A063200 Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 15 ).

1, 2, 4, 4, 6, 8, 8, 10, 12, 12, 14, 16, 16, 18, 20, 20, 22, 24, 24, 26, 28, 28, 30, 32, 32, 34, 36, 36, 38, 40, 40, 42, 44, 44, 46, 48, 48, 50, 52, 52, 54, 56, 56, 58, 60, 60, 62, 64, 64, 66

Offset: 1

Author

N. J. A. Sloane, Jul 10 2001

Keywords

• nonn

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• 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, 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 (1,0,1,-1).

Programs

Mathematica

LinearRecurrence[{1, 0, 1, -1}, {1, 2, 4, 4, 6}, 100] (* Paolo Xausa, Jan 29 2024 *)

Python

def A063200(n): return n-1+sum(divmod(n-1,3)) if n > 1 else 1 # Chai Wah Wu, Jan 29 2023

Formula

G.f.: x +2*x^2*(1+x) / ( (1+x+x^2)*(x-1)^2 ). - R. J. Mathar, Jul 15 2015

Sum_{n>=1} (-1)^(n+1)/a(n) = 1- Pi/8. - Amiram Eldar, Jan 12 2024