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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. 1
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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) = 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-OesterlĂ© dimension formula", MathOverflow, 2014. 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 *) PROG (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]]; CROSSREFS Cf. A159630, A159631, A159632, A159634, A159635, A159636. - Steven Finch, Apr 22 2009 Cf. A001616. Sequence in context: A074103 A051785 A144825 * A049044 A102284 A273098 Adjacent sequences:  A159630 A159631 A159632 * A159634 A159635 A159636 KEYWORD nonn AUTHOR Steven Finch, Apr 17 2009 STATUS approved

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Last modified April 18 07:59 EDT 2021. Contains 343084 sequences. (Running on oeis4.)