A127788 Dimension of the space of newforms of weight 2 and level n.
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 0, 2, 1, 0, 2, 1, 2, 1, 1, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 1, 1, 1, 4, 1, 1, 2, 3, 1, 4, 2, 3, 2, 3, 2, 5, 0, 4, 3, 3, 1, 5, 3, 5, 2, 3, 1, 6, 1, 5, 4, 3, 1, 5, 1, 6, 2, 2, 3, 7, 2, 5, 4, 5, 3, 7, 3, 7, 2, 5, 3, 7, 2, 7, 3, 4, 1, 8, 3
Offset: 1
Keywords
Examples
a(p) = A001617(p) for any prime p. G.f. = x^11 + x^14 + x^15 + x^17 + x^19 + x^20 + x^21 + 2*x^23 + x^24 + ...
References
- H. Cohen, Number Theory. Vol. II. Analytic and Modern Tools. Springer, 2007, pp. 496-497.
- Toshitsune Miyake, Modular Forms, Springer-Verlag, 1989. See Table A.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..10000
- E. Halberstadt and A. Kraus, Courbes de Fermat: résultats et problèmes, J. Reine Angew. Math. 548 (2002) 167-234. [_Steven Finch_, Mar 27 2009]
- Xian-Jin Li, An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials, arXiv:math/0403148 [math.NT], 2004; J. Number Theory 113 (2005) 175-200. See Formula (5.8).
- G. Martin, Dimensions of the spaces of cusp forms and newforms on Gamma_0(N) and Gamma_1(N), J. Numb. Theory 112 (2005) 298-331. [_Steven Finch_, Mar 27 2009]
Programs
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Maple
seq( g0star(2,N),N=1..80); # using the source in A063195 - R. J. Mathar, Jul 15 2015
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Mathematica
A001617[n_] := If[n < 1, 0, 1 + Sum[MoebiusMu[d]^2 n/d/12 - EulerPhi[GCD[d, n/d]]/2, {d, Divisors@n}] - Count[(#^2 - # + 1)/n & /@ Range[n], ?IntegerQ]/3 - Count[(#^2 + 1)/n & /@ Range[n], ?IntegerQ]/4]; a[n_ /; n < 10] = 0; a[n_] := a[n] = A001617[n] - Sum[a[m]*DivisorSigma[0, n/m], {m, Divisors[n][[2 ;; -2]]}]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Sep 07 2015, A001617 code due to Michael Somos *)
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PARI
{a(n) = my(v = [1, 3, 4, 6], A, p, e); if( n<1, 0, A = factor(n); for( k=1, matsize(A)[1], [p, e] = A[k,]; v[1] *= if( e==1, p-1, e==2, p^2-p-1, p^(e-3) * (p+1) * (p-1)^2); v[2] *= if( p==2, (e==3) - (e<3), e==1, kronecker(-4, p) - 1, e==2, -kronecker(-4, p)); v[3] *= if( p==3, (e==3) - (e<3), e==1, kronecker(-3, p) - 1, e==2, -kronecker(-3, p)); v[4] *= if( e%2, 0, e==2, p-2, p^(e/2-2) * (p-1)^2)); moebius(n) + (v[1] - v[2] - v[3] - v[4]) / 12 )}; /* Michael Somos, Jun 06 2015 */
Formula
a(n) = A001617(n) - sum a(m)*d(n/m), where the summation is over all divisors 1 < m < n of n and d is the divisor function.
Comments