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Dimension of the space of Siegel cusp forms of genus 2 and weight n.
4

%I #37 Mar 10 2022 08:28:39

%S 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,

%T 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,

%U 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

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

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

%H Andy Huchala, <a href="/A165685/b165685.txt">Table of n, a(n) for n = 1..20000</a>

%H Andy Huchala, <a href="/A165685/a165685_1.pdf">Proof of generating function</a>.

%F 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

%F a(2n) = A165684(n) and a(2n+35) = A029143(n). - _Andy Huchala_, Mar 04 2022

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

%t 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;

%t 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;

%t 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;

%t DimSk[k_] := If[k >= 7, N1[k] + N2[k] + N3[k] + N4[k], 0];

%t Table[ DimSk[k], {k, 1, 100}]

%t (* second program: *)

%t 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};

%t 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};

%t ans = LinearRecurrence[ker, init, 100];

%t ans[[3]] = 0 ; ans (* _Andy Huchala_, Mar 03 2022 *)

%o (Sage)

%o R.<x> = PowerSeriesRing(ZZ, 100);

%o 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;

%o 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;

%o (x^3 + p/q).list()[1:] # _Andy Huchala_, Mar 03 2022

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

%K nonn

%O 1,16

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

%E a(73) corrected by _Andy Huchala_, Mar 02 2022