OFFSET
1,1
COMMENTS
Denote dim{M_k(Gamma_0(N))} by m(k,N) and dim{S_k(Gamma_0(N))} by s(k,N).
We have:
m(3/2,N)-s(3/2,N)+m(1/2,N)-s(1/2,N) =
m(5/2,N)-s(5/2,N) = m(7/2,N)-s(7/2,N) =
m(9/2,N)-s(9/2,N) = m(11/2,N)-s(11/2,N) = ...
m(k/2,N)-s(k/2,N) = ...
where N is any positive multiple of 4 and k>=5 is odd.
Conjecture: a(n) = 2*chi(n) - if(mod(n+2,4)=0, chi(n)/2, 0) with chi(n) = Sum(d|n; phi(gcd(d,n/d)); checked up to n=1024. - Wouter Meeussen, Apr 02 2014
REFERENCES
K. Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and q-series. American Mathematical Society, 2004 (p. 16, theorem 1.56).
LINKS
H. Cohen and J. Oesterle, Dimensions des espaces de formes modulaires, Modular Functions of One Variable. VI, Proc. 1976 Bonn conf., Lect. Notes in Math. 627, Springer-Verlag, 1977, pp. 69-78.
S. R. Finch, Primitive Cusp Forms, April 27, 2009. [Cached copy, with permission of the author]
Peter Humphries, Answer to: "A conjecture related to the Cohen-Oesterlé dimension formula", MathOverflow, 2014.
MATHEMATICA
(* see link, conjecture proved by P. Humphries *)
chi[n_Integer]:=Sum[EulerPhi[GCD[d, n/d]], {d, Divisors[n]}];
2 chi[#] - If[Mod[# + 2, 4] == 0, chi[#]/2, 0] & /@ Range[89]
(* Wouter Meeussen, Apr 06 2014 *)
PROG
(Magma) [[4*n, Dimension(HalfIntegralWeightForms(4*n, 5/2))-Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n, 5/2)))] : n in [1..100]]; [[4*n, Dimension(HalfIntegralWeightForms(4*n, 7/2))-Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n, 7/2)))] : n in [1..100]]; [[4*n, Dimension(HalfIntegralWeightForms(4*n, 3/2))-Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n, 3/2)))+Dimension(HalfIntegralWeightForms(4*n, 1/2))-Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n, 1/2)))] : n in [1..100]];
CROSSREFS
KEYWORD
nonn
AUTHOR
Steven Finch, Apr 17 2009
STATUS
approved