A029143 -id:A029143 - cp's OEIS frontend

A165684 Dimension of the space of Siegel cusp forms of genus 2 and dimension 2n (associated with full modular group Gamma_2).

0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 9, 11, 13, 13, 17, 18, 20, 23, 26, 27, 32, 34, 37, 41, 46, 47, 54, 57, 61, 67, 73, 75, 84, 88, 94, 101, 109, 112, 123, 129, 136, 145, 155, 159, 173, 180, 189, 200, 212, 218, 234, 243, 254, 267, 282, 289, 308, 319

Offset: 1

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

			a(5)=1 as the space of Siegel cusp forms of genus 2 and weight 10 is one-dimensional.

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,0,1,-1,-2,-1,1,0,0,1,1,0,-1).

Cf. A029143 (dimension of the full space of Siegel modular forms of genus 2).

N1[k_] := 2^(-7)*3^(-3)*5^(-1) (2 k^3 + 96 k^2 - 52 k - 3231); N2[k_] := 2^(-5)*3^(-3)*(17 k - 294) /; Mod[k, 12] == 0; N2[k_] := 2^(-5)*3^(-3)*(-25 k + 254) /; Mod[k, 12] == 2; N2[k_] := 2^(-5)*3^(-3)*(17 k - 86) /; Mod[k, 12] == 4; N2[k_] := 2^(-5)*3^(-3)*(-k - 42) /; Mod[k, 12] == 6; N2[k_] := 2^(-5)*3^(-3)*(-7 k + 2) /; Mod[k, 12] == 8; N2[k_] := 2^(-5)*3^(-3)*(-k + 166) /; Mod[k, 12] == 10; N3[k_] := 2^(-7)*3^(-3)*1131 /; Mod[k, 12] == 0; N3[k_] := 2^(-7)*3^(-3)*(-229) /; Mod[k, 12] == 2; N3[k_] := 2^(-7)*3^(-3)*427 /; Mod[k, 12] == 4; N3[k_] := 2^(-7)*3^(-3)*123 /; Mod[k, 12] == 6; N3[k_] := 2^(-7)*3^(-3)*203 /; Mod[k, 12] == 8; N3[k_] := 2^(-7)*3^(-3)*571 /; Mod[k, 12] == 10; 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_] := N1[k] + N2[k] + N3[k] + N4[k]/;Mod[k,2]==0; Table[DimSk[2k],{k,1,100}]
CoefficientList[Series[-x^4*(x^6 - x - 1)/((1 - x^2)*(1 - x^3)*(1 - x^5)*(1 - x^6)), {x, 0, 50}], x] (* Wesley Ivan Hurt, Jun 20 2014 *)
LinearRecurrence[{0,1,1,0,0,1,-1,-2,-1,1,0,0,1,1,0,-1},{0,0,0,0,1,1,1,2,2,3,4,5,5,7,8,9},70] (* Harvey P. Dale, Dec 25 2016 *)

G.f.: -x^5*(x^6-x-1) / ((1-x^2)*(1-x^3)*(1-x^5)*(1-x^6)). - Colin Barker, Mar 30 2013

a(n) = 1/1080*n^3 + 1/45*n^2 + O(n). (from g.f.) - Ralf Stephan, Jun 20 2014

More terms from Wesley Ivan Hurt, Jun 20 2014

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

Original entry on oeis.org

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

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

nonn

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

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

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

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

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 *)

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

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

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

A352095 Dimension of the space of Siegel cusp forms of genus 3 and weight 2n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 6, 9, 14, 17, 27, 34, 46, 61, 82, 99, 135, 165, 208, 261, 325, 389, 490, 584, 708, 852, 1023, 1200, 1445, 1687, 1984, 2327, 2717, 3133, 3663, 4199, 4838, 5557, 6360, 7225, 8267, 9344, 10587, 11968, 13489, 15126, 17037, 19023

Offset: 0

Andy Huchala, Mar 09 2022

nonn

There are no nonzero Siegel cusp forms of genus 3 and odd weight.

Sequence satisfies linear recurrence of order 54 for a(n) when n > 57.

			The space of weight 18 Siegel cusp forms of genus 3 has dimension 4.

