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Search: a047628 -id:a047628
Displaying 1-7 of 7 results found.
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A362875
Theta series of 15-dimensional lattice Kappa_15.
+10
7
1, 0, 1746, 21456, 147150, 607536, 2036334, 5410800, 13282866, 27563184, 56679732, 102040272, 184563384, 302221728, 504866340, 763016400, 1202127174, 1728479808, 2575653198, 3561176016, 5127122304, 6797385072, 9531403128, 12329627616, 16701654486, 21199654080
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Theta series is an element of the space of modular forms on Gamma_1(48) with Kronecker character 12 in modulus 48, weight 15/2, and dimension 58 over the integers.
REFERENCES
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.
LINKS
Andy Huchala, Table of n, a(n) for n = 0..20000
G. Nebe and N. J. A. Sloane, Home page for this lattice.
EXAMPLE
G.f. = 1 + 1746*q^4 + 21456*q^6 + 147150*q^8 + ...
PROG
(Magma)
prec := 70;
S := SymmetricMatrix([4, 2, 4, 0, -2, 4, 0, -2, 0, 4, 0, 0, -2, 0, 4, -2, -2, 0, 0, 0, 4, -2, -1, 1, 0, 0, 0, 4, -2, -1, 0, -1, 1, 2, 2, 4, -2, -2, 0, 1, 1, 2, 2, 2, 4, -2, 0, -2, 0, 1, 1, 0, 0, 0, 4, 1, 1, 0, 0, 0, -2, 0, -1, -1, -2, 4, -2, -1, 0, 0, 0, 1, 1, 1, 1, 1, -2, 4, 0, -1, 1, 1, 0, -1, 1, 0, 0, -1, 1, -1, 4, 0, 0, 0, 0, 0, 0, 1, 0, 1, -1, 1, -1, 1, 4, 0, 0, 0, 0, -1, 1, -1, 0, 0, 0, -1, 0, 0, -1, 4]);
ls := [1, 0, 1746, 21456, 147150, 607536, 2036334, 5410800, 13282866, 27563184, 56679732, 102040272, 184563384, 302221728, 504866340, 763016400, 1202127174, 1728479808, 2575653198, 3561176016, 5127122304, 6797385072, 9531403128, 12329627616, 16701654486, 21199654080, 28230179220, 34817427648, 45678519396, 55628679312, 71267532432, 85814825328, 108809427618, 128313065808, 161435864196, 188866349856, 233000967122, 271038881664, 332652360024, 380052936000, 464058384948, 528207272064, 634933480440, 719891109360, 862226645076, 963402396336, 1151630548200, 1283383148256, 1511712192624, 1682610190272, 1980149372586, 2173335020640, 2553938906832, 2802302452080, 3252053197962, 3565107859680, 4134281599332, 4478370612624];
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
Coefficients(&+[ls[i] * B[i] : i in [1..58]]);
CROSSREFS
Cf. A029897, A047628, A362876, A362877, A362878, A362879, A362880.
KEYWORD
nonn
AUTHOR
Andy Huchala, May 07 2023
STATUS
approved
A362876
Theta series of 16-dimensional lattice Kappa_16.
+10
7
1, 0, 2772, 42624, 335052, 1545984, 5698860, 16297344, 42785244, 94440960, 204094296, 385391232, 730053060, 1240934400, 2151268128, 3374469504, 5476016700, 8115545088, 12477938100, 17677480320, 26111897640, 35570481408, 50909418000, 67336722432, 93433877268
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Theta series is an element of the space of modular forms on Gamma_0(12) of weight 8 and dimension 17 over the integers.
REFERENCES
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.
LINKS
Andy Huchala, Table of n, a(n) for n = 0..20000
G. Nebe and N. J. A. Sloane, Home page for this lattice.
EXAMPLE
G.f. = 1 + 2772*q^4 + 42624*q^6 + ...
PROG
(Magma)
prec := 40;
S := SymmetricMatrix([4, 2, 4, 0, -2, 4, 0, -2, 0, 4, 0, 0, -2, 0, 4, -2, -2, 0, 0, 0, 4, -2, -1, 1, 0, 0, 0, 4, -2, -1, 0, -1, 1, 2, 2, 4, -2, -2, 0, 1, 1, 2, 2, 2, 4, -2, 0, -2, 0, 1, 1, 0, 0, 0, 4, 1, 1, 0, 0, 0, -2, 0, -1, -1, -2, 4, -2, -1, 0, 0, 0, 1, 1, 1, 1, 1, -2, 4, 0, -1, 1, 1, 0, -1, 1, 0, 0, -1, 1, -1, 4, 0, 0, 0, 0, 0, 0, 1, 0, 1, -1, 1, -1, 1, 4, 0, 0, 0, 0, -1, 1, -1, 0, 0, 0, -1, 0, 0, -1, 4, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, -1, 4]);
L := LatticeWithGram(S);
T<q> := ThetaSeries(L, 32);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
Coefficients(&+[Coefficients(T)[2*i-1]*B[i] : i in [1..17]]);
CROSSREFS
Cf. A029897, A047628, A362875, A362877, A362878, A362879, A362880.
