Characteristic Masses of Lattices

By Gaëtan Chenevier

This website is a companion to the paper Characteristic Masses of Niemeier Lattices by the author.

  • Documented source code of our PARI/GP implementation of the algorithms of this paper: (tutorial).
  • 1. Dimension of spaces of level 1 orthogonal automorphic forms

    Nonzero dimensions for: O7, O8, O9, O15, O16, O17, O23, O24, O25.


    Masses: O7, O8, O9, O15, O16, O17, O23, O24, O25.


    2. Characteristic masses of even lattices of determinant 1 or 2

    Dimension 7 (Gram): E7.

    Dimension 8 (Gram): E8.

    Dimension 9 (Gram): A1_E8.

    Dimension 15 (Gram): E7_E8, A1_D14.

    Dimension 16 (Gram): 2E8, D16.

    Dimension 17 (Gram): A1_2E8, A1_D16, A17, D10_E7.

    Dimension 23 (Gram): A1_D22, A1_D14_E8, D16_E7, E7_2E8, A22, A1_D10_D12, A15_E7, A17_D6, A1_D8_2E7, D10_D6_E7, A13_D9, A1_A15_D7, A1_D6_2D8, A10_A12, A9_D7_E6, A1_A11_D5_E6, A11_A5_D7, A5_3E6, A7_A9_D6, A1_2A9_D4, A1_D4_3D6, A6_2A8, A5_A7_2D5, A1_A3_2A7_D5, A4_3A6, A3_3A5_D4, 3A1_4A5, 3A1_5D4, A2_5A4, A1_7A3, 11A2, 23A1.

    Dimension 24 (Niemeier lattices, Gram): D24, D16_E8, 3E8, A24, 2D12, A17_E7, D10_2E7, A15_D9, 3D8, 2A12, A11_D7_E6, 4E6, 2A9_D6, 4D6, 3A8, 2A7_2D5, 4A6, 4A5_D4, 6D4, 6A4, 8A3, 12A2, 24A1, Leech.

    Dimension 25 (Gram): A1, A2, 9A1, 12A1_A2, 25A1, 9A1_4A2, 15A1_A3, 9A2, 6A1_5A2_A3, A1_12A2, 3A1_4A2_3A3, 9A1_4A3, 5A1_6A2_A4, 21A1_D4, A2_6A3, 2A1_3A2_3A3_A4, 9A2_D4, A1_8A3, A1_4A3_2A4, 3A1_2A2_A3_3A4, 3A1_5A3_D4, 4A2_3A3_A5, 6A1_4A3_A5, 3A2_3A4_D4, A1_A2_2A3_2A4_A5, A1_6A4, 9A1_4D4, 3A3_A5_2D4, A1_A3_2A4_2A5, 3A1_2A3_2A5_D4, 4A2_3A5, A1_6A3_D5, A1_A2_2A3_2A4_A6, A2_4A5, A2_4A4_D5, 3A4_A6_D4, 2A2_A3_2A5_A6, A1_4A5_D4, A1_A3_2A5_D4_D5, 4A1_3A5_D5, A1_6D4, 2A1_A4_A5_2A6, A3_2A4_A5_A7, 2A1_2A3_A5_A7_D4, A2_A4_2A6_D5, 3A6_D4, A1_A4_A5_A6_A7, A1_4A6, A3_A5_3D5, 3A1_4D4_D6, A3_3A5_D6, A1_2A3_A7_2D5, A1_2A7_2D4, 3A5_A8, A2_A3_A5_A6_A8, A2_3A7, A4_A6_A8_D5, A1_2A7_2D5, 3A5_D4_E6, A5_A7_D5_D6, A1_A3_2A7_D6, A1_2A7_A8, A1_A5_A9_D4_D5, 2A4_A7_A9, 3A6_E6, A2_A5_A8_A9, A1_3A8, 3A1_D4_3D6, A1_A7_2D5_E6, A5_A7_D5_D7, A1_A3_2A7_D7, 2A1_A7_A9_D6, A1_A10_A6_A7, A2_2A8_E6, A10_A8_D5, A1_2A9_D6, A11_A3_A5_D6, A11_A5_A8, A1_4D6, A3_A9_D6_E6, A5_D7_2E6, A1_D4_2D6_D8, A9_2D7, A7_A9_D8, A1_A11_D5_D7, 2A11_A2, A12_A6_E6, A1_A11_D7_E6, A13_D5_D6, A1_4E6, A1_A10_A13, A1_2A12, 3D6_E7, 2A9_E7, A9_D9_E6, A11_A5_D9, A14_A2_A9, A11_E6_E7, A1_3D8, 2A1_2D8_E7, A1_D10_D6_D8, A1_A15_D8, A1_A15_D9, A15_A3_E7, A17_A8, D4_3E7, A13_D11, A18_E6, A1_A17_E7, A1_D10_2E7, D12_D6_E7, A1_2D12, D10_E7_E8, A1_D10_D14, A17_E8, A2_A23, A1_A24, A1_D16_E8, A1_3E8, D18_E7, A1_D24.