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A159633
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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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1
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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K. Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and q-series. American Mathematical Society, 2004 (p. 16, theorem 1.56).
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LINKS
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MATHEMATICA
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(* 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]
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PROG
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(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]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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