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A127788 Dimension of the space of newforms of weight 2 and level n. 3
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 (list; graph; refs; listen; history; text; internal format)
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

E. Halberstadt and A. Kraus, Courbes de Fermat: resultats et problemes, J. Reine Angew. Math. 548 (2002) 167-234. - Steven Finch, Mar 27 2009

G. Martin, Dimensions of the spaces of cusp forms and newforms on Gamma_0(N) and Gamma_1(N), J. Number Theory 112 (2005) 298-331. - Steven Finch, Mar 27 2009

H. Cohen, Number Theory. Vol. II. Analytic and Modern Tools. Springer, 2007, pp. 496-497. - Steven Finch, Mar 27 2009

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula (5.8) in Xian-Jin Li, An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials, J. Number Theory 113 (2005) 175-200

Table A in Toshitsune Miyake, Modular Forms, Springer-Verlag, 1989

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-Fran├žois 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

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Last modified October 23 22:26 EDT 2017. Contains 293833 sequences.