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A159634
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Coefficient for dimensions of spaces of modular & cusp forms of weight k/2, level 4*n and trivial character, where k>=5 is odd.
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5
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1, 2, 4, 4, 6, 8, 8, 8, 12, 12, 12, 16, 14, 16, 24, 16, 18, 24, 20, 24, 32, 24, 24, 32, 30, 28, 36, 32, 30, 48, 32, 32, 48, 36, 48, 48, 38, 40, 56, 48, 42, 64, 44, 48, 72, 48, 48, 64, 56, 60, 72, 56, 54, 72, 72, 64, 80, 60, 60, 96, 62, 64, 96, 64, 84, 96, 68, 72, 96, 96
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OFFSET
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1,2
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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(7/2,N)+s(5/2,N) = m(5/2,N)+s(7/2,N) =
(m(11/2,N)+s(9/2,N))/2 = (m(9/2,N)+s(11/2,N))/2 =
(m(15/2,N)+s(13/2,N))/3 = (m(13/2,N)+s(15/2,N))/3 = ...
(m((4j+3)/2,N)+s((4j+1)/2,N))/j = (m((4j+1)/2,N)+s((4j+3)/2,N))/j = ...
where N is any positive multiple of 4 and j>=1.
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REFERENCES
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Ken 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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FORMULA
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a(n)= n*Product_{p|n, p odd prime} (1 + 1/p).
a(n) = Sum_{d|n, d odd} mu(d)^2*n/d, where mu(n) = A008683(n) is the Möbius function.
If n = m*2^k , where 2^k is the largest power of 2 dividing n, then
a(n) = (2^k)*a(m) = 2^k * Sum_{d^2|m} mu(d)*sigma(m/d^2), where sigma(n) = A000203(n) is the sum of the divisors of n, and also
a(n) = 2^k * Sum_{d|m} 2^omega(d)*phi(m/d), where omega(n) = A001221(n) is the number of different primes dividing n and phi(n) = A000010 is the Euler totient function.
O.g.f.: Sum_{n >= 1} mu(2*n-1)^2*x^(2^n-1)/(1 - x^(2*n-1))^2. (End)
Multiplicative with a(2^e) = 2^e, and a(p^e) = (p+1)*p^(e-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 6/Pi^2 = 0.607927... (A059956). (End)
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MATHEMATICA
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dedekindPsi[n_Integer]:=n Apply[Times, 1+1/Map[First, FactorInteger[n]]];
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PROG
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(Magma) [[4*n, (Dimension(HalfIntegralWeightForms(4*n, 7/2))+ Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n, 5/2))))/2] : n in [1..70]]; [[4*n, (Dimension(HalfIntegralWeightForms(4*n, 5/2))+ Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n, 7/2))))/2] : n in [1..70]];
(PARI) a(n) = 2*n*sumdiv( 2*n, d, moebius(d)^2 / d)/3; \\ Andrew Howroyd, Aug 08 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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