KEYWORD
nonn
AUTHOR
Andy Huchala, May 07 2023
STATUS
approved
A362878
Theta series of 18-dimensional lattice Kappa_18.
+10
7
1, 0, 6480, 157680, 1596510, 9488016, 40681440, 140492880, 406046520, 1047312720, 2426695200, 5208293520, 10421250750, 19873356480, 35716191840, 62355291696, 104234541390, 169488573120, 267064691760, 413777075760, 619573504896, 920235334320, 1331744781600
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Theta series is an element of the space of modular forms on Gamma_1(3) with Kronecker character -3, weight 9, and dimension 4 over the integers.
REFERENCES
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.
LINKS
Andy Huchala, Table of n, a(n) for n = 0..20000
G. Nebe and N. J. A. Sloane, Home page for this lattice.
EXAMPLE
G.f. = 1 + 6480*q^4 + 157680*q^6 + ...
PROG
(Magma)
prec := 20;
ls := [4, 2, 4, 0, -2, 4, 0, -2, 0, 4, 0, 0, -2, 0, 4, -2, -2, 0, 0, 0, 4, -2, -1, 1, 0, 0, 0, 4, -2, -1, 0, -1, 1, 2, 2, 4, -2, -2, 0, 1, 1, 2, 2, 2, 4, -2, 0, -2, 0, 1, 1, 0, 0, 0, 4, 1, 1, 0, 0, 0, -2, 0, -1, -1, -2, 4, -2, -1, 0, 0, 0, 1, 1, 1, 1, 1, -2, 4, 0, -1, 1, 1, 0, -1, 1, 0, 0, -1, 1, -1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 4, 0, -1, 0, 0, 1, 1, 0, 1, 1, -1, 0, 0, 1, -1, 4, 0, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 0, 1, 4];
S := SymmetricMatrix(ls);
L := LatticeWithGram(S);
T<q> := ThetaSeries(L, 8);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
Coefficients(&+[Coefficients(T)[2*i-1]*B[i] :i in [1..4]]);
CROSSREFS
Cf. A029897, A047628, A362875, A362876, A362877, A362879, A362880.
KEYWORD
nonn
AUTHOR
Andy Huchala, May 08 2023
STATUS
approved
A362879
Theta series of 19-dimensional lattice Kappa_19.
+10
7
1, 0, 9396, 284528, 3309660, 21996036, 103632480, 384538752, 1195104618, 3253783500, 7971340896, 17905302720, 37530681590, 74139276672, 139067432280, 250102136592, 433070833500, 724358442744, 1178016364548, 1866143480400, 2883345017508, 4367172766500
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Theta series is an element of the space of modular forms on Gamma_0(12) of weight 19/2 and dimension 19 over the integers.
REFERENCES
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.
LINKS
Andy Huchala, Table of n, a(n) for n = 0..20000
G. Nebe and N. J. A. Sloane, Home page for this lattice.
EXAMPLE
G.f. = 1 + 9396*q^4 + 284528*q^6 + ...
PROG
(Magma)
prec := 30;
coeffs := [1, 0, 9396, 284528, 3309660, 21996036, 103632480, 384538752, 1195104618, 3253783500, 7971340896, 17905302720, 37530681590, 74139276672, 139067432280, 250102136592, 433070833500, 724358442744, 1178016364548];
ls := [4, 2, 4, 0, -2, 4, 0, -2, 0, 4, 0, 0, -2, 0, 4, -2, -2, 0, 0, 0, 4, -2, -1, 1, 0, 0, 0, 4, -2, -1, 0, -1, 1, 2, 2, 4, -2, -2, 0, 1, 1, 2, 2, 2, 4, -2, 0, -2, 0, 1, 1, 0, 0, 0, 4, 1, 1, 0, 0, 0, -2, 0, -1, -1, -2, 4, -2, -1, 0, 0, 0, 1, 1, 1, 1, 1, -2, 4, 0, -1, 1, 1, 0, -1, 1, 0, 0, -1, 1, -1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 4, 0, -1, 0, 0, 1, 1, 0, 1, 1, -1, 0, 0, 1, -1, 4, 0, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 0, 1, 4, 1, 0, -1, 1, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, 4];
S := SymmetricMatrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
Coefficients(&+[coeffs[i]*B[i] :i in [1..19]]);
CROSSREFS
Cf. A029897, A047628, A362875, A362876, A362877, A362878, A362880.
KEYWORD
nonn
AUTHOR
Andy Huchala, May 08 2023
STATUS
approved
A362880
Theta series of 20-dimensional lattice Kappa_20.
+10
7
1, 0, 15390, 575160, 7712820, 57281580, 296150580, 1184012640, 3944197800, 11364334080, 29395745478, 69157229760, 151652810580, 311116423500, 607158951120, 1127694969072, 2020055770530, 3478103852940, 5829999042420, 9467119804680, 15046034533560
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Theta series is an element of the space of modular forms on Gamma_0(9) of weight 10 and dimension 11 over the integers.
REFERENCES
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.
LINKS
Andy Huchala, Table of n, a(n) for n = 0..20000
G. Nebe and N. J. A. Sloane, Home page for this lattice.
EXAMPLE
G.f. = 1 + 15390*q^4 + 575160*q^6 + ...
PROG
(Magma)
prec := 40;
ls := [4, 2, 4, 0, -2, 4, 0, -2, 0, 4, 0, 0, -2, 0, 4, -2, -2, 0, 0, 0, 4, -2, -1, 1, 0, 0, 0, 4, -2, -1, 0, -1, 1, 2, 2, 4, -2, -2, 0, 1, 1, 2, 2, 2, 4, -2, 0, -2, 0, 1, 1, 0, 0, 0, 4, 1, 1, 0, 0, 0, -2, 0, -1, -1, -2, 4, -2, -1, 0, 0, 0, 1, 1, 1, 1, 1, -2, 4, 0, -1, 1, 1, 0, -1, 1, 0, 0, -1, 1, -1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 4, 0, -1, 0, 0, 1, 1, 0, 1, 1, -1, 0, 0, 1, -1, 4, 0, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 0, 1, 4, 1, 0, -1, 1, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 1, 0, 1, 0, 4];
S := SymmetricMatrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
coeffs := [1, 0, 15390, 575160, 7712820, 57281580, 296150580, 1184012640, 3944197800, 11364334080, 29395745478];
Coefficients(&+[coeffs[i]*B[i] :i in [1..11]]);
CROSSREFS
Cf. A029897, A047628, A362875, A362876, A362877, A362878, A362879.
KEYWORD
nonn
AUTHOR
Andy Huchala, May 08 2023
STATUS
approved
A362877
Theta series of 17-dimensional lattice Kappa_17.
+10
6
1, 0, 4266, 81792, 737862, 3809280, 15406210, 47505792, 133390290, 312588288, 711232812, 1408787328, 2789963820, 4931371008, 8870944884, 14417119872, 24144502662, 36878456832, 58393537998, 84926534016
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Theta series is an element of the space of modular forms on Gamma_1(48) with Kronecker character 12 in modulus 48, weight 17/2, and dimension 66 over the integers.
REFERENCES
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.
LINKS
Table of n, a(n) for n=0..19.
G. Nebe and N. J. A. Sloane, Home page for this lattice.
EXAMPLE
G.f. = 1 + 4266*q^4 + 81792*q^6 + ...
PROG
(Magma)
prec := 10;
S := SymmetricMatrix([4, 2, 4, 0, -2, 4, 0, -2, 0, 4, 0, 0, -2, 0, 4, -2, -2, 0, 0, 0, 4, -2, -1, 1, 0, 0, 0, 4, -2, -1, 0, -1, 1, 2, 2, 4, -2, -2, 0, 1, 1, 2, 2, 2, 4, -2, 0, -2, 0, 1, 1, 0, 0, 0, 4, 1, 1, 0, 0, 0, -2, 0, -1, -1, -2, 4, -2, -1, 0, 0, 0, 1, 1, 1, 1, 1, -2, 4, 0, -1, 1, 1, 0, -1, 1, 0, 0, -1, 1, -1, 4, 0, 0, 0, 0, 0, 0, 1, 0, 1, -1, 1, -1, 1, 4, 0, 0, 0, 0, -1, 1, -1, 0, 0, 0, -1, 0, 0, -1, 4, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, -1, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 1, -1, 1, 0, 1, -1, 4]);
L := LatticeWithGram(S);
T<q> := ThetaSeries(L, 2*prec);
[Coefficients(T)[2*i-1] : i in [1..prec]];
CROSSREFS
Cf. A029897, A047628, A362875, A362876, A362878, A362879, A362880.
KEYWORD
nonn
AUTHOR
Andy Huchala, May 07 2023
STATUS
approved
A015235
Theta series of lattice Kappa_8.
+10
1
1, 0, 132, 192, 828, 1152, 2796, 2880, 6828, 5376, 14904, 10944, 20772, 18432, 40224, 25920, 53964, 41472, 76452, 58176, 107784, 69504, 156816, 101376, 163284, 131328, 259032, 147072, 295200, 206208, 357480, 250560, 432780, 269568, 576072, 365184, 555804, 426240
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 161.
LINKS
Andy Huchala, Table of n, a(n) for n = 0..20000
G. Nebe and N. J. A. Sloane, Home page for this lattice
EXAMPLE
G.f. = 1 + 132*q^4 + 192*q^6 + ...
PROG
(Sage)
L = [1, 0, 132, 192, 828, 1152, 2796, 2880, 6828, 5376]
M = ModularForms(Gamma0(12), 4)
bases = [_.q_expansion(35) for _ in M.integral_basis()]
f = sum(x*y for (x, y) in zip(bases, L)); list(f) # Andy Huchala, Jul 23 2021
CROSSREFS
Cf. A015236 (K_7), A015233 (K_9), A015232 (K_10), A015229 (K_11), A004010 (K_12), A029897 (K_13), A047628 (K_14).
KEYWORD
nonn
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Sean A. Irvine, Feb 26 2020
STATUS
approved
page 1

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Last modified November 7 13:46 EST 2024. Contains 377535 sequences.
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