–Interscience,New York (1998).[5] H. Cohen and F. Str¨omberg, Modular Forms A Classical Approach, Graduate studies inmathematics, American Mathematical Society, Providence, Rhode Island (2017).[6] S. Cooper, Ramanujan’s Theta Functions, Springer International Publishing AG, Switzerland(2017).[7] S. Cooper, B. Kane and D. Ye, Analogues of the Ramanujan–Mordell theorem, J. Math.Anal. Appl., 446, 568–579 (2017).[8] J.E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cam-bridge (1992).[9] F. Diamond and J. Shurman, A First Course in Modular Forms, Graduate Texts in Mathe-matics 228, Springer-Verlag (2004).[10] N. Fine, Basic Hypergeometric Series and Applications, American Mathematical Society,Providence, RI (1988).[11] H. Iwaniec, Topics in Classical Automorphic Forms, Grad. Stud. Math., vol. 17, AmericanMathematical Society, Providence, RI (1997).[12] G. K¨ohler, Eta Products and Theta Series Identities, Springer Monographs in Mathematics,Springer (2011).[13] T. Miyake, Modular Forms, Springer-Verlag, Berlin (1989), translated from the Japaneseby Yoshitaka Maeda.[14] L.J. Mordell, On the representations of numbers as a sum of 2 r squares, Quart. J. PureAppl. Math. 48 (1917), 93–104. ,309–356 (1998). Appendix A. The SAGE functions for computing the constantterms of eta quotients at a given cusp Let r d ∈ Z , not all zeros, N ∈ N and define f ( z ) = Y d | N η r d ( dz ) . Assuming f ( z ) to be a modular form the following SAGE functions (written usingversion 9.1 of the software [17]) help computing [0] a/c f , the constant term of f ( z )at the cusp a/c . def v e t a 1 ( a , b , c , d ) : i f c%2==1: return k r o n e c k e r s y m b o l ( d , abs ( c ) ) i f c%2==0: return k r o n e c k e r s y m b o l ( c , abs ( d ) ) def v e t a 2 ( a , b , c , d ) : i f c%2==1: return i f c%2==0: return ( − ∗ ( sgn ( c ) − ∗ ( sgn ( d) − def v e t a 3 ( a , b , c , d ) : i f c%2==1: return ( 1 / 2 4 ∗ ( ( a+d ) ∗ c − b ∗ d ∗ ( cˆ2 − − ∗ c ) ) i f c%2==0: return ( 1 / 2 4 ∗ ( ( a+d ) ∗ c − b ∗ d ∗ ( c ˆ2 − ∗ d − − ∗ c ∗ d ) )27 ef L c o n s t r (m, d , c ) : x1=m ∗ d/ gcd ( c ,m)u1= − c / gcd ( c ,m)y1=0v1=0 for i 1 in range ( − abs ( x1 ∗ u1 ) , abs ( x1 ∗ u1 ) ) : i f gcd ( i1 , x1)==1 and (1+ i 1 ∗ u1)%x1==0 and ((1+ i 1 ∗ u1 )/ x1)%2==1:y1=i 1v1=(1+ i 1 ∗ u1 )/ x1 return [ x1 , y1 , u1 , v1 ] breakdef A f i n d ( d , c ) : for b in range ( abs ( d ∗ c ) ) : i f gcd ( b , d)==1 and (1+b ∗ c)%d==0:a=(1+b ∗ c )/ d return [ a , b , c , d ] breakdef f c o f e t a (m, d , c ) : − d/ c A=A f i n d ( d , c )L=L c o n s t r (m, d , c )a=A [ 0 ]b=A [ 1 ]c=A [ 2 ]d=A [ 3 ]x=L [ 0 ]y=L [ 1 ]u=L [ 2 ]v=L [ 3 ]vv= − m ∗ b ∗ v − y ∗ aOP1=v e t a 1 ( x , y , u , v )OP2=v e t a 2 ( x , y , u , v )OP3=v e t a 3 ( x , y , u , v )OP4=(1/24/m ∗ vv ∗ gcd ( c ,m) )OP5=(gcd ( c ,m)/m) ˆ ( 1 / 2 ) return [ OP1, OP2, OP3, OP4, OP5 ] def f i r s t c o e f f o f e t a q (N, etaq , a , c ) : d= − ad i v s=d i v i s o r s (N)L= len ( etaq ) i f sum (1/24/ d i v s [ i 2 ] ∗ ( gcd ( c , d i v s [ i 2 ] ) ) ˆ 2 ∗ etaq [ i 2 ] for i 2 in range (L)) > return e l s e : 28V1=prod ( ( f c o f e t a ( d i v s [ i 1 ] , d , c ) [ 0 ] ) ˆ ( etaq [ i 1 ] ) for i 1 in range (L ) )VV2=prod ( ( f c o f e t a ( d i v s [ i 1 ] , d , c ) [ 1 ] ) ˆ ( etaq [ i 1 ] ) for i 1 in range (L ) )SS1= sum ( ( f c o f e t a ( d i v s [ i 1 ] , d , c ) [ 2 ] ) ∗ ( etaq [ i 1 ] ) for i 1 in range (L ) )SS2= sum ( ( f c o f e t a ( d i v s [ i 1 ] , d , c ) [ 3 ] ) ∗ ( etaq [ i 1 ] ) for i 1 in range (L ) )VV3=prod ( ( f c o f e t a ( d i v s [ i 1 ] , d , c ) [ 4 ] ) ˆ ( etaq [ i 1 ] ) for i 1 in range (L ) )VV4=e ˆ(2 ∗ p i ∗ I ∗ ( SS1+SS2 ) )kk= sum ( r for r in etaq )/2 return ( − ∗ VV1 ∗ VV2 ∗ VV3 ∗ VV4 By Lemma 6.1 it will be sufficient to compute the constant terms of the etaquotient f k ( z ) defined by (3.1) at a set of inequivalent cusps of Γ (24), which isdone below with the help of this code. The set { / , / , / , / , / , / , / , / } gives a complete set of inequivalent cusps of Γ (24), see [5, Corollary 6.3.23]. Notethat if k is fixed then the code can handle the vanishing order analysis. For instancethe output for the codek=3e t a q =[ − ∗ k − ∗ k +1 ,2 ∗ k + 1 , 0 , 0 , 2 ∗ k +1 ,2 ∗ k+1, − ∗ k − print ( f i r s t c o e f f o f e t a q ( 2 4 , e t a q , 1 , 2 ) )will be 0. However, here we are working with a general k , and therefore the orderanalysis has to be done manually. When k ≥ 1, the vanishing orders of f k ( z ) isgreater than 0 at cusps { / , / , / , / , / , / } . Thus we have[0] / f k = 0 , [0] / f k = 0 , [0] / f k = 0 , [0] / f k = 0 , [0] / f k = 0 , [0] / f k = 0 . To compute [0] / f k and [0] / f k we run the following code:k=v a r ( ’ k ’ )assume ( k , ’ i n t e g e r ’ )e t a =[ − ∗ k − ∗ k +1 ,2 ∗ k + 1 , 0 , 0 , 2 ∗ k +1 ,2 ∗ k+1, − ∗ k − print ( f i r s t c o e f f o f e t a q ( 2 4 , e t a , 1 , 1 ) . s i m p l i f y ( ) ) print ( f i r s t c o e f f o f e t a q ( 2 4 , e t a , 1 , 2 4 ) . s i m p l i f y ( ) )The output will be: − I ∗ ∗ − ∗ k − ∗ − ∗ k − ∗ ( − / f k = − i k +1 √ k +1 k +2 , [0] / f k = 1 . Putting everything together, for all k ≥ / f k = − i k +1 √ k +1 k +2 , [0] / f k = 0 , [0] / f k = 0 , [0] / f k = 0 , / f k = 0 , [0] / f k = 0 , [0] / f k = 0 , [0] / f k = 1 . Department of Mathematics and Statistics, University of Calgary, Calgary,AB T2N 1N4, Canada Email address : [email protected]@gmail.com