Andy Huchala, Table of n, a(n) for n = 0..20000
O. Taibi, Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula, arXiv 1406.4247 [math.NT] (2014), 64-65.
S. Tsuyumine, On Siegel modular forms of degree three, Amer. J. Math., 108 (1986), 831-832.
Index entries for linear recurrences with constant coefficients signature (1,0,0,0,0,2,-1,-1,1,0,-1,-1,-1,2,-1,-2,2,1,0,0,-1,3,0,-3,2,0,0,0,-2,3,0,-3,1,0,0,-1,-2,2,1,-2,1,1,1,0,-1,1,1,-2,0,0,0,0,-1,1)

Cf. A027634, A029143.

R. = PowerSeriesRing(ZZ,100)
p = -x^56 + x^55 - x^54 - x^51 - 3*x^48 + x^47 - 3*x^46 - 2*x^45 - 2*x^44 - 3*x^43 - 4*x^42 - 2*x^41 - 7*x^40 - 3*x^39 - 8*x^38 - 4*x^37 - 10*x^36 - 6*x^35 - 10*x^34 - 9*x^33 - 9*x^32 - 9*x^31 - 13*x^30 - 5*x^29 - 15*x^28 - 6*x^27 - 11*x^26 - 10*x^25 - 8*x^24 - 8*x^23 - 11*x^22 - 4*x^21 - 10*x^20 - 5*x^19 - 6*x^18 - 5*x^17 - 6*x^16 - 2*x^15 - 6*x^14 - 2*x^13 - 3*x^12 - 3*x^11 - 2*x^10 - x^9 - 2*x^8 - x^6;
q = x^54 - x^53 - 2*x^48 + x^47 + x^46 - x^45 + x^43 + x^42 + x^41 - 2*x^40 + x^39 + 2*x^38 - 2*x^37 - x^36 + x^33 - 3*x^32 + 3*x^30 - 2*x^29 + 2*x^25 - 3*x^24 + 3*x^22 - x^21 + x^18 + 2*x^17 - 2*x^16 - x^15 + 2*x^14 - x^13 - x^12 - x^11 + x^9 - x^8 - x^7 + 2*x^6 + x - 1;
(p/q).list()[:30]

G.f.: p/q with p,q given in Sage program.

a(n) = A027634(n) - A029143(n).

A165686 Dimension of the space of Siegel cusp forms of genus 2 and weight 2k which are not Saito-Kurokawa lifts of forms of genus 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 4, 5, 6, 8, 8, 11, 12, 14, 16, 19, 20, 24, 26, 29, 32, 37, 38, 44, 47, 51, 56, 62, 64, 72, 76, 82, 88, 96, 99, 109, 115, 122, 130, 140, 144, 157, 164, 173, 183, 195, 201, 216, 225, 236, 248, 263, 270, 288, 299, 312, 327, 344, 353, 374

Offset: 1

Kilian Kilger (kilian(AT)nihilnovi.de), Sep 26 2009

Also the dimension of the largest Hecke-closed subspace of forms in S_k(Gamma_2) which satisfy the Ramanujan-Petersson conjecture. These forms are also characterized by the property that their (Andrianov) spinor zeta function does not have any pole.

			a(20)=1 because there is exactly one Siegel modular form of genus 2 and weight 20 which is not a lift of some form of genus 1.

M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhaeusser, 1985.
T. Oda, On the poles of Andrianov L-functions, Math. Ann. 256(3), p. 323-340, 1981.
R. Weissauer, The Ramanujan conjecture for genus two Siegel modular forms (an application of the trace formula). Preprint, Mannheim (1993)

Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,0,1,-1,-2,-1,1,0,0,1,1,0,-1).

Cf. A165684 for the full space of Siegel cusp forms. See also A029143, A027640, A165685.

For k > 1 we have a(k) = A165684(k) - A008615(2k-5).

Conjectured G.f.: -x^10*(x^7+x^6-x^2-x-1) / ((1-x^2)*(1-x^3)*(1-x^5)*(1-x^6)). - Colin Barker, Mar 30 2013

cp's OEIS Frontend

A165684 Dimension of the space of Siegel cusp forms of genus 2 and dimension 2n (associated with full modular group Gamma_2).

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

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A352095 Dimension of the space of Siegel cusp forms of genus 3 and weight 2n.

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A165686 Dimension of the space of Siegel cusp forms of genus 2 and weight 2k which are not Saito-Kurokawa lifts of forms of genus 1.

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