A027640 Poincaré series [or Poincare series] for ring of modular forms of genus 2.

1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 2, 0, 4, 0, 4, 0, 5, 0, 6, 0, 8, 0, 7, 0, 10, 0, 11, 0, 12, 0, 14, 1, 17, 0, 16, 1, 21, 1, 22, 1, 24, 2, 27, 3, 31, 2, 31, 4, 37, 4, 39, 5, 42, 6, 46, 8, 52, 7, 52, 10, 60, 11, 63, 12, 67, 14, 73, 17, 80, 16, 81, 21, 91, 22, 95

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(k) for k>0 is the dimension of the space of Siegel modular forms of genus 2 and weight k (for the full modular group Gamma_2). - Kilian Kilger (kilian(AT)nihilnovi.de), Sep 24 2009

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• J. Igusa, On Siegel modular forms of genus 2 (II), Amer. J. Math., 86 (1964), 392-412, esp. p. 402.
• Bernhard Runge, On Siegel modular forms. Part I, J. Reine Angew. Math., 436 (1993), 57-85.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,1,1,0,0,-1,-1,-1,1,0,0,1,-1,-1,-1,0,0,1,1,1,0,0,0,-1).

Crossrefs

Cf. A165685 for the corresponding dimension of the space of cusp forms. - Kilian Kilger (kilian(AT)nihilnovi.de), Sep 24 2009

Programs

Magma

R:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (1+x^35)/((1-x^4)*(1-x^6)*(1-x^10)*(1-x^12)) )); // G. C. Greubel, Aug 04 2022

Mathematica

Table[SeriesCoefficient[Series[(1+t^(35))/((1-t^4) (1-t^6)(1-t^(10)) (1-t^(12))), {t, 0,100}], i], {i, 0, 100}] (* Kilian Kilger (kilian(AT)nihilnovi.de), Sep 24 2009 *)

PARI

Vec((1+x^35)/((1-x^4)*(1-x^6)*(1-x^10)*(1-x^12)) + O(x^100)) \\ Colin Barker, Jul 27 2019

Sage

def A027640_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^35)/((1-x^4)*(1-x^6)*(1-x^10)*(1-x^12)) ).list()
A027640_list(100) # G. C. Greubel, Aug 04 2022

Formula

G.f.: (1+x^35)/((1-x^4)*(1-x^6)*(1-x^10)*(1-x^12)).