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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.

%I A159633 #36 Oct 06 2024 20:43:46
%S A159633 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,
%T A159633 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,
%U A159633 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
%N 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.
%C A159633 Denote dim{M_k(Gamma_0(N))} by m(k,N) and dim{S_k(Gamma_0(N))} by s(k,N).
%C A159633 We have:
%C A159633 m(3/2,N)-s(3/2,N)+m(1/2,N)-s(1/2,N) =
%C A159633 m(5/2,N)-s(5/2,N) = m(7/2,N)-s(7/2,N) =
%C A159633 m(9/2,N)-s(9/2,N) = m(11/2,N)-s(11/2,N) = ...
%C A159633 m(k/2,N)-s(k/2,N) = ...
%C A159633 where N is any positive multiple of 4 and k>=5 is odd.
%C A159633 a(n) = A159635(n) - A159636(n). - _Steven Finch_, Apr 22 2009
%C A159633 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
%D A159633 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 A159633 H. Cohen and J. Oesterle, <a href="http://dx.doi.org/10.1007/BFb0065297">Dimensions des espaces de formes modulaires</a>, Modular Functions of One Variable. VI, Proc. 1976 Bonn conf., Lect. Notes in Math. 627, Springer-Verlag, 1977, pp. 69-78.
%H A159633 S. R. Finch, <a href="/A001616/a001616.pdf">Primitive Cusp Forms</a>, April 27, 2009. [Cached copy, with permission of the author]
%H A159633 Peter Humphries, <a href="http://mathoverflow.net/questions/161948/">Answer to: "A conjecture related to the Cohen-Oesterlé dimension formula"</a>, MathOverflow, 2014.
%H A159633 <a href="http://magma.maths.usyd.edu.au/calc/">MAGMA Calculator</a>.
%t A159633 (* see link, conjecture proved by P. Humphries *)
%t A159633 chi[n_Integer]:=Sum[EulerPhi[GCD[d,n/d]],{d,Divisors[n]}];
%t A159633 2 chi[#] - If[Mod[# + 2, 4] == 0, chi[#]/2, 0] & /@ Range[89]
%t A159633 (* _Wouter Meeussen_, Apr 06 2014 *)
%o A159633 (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]];
%Y A159633 Cf. A159630, A159631, A159632, A159634, A159635, A159636. - _Steven Finch_, Apr 22 2009
%Y A159633 Cf. A001616.
%K A159633 nonn
%O A159633 1,1
%A A159633 _Steven Finch_, Apr 17 2009