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

Author

Steven Finch, Apr 17 2009

Keywords

• nonn

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.

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

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.
• MAGMA Calculator.

Crossrefs

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

Cf. A001616.

Programs

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]];

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 *)