A165685 Dimension of the space of Siegel cusp forms of genus 2 and weight n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 4, 0, 5, 0, 5, 0, 7, 0, 8, 0, 9, 0, 11, 1, 13, 0, 13, 1, 17, 1, 18, 1, 20, 2, 23, 3, 26, 2, 27, 4, 32, 4, 34, 5, 37, 6, 41, 8, 46, 7, 47, 10, 54, 11, 57, 12, 61, 14, 67, 17, 73, 16, 75, 21, 84, 22, 88, 24, 94, 27, 101, 31, 109, 31, 112

Offset: 1

Author

Kilian Kilger (kilian(AT)nihilnovi.de), Sep 24 2009

Keywords

• nonn

Examples

			a(35)=1 as the dimension of the space of Siegel cusp form of genus 2 and weight 35 is 1.
		

References

• M. Eie, Dimensions of spaces of Siegel cusp forms of degree two and three, AMS, 1984, p. 44-45.

Links

• Andy Huchala, Table of n, a(n) for n = 1..20000
• Andy Huchala, Proof of generating function.

Crossrefs

Cf. A008615, A029143. A165684 gives only the even weights.

Programs

Mathematica

N1[k_] := 2^(-7)*3^(-3)*5^(-1) (2 k^3 + 96 k^2 - 52 k - 3231) /; Mod[k, 2] == 0; N1[k_] := 2^(-7)*3^(-3)*5^(-1)*(2 k^3 - 114 k^2 + 2018 k - 9051) /; Mod[k, 2] == 1; N2[k_] := 2^(-5)*3^(-3)*(17 k - 294) /; Mod[k, 12] == 0; N2[k_] := 2^(-5)*3^(-3)*(-25 k + 325) /; Mod[k, 12] == 1; N2[k_] := 2^(-5)*3^(-3)*(-25 k + 254) /; Mod[k, 12] == 2; N2[k_] := 2^(-5)*3^(-3)*(17 k - 261) /; Mod[k, 12] == 3; N2[k_] := 2^(-5)*3^(-3)*(17 k - 86) /; Mod[k, 12] == 4; N2[k_] := 2^(-5)*3^(-3)*(-k + 53) /; Mod[k, 12] == 5; N2[k_] := 2^(-5)*3^(-3)*(-k - 42) /; Mod[k, 12] == 6; N2[k_] := 2^(-5)*3^(-3)*(-7 k + 91) /; Mod[k, 12] == 7; N2[k_] := 2^(-5)*3^(-3)*(-7 k + 2) /; Mod[k, 12] == 8; N2[k_] := 2^(-5)*3^(-3)*(-k - 27) /; Mod[k, 12] == 9;
N2[k_] := 2^(-5)*3^(-3)*(-k + 166) /; Mod[k, 12] == 10; N2[k_] := 2^(-5)*3^(-3)*(17 k - 181) /; Mod[k, 12] == 11; N3[k_] := 2^(-7)*3^(-3)*1131 /; Mod[k, 12] == 0; N3[k_] := 2^(-7)*3^(-3)*229 /; Mod[k, 12] == 1; N3[k_] := 2^(-7)*3^(-3)*(-229) /; Mod[k, 12] == 2; N3[k_] := 2^(-7)*3^(-3)*(-1131) /; Mod[k, 12] == 3; N3[k_] := 2^(-7)*3^(-3)*427 /; Mod[k, 12] == 4; N3[k_] := 2^(-7)*3^(-3)*(-571) /; Mod[k, 12] == 5;
N3[k_] := 2^(-7)*3^(-3)*123 /; Mod[k, 12] == 6; N3[k_] := 2^(-7)*3^(-3)*(-203) /; Mod[k, 12] == 7; N3[k_] := 2^(-7)*3^(-3)*203 /; Mod[k, 12] == 8; N3[k_] := 2^(-7)*3^(-3)*(-123) /; Mod[k, 12] == 9; N3[k_] := 2^(-7)*3^(-3)*571 /; Mod[k, 12] == 10; N3[k_] := 2^(-7)*3^(-3)*(-427) /; Mod[k, 12] == 11; N4[k_] := 5^(-1) /; Mod[k, 5] == 0; N4[k_] := -5^(-1) /; Mod[k, 5] == 3; N4[k_] := 0 /; Mod[k, 5] == 1 || Mod[k, 5] == 2 || Mod[k, 5] == 4;
DimSk[k_] := If[k >= 7, N1[k] + N2[k] + N3[k] + N4[k], 0];
Table[ DimSk[k], {k, 1, 100}]
(* second program: *)
init = {0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 4, 0, 5, 0, 5, 0};
ker = {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};
ans = LinearRecurrence[ker, init, 100];
ans[[3]] = 0 ; ans (* Andy Huchala, Mar 03 2022 *)

Sage

R. = PowerSeriesRing(ZZ, 100);
p = x^26 + x^24 - x^21 - x^19 + x^18 - x^17 - x^14 - x^13 + x^10 + x^9 + x^8 + x^7 - x^3;
q = x^27 - x^23 - x^22 - x^21 + x^18 + x^17 + x^16 - x^15 - x^12 + x^11 + x^10 + x^9 - x^6 - x^5 - x^4 + 1;
(x^3 + p/q).list()[1:] # Andy Huchala, Mar 03 2022

Formula

G.f.: x^10 (1+x^2-x^5-x^7+x^10-x^15+x^20) / ((-1+x)^4 (1+x)^3 (1+2x^2+2x^4+x^6)^2 (1+x+x^4+x^7+x^8)). - Andy Huchala, Mar 03 2022

a(2n) = A165684(n) and a(2n+35) = A029143(n). - Andy Huchala, Mar 04 2022

Extensions

a(73) corrected by Andy Huchala, Mar 02 2022