A159632 -id:A159632 - cp's OEIS frontend

A159633 Dimension of Eisenstein subspace of the space of modular forms of weight k/2, level 4*n and trivial character, where k>=5 is odd.

2, 3, 4, 6, 4, 6, 4, 8, 8, 6, 4, 12, 4, 6, 8, 12, 4, 12, 4, 12, 8, 6, 4, 16, 12, 6, 12, 12, 4, 12, 4, 16, 8, 6, 8, 24, 4, 6, 8, 16, 4, 12, 4, 12, 16, 6, 4, 24, 16, 18, 8, 12, 4, 18, 8, 16, 8, 6, 4, 24, 4, 6, 16, 24, 8, 12, 4, 12, 8, 12, 4, 32, 4, 6, 24, 12, 8, 12, 4, 24, 24, 6, 4, 24, 8, 6, 8, 16, 4

Offset: 1

Steven Finch, Apr 17 2009

nonn

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.
a(n) = A159635(n) - A159636(n). - Steven Finch, Apr 22 2009
Conjecture: a(n) = 2*chi(n) - if(mod(n+2,4)=0, chi(n)/2, 0) with chi(n) = A001616(n) = Sum_{d|n} phi(gcd(d,n/d)); checked up to n=1024. - Wouter Meeussen, Apr 02 2014

K. Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and q-series. American Mathematical Society, 2004 (p. 16, theorem 1.56).

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.
MAGMA Calculator.

Cf. A159630, A159631, A159632, A159634, A159635, A159636. - Steven Finch, Apr 22 2009

Cf. A001616.

[[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]];

(* 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 *)

cp's OEIS Frontend

A159633 Dimension of Eisenstein subspace of the space of modular forms of weight k/2, level 4*n and trivial character, where k>=5 is odd.

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