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Properties

Label 81.6
Level 81
Weight 6
Dimension 932
Nonzero newspaces 4
Newform subspaces 16
Sturm bound 2916
Trace bound 1

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Defining parameters

Level: \( N \) = \( 81 = 3^{4} \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 16 \)
Sturm bound: \(2916\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(81))\).

Total New Old
Modular forms 1269 988 281
Cusp forms 1161 932 229
Eisenstein series 108 56 52

Trace form

\( 932 q - 12 q^{2} - 18 q^{3} - 52 q^{4} + 75 q^{5} - 18 q^{6} - 107 q^{7} - 591 q^{8} - 18 q^{9} + O(q^{10}) \) \( 932 q - 12 q^{2} - 18 q^{3} - 52 q^{4} + 75 q^{5} - 18 q^{6} - 107 q^{7} - 591 q^{8} - 18 q^{9} - 69 q^{10} - 129 q^{11} - 18 q^{12} + 1717 q^{13} + 1623 q^{14} - 18 q^{15} - 3328 q^{16} - 3483 q^{17} + 8658 q^{18} - 1295 q^{19} - 26709 q^{20} - 13761 q^{21} - 228 q^{22} + 7401 q^{23} + 16398 q^{24} + 8900 q^{25} + 44301 q^{26} + 17505 q^{27} + 31303 q^{28} + 24153 q^{29} - 15354 q^{30} - 12287 q^{31} - 112554 q^{32} - 36765 q^{33} - 51318 q^{34} - 55299 q^{35} + 17838 q^{36} + 23995 q^{37} + 95934 q^{38} - 18 q^{39} + 44061 q^{40} + 163749 q^{41} + 52137 q^{42} - 15011 q^{43} - 268815 q^{44} - 136044 q^{45} - 181401 q^{46} - 188409 q^{47} - 66447 q^{48} - 14586 q^{49} + 249522 q^{50} + 126297 q^{51} + 250807 q^{52} + 358083 q^{53} + 347112 q^{54} + 210855 q^{55} + 362337 q^{56} + 41508 q^{57} - 114159 q^{58} - 216987 q^{59} - 329841 q^{60} - 113039 q^{61} - 707523 q^{62} - 266292 q^{63} - 324967 q^{64} - 543729 q^{65} - 324666 q^{66} + 16873 q^{67} + 974808 q^{68} + 717912 q^{69} - 233481 q^{70} + 127323 q^{71} + 981486 q^{72} - 81926 q^{73} - 58029 q^{74} - 