Abstract
We prove that the number of quartic
S
4
--extensions of the rationals of given discriminant
d
is $O_\eps(d^{1/2+\eps})$ for all $\eps>0$. For a prime number
p
we derive that the dimension of the space of octahedral modular forms of weight 1 and conductor
p
or
p
2
is bounded above by
O(
p
1/2
log(p
)
2
)
.