A063195 -id:A063195 - cp's OEIS frontend

A063226 Dimension of the space of weight 2n cuspidal newforms for Gamma_0(63).

3, 7, 13, 17, 23, 27, 33, 37, 43, 47, 53, 57, 63, 67, 73, 77, 83, 87, 93, 97, 103, 107, 113, 117, 123, 127, 133, 137, 143, 147, 153, 157, 163, 167, 173, 177, 183, 187, 193, 197, 203, 207, 213, 217, 223, 227, 233, 237, 243, 247

Offset: 1

N. J. A. Sloane, Jul 10 2001

Also, dimension of the space of weight 2n cuspidal newforms for Gamma_0(88). - N. J. A. Sloane, Nov 24 2016
First differences are 4,6,4,6,4,6.... Also  values of k such that k^(10*n) mod 10 = 8*(n mod 2)+1. - Gary Detlefs, Jul 04 2014
In other words, numbers n such that n^(2+4*k) + 1 is divisible by 10, for k >= 0. - Altug Alkan, Mar 30 2016
The rational generating function, the periodic first differences and Greubel's closed form are an immediate consequence of the structure of formula given by [Martin]. - R. J. Mathar, Apr 09 2016
A quasipolynomial of order 2 and degree 1: a(n) = 5n - 3 if n is even and 5n - 2 if n is odd. - Charles R Greathouse IV, Nov 03 2021
Numbers that are congruent to {3, 7} mod 10. - Amiram Eldar, Nov 23 2024

G. C. Greubel, Table of n, a(n) for n = 1..10000
Greg 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, Theorem 1.
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A017305 (bisection), A017353 (bisection), A019934, A182007.

Maple
```
# see A063195
```

Table[4 Floor[n/2] + 6 Floor[(n - 1)/2] + 3, {n, 50}] (* or *)
Table[SeriesCoefficient[3 x - x^2 (-7 - 6 x + 3 x^2)/((1 + x) (x - 1)^2), {x, 0, n}], {n, 50}] (* Michael De Vlieger, Mar 30 2016 *)
LinearRecurrence[{1, 1, -1}, {3, 7, 13}, 100] (* G. C. Greubel, Mar 30 2016 *)

my(x='x+O('x^99)); Vec(3*x-x^2*(-7-6*x+3*x^2)/((1+x)*(x-1)^2)) \\ Altug Alkan, Mar 31 2016

a(n)=5*n-3+n%2 \\ Charles R Greathouse IV, Mar 31 2016

a(n) = 4*floor(n/2) + 6*floor((n-1)/2) + 3. - Gary Detlefs, Jul 04 2014
G.f.: 3*x - x^2*(-7-6*x+3*x^2)/((1+x)*(x-1)^2). - R. J. Mathar, Jul 15 2015
From G. C. Greubel, Mar 30 2016: (Start)
a(n) = (1/2)*(10*n - 5 - (-1)^n).
E.g.f.: (5*x + 3)*cosh(x) + (5*x + 2)*sinh(x). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(5-2*sqrt(5))*Pi/10. - Amiram Eldar, Sep 26 2022
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*sin(Pi/5) (A182007).
Product_{n>=1} (1 + (-1)^n/a(n)) = tan(Pi/5) (A019934). (End)

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

Original entry on oeis.org

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.

A063279 Dimension of the space of weight n cuspidal newforms for Gamma_1( 6 ).

Original entry on oeis.org

-1, 0, 0, 1, 2, 1, 2, 1, 2, 1, 4, 3, 4, 1, 4, 3, 6, 3, 6, 3, 6, 3, 8, 5, 8, 3, 8, 5, 10, 5, 10, 5, 10, 5, 12, 7, 12, 5, 12, 7, 14, 7, 14, 7, 14, 7, 16, 9, 16, 7, 16, 9, 18, 9, 18, 9, 18, 9, 20, 11, 20, 9, 20, 11, 22, 11, 22, 11, 22, 11, 24, 13, 24, 11, 24, 13, 26, 13

Offset: 2

N. J. A. Sloane, Jul 14 2001

sign

William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_1(N))
William A. Stein, The modular forms database
Index entries for linear recurrences with constant coefficients, signature (1, 0, -1, 2, -1, 0, 1, -1).

Cf. A302402 (bisection), A063195 (bisection)

g.f.: x^2*(x^9-2*x^8+2*x^6-2*x^5+3*x^4+x-1) / ((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+1)). - Colin Barker, Feb 24 2015

A260088 Dimension of the space of newforms of weight 4 and level n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 2, 1, 4, 1, 4, 1, 4, 3, 5, 1, 3, 3, 4, 2, 7, 2, 7, 3, 6, 4, 6, 1, 9, 5, 6, 3, 10, 2, 10, 3, 5, 6, 11, 3, 8, 5, 8, 3, 13, 4, 10, 4, 10, 7, 14, 2, 15, 8, 7, 5, 12, 4, 16, 4, 12, 6, 17, 4, 18, 9, 10, 5, 16, 6, 19, 6

Offset: 1

R. J. Mathar, Jul 15 2015

Cf. A127788, A063195 - A063247.

seq(g0star(4,N),N=1..80); # using functions coded in A063195

cp's OEIS Frontend

A063226 Dimension of the space of weight 2n cuspidal newforms for Gamma_0(63).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

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

A063279 Dimension of the space of weight n cuspidal newforms for Gamma_1( 6 ).

Views

Author

Keywords

Links

Crossrefs

Formula

A260088 Dimension of the space of newforms of weight 4 and level n.

Views

Author

Keywords

Crossrefs

Programs

Maple