cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A362875 Theta series of 15-dimensional lattice Kappa_15.

Original entry on oeis.org

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
Offset: 0

Views

Author

Andy Huchala, May 07 2023

Keywords

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.

Examples

			G.f. = 1 + 1746*q^4 + 21456*q^6 + 147150*q^8 + ...

References

  • J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.

Links

Crossrefs

Cf. A029897, A047628, A362876, A362877, A362878, A362879, A362880.

Programs

  • 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]]);

A362876 Theta series of 16-dimensional lattice Kappa_16.

Original entry on oeis.org

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
Offset: 0

Views

Author

Andy Huchala, May 07 2023

Keywords

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.

Examples

			G.f. = 1 + 2772*q^4 + 42624*q^6 + ...

References

  • J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.

Links

Crossrefs

Cf. A029897, A047628, A362875, A362877, A362878, A362879, A362880.

Programs

  • 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 := ThetaSeries(L, 32);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M,prec);
Coefficients(&+[Coefficients(T)[2*i-1]*B[i] : i in [1..17]]);

A362878 Theta series of 18-dimensional lattice Kappa_18.

Original entry on oeis.org

1, 0, 6480, 157680, 1596510, 9488016, 40681440, 140492880, 406046520, 1047312720, 2426695200, 5208293520, 10421250750, 19873356480, 35716191840, 62355291696, 104234541390, 169488573120, 267064691760, 413777075760, 619573504896, 920235334320, 1331744781600
Offset: 0

Views

Author

Andy Huchala, May 08 2023

Keywords

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.

Examples

			G.f. = 1 + 6480*q^4 + 157680*q^6 + ...

References

  • J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.

Links

Crossrefs

Cf. A029897, A047628, A362875, A362876, A362877, A362879, A362880.

Programs

  • 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 := ThetaSeries(L, 8);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
Coefficients(&+[Coefficients(T)[2*i-1]*B[i] :i in [1..4]]);

A362879 Theta series of 19-dimensional lattice Kappa_19.

Original entry on oeis.org

1, 0, 9396, 284528, 3309660, 21996036, 103632480, 384538752, 1195104618, 3253783500, 7971340896, 17905302720, 37530681590, 74139276672, 139067432280, 250102136592, 433070833500, 724358442744, 1178016364548, 1866143480400, 2883345017508, 4367172766500
Offset: 0

Views

Author

Andy Huchala, May 08 2023

Keywords

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.

Examples

			G.f. = 1 + 9396*q^4 + 284528*q^6 + ...

References

  • J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.

Links

Crossrefs

Cf. A029897, A047628, A362875, A362876, A362877, A362878, A362880.

Programs

  • 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]]);

A362880 Theta series of 20-dimensional lattice Kappa_20.

Original entry on oeis.org

1, 0, 15390, 575160, 7712820, 57281580, 296150580, 1184012640, 3944197800, 11364334080, 29395745478, 69157229760, 151652810580, 311116423500, 607158951120, 1127694969072, 2020055770530, 3478103852940, 5829999042420, 9467119804680, 15046034533560
Offset: 0

Views

Author

Andy Huchala, May 08 2023

Keywords

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.

Examples

			G.f. = 1 + 15390*q^4 + 575160*q^6 + ...

References

  • J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.

Links

Crossrefs

Cf. A029897, A047628, A362875, A362876, A362877, A362878, A362879.

Programs

  • 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]]);

A362877 Theta series of 17-dimensional lattice Kappa_17.

Original entry on oeis.org

1, 0, 4266, 81792, 737862, 3809280, 15406210, 47505792, 133390290, 312588288, 711232812, 1408787328, 2789963820, 4931371008, 8870944884, 14417119872, 24144502662, 36878456832, 58393537998, 84926534016
Offset: 0

Views

Author

Andy Huchala, May 07 2023

Keywords

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.

Examples

			G.f. = 1 + 4266*q^4 + 81792*q^6 + ...

References

  • J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chap. 6.

Links

Crossrefs

Cf. A029897, A047628, A362875, A362876, A362878, A362879, A362880.

Programs

  • 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 := ThetaSeries(L, 2*prec);
[Coefficients(T)[2*i-1] : i in [1..prec]];

A015235 Theta series of lattice Kappa_8.

Original entry on oeis.org

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
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			G.f. = 1 + 132*q^4 + 192*q^6 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 161.

Links

Crossrefs

Cf. A015236 (K_7), A015233 (K_9), A015232 (K_10), A015229 (K_11), A004010 (K_12), A029897 (K_13), A047628 (K_14).

Programs

  • 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

Extensions

More terms from Sean A. Irvine, Feb 26 2020
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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