A116563 -id:A116563 - cp's OEIS frontend

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

Steven Finch, Apr 04 2007

nonn

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

			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 + ...

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.

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]

Cf. A001617, A116563.

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

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 *)

{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 */

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.

A116569 a(n) = (x^3 - x) / 6, where x is the genus of the modular curve X_0(p) for p = prime(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 4, 4, 10, 10, 20, 10, 20, 35, 20, 35, 56, 56, 56, 84, 84, 120, 84, 120, 165, 220, 220, 220, 286, 286, 286, 364, 455, 455, 560, 455, 680, 560, 680, 680, 816, 969, 1140, 969

Offset: 1

Roger L. Bagula, Mar 18 2006

nonn

From Mia Boudreau, Jul 29 2025: (Start)
Previously named "Ono prime weight function divided by 6.".
See A001617 and A116563 for definition of genus of modular curve for X_0(n). (End)

			a(415) = 2218636 = (A116563(415)^3 - A116563(415)) / 6.

Mia Boudreau, Table of n, a(n) for n = 1..10000
Ken Ono and Scott Ahlgren, Weierstrass points on X0(p) and supersingular j-invariants, Mathematische Annalen 325, 2003, pp. 355-368.

Cf. A116563, A001617, A000040.

long a(int n){
 long p = prime(n);
 long k = (p - switch((int)(p % 12)){
  case 1 -> 13; case 2 -> 5; case 3 -> 7; default -> -1;}) / 12;
 return k * (k - 1) * (k + 1) / 6;} // Mia Boudreau, Jul 29 2025

g[1] = 1; g[2] = 1;
g[n_] := (Prime[n] - 13)/12 /; Mod[Prime[n], 12] - 1 == 0;
g[n_] := (Prime[n] - 5)/12 /; Mod[Prime[n], 12] - 5 == 0;
g[n_] := (Prime[n] - 7)/12 /; Mod[Prime[n], 12] - 7 == 0;
g[n_] := (Prime[n] + 1)/12 /; Mod[Prime[n], 12] - 11 == 0;
Table[g[n]*(g[n]^2 - 1)/6, {n, 1, 50}]

a(n) = {if (n < 3, g = 1, p = prime(n); m = p % 12; g = if (m==1, (p-13)/12, if (m==5, (p-5)/12, if (m==7, (p-7)/12, if (m==11, (p+1)/12))))); g*(g^2-1)/6;} \\ Michel Marcus, Apr 06 2018

a(n) = (A116563(n)^3 - A116563(n)) / 6. - Mia Boudreau, Jul 29 2025

Offset corrected by Michel Marcus, Apr 06 2018

cp's OEIS Frontend

A127788 Dimension of the space of newforms of weight 2 and level n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula