A107505 Theta series of quadratic form with Gram matrix [ 2, 1, 0, 1; 1, 4, 1, 0; 0, 1, 4, -2; 1, 0, -2, 8].

1, 2, 6, 8, 14, 12, 24, 16, 30, 26, 36, 24, 56, 2, 48, 48, 62, 36, 78, 40, 84, 64, 72, 48, 120, 62, 6, 80, 112, 60, 144, 64, 126, 96, 108, 96, 182, 76, 120, 8, 180, 84, 192, 88, 168, 156, 144, 96, 248, 114, 186, 144, 14, 108, 240, 144, 240, 160, 180, 120

Offset

0,2

Comments

Coefficients of a theta series associated with a certain "Haupt-form" of rank 4 and level 13.

The Gram matrix is denoted by A in Parry 1979 on page 165.

Links

Table of n, a(n) for n=0..59.

W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.

Formula

a(n) = 2 * b(n) where b() is multiplicative and b(13^e) = 1, b(p^e) = (p^(e+1) - 1) / (p - 1) otherwise. - Michael Somos, Mar 23 2012

G.f. is a period 1 Fourier series which satisfies f(-1 / (13 t)) = 13 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Mar 23 2012

a(n) = 2 * A284587(n) if n>1. - Michael Somos, Oct 23 2019

Example

G.f. = 1 + 2*q + 6*q^2 + 8*q^3 + 14*q^4 + 12*q^5 + 24*q^6 + 16*q^7 + 30*q^8 + ...

Mathematica

a[n_] := If[n == 0, 1, 2 DivisorSigma[1, n/13^IntegerExponent[n, 13]]];
a /@ Range[0, 59] (* Jean-François Alcover, Oct 23 2019, after Michael Somos *)
a[n_] := If[n == 0, 1, 2 DivisorSum[n, Boole[!Divisible[#, 13]] # &]];
a /@ Range[0, 59] (* Jean-François Alcover, Oct 23 2019 *)

Prog

(PARI) {a(n) = if( n<1, n==0, 1, 2 * sigma(n / 13^valuation(n, 13)))}; /* Michael Somos, Mar 23 2012 */
(PARI) {a(n) = my(G); if( n<0, 0, G = [2, 1, 0, 1; 1, 4, 1, 0; 0, 1, 4, -2; 1, 0, -2, 8]; polcoeff( 1 + 2 * x * Ser(qfrep( G, n, 1)), n))}; /* Michael Somos, Mar 23 2012 */
(Sage) ModularForms( Gamma0(13), 2, prec=100).0; # Michael Somos, Jun 27 2013
(Magma) Basis( ModularForms( Gamma0(13), 2), 100) [1]; /* Michael Somos, Aug 15 2016 */
(Magma) [Coefficient(Basis(ModularForms(Gamma0(13), 2))[1], n) : n in [0..100] ]; // Vincenzo Librandi, Jun 27 2017

Crossrefs

Cf. A284587.

Sequence in context: A289753 A002511 A074383 * A074400 A264598 A165607

Adjacent sequences: A107502 A107503 A107504 * A107506 A107507 A107508

Keyword

nonn

Author

N. J. A. Sloane, May 28 2005

Status

approved