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

1,23

Comments

"Newform" is meant in the sense of Atkin-Lehner, that is, a primitive Hecke eigenform relative to the subgroup Gamma_0(n).

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]

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.

Example

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 + ...

Maple

seq( g0star(2, N), N=1..80); # using the source in A063195 - R. J. Mathar, Jul 15 2015

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 *)

Prog

(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 */

Crossrefs

Cf. A001617, A116563.

Sequence in context: A025854 A190766 A025857 * A025656 A194517 A110658

Adjacent sequences: A127785 A127786 A127787 * A127789 A127790 A127791

Keyword

nonn

Author

Steven Finch, Apr 04 2007

Status

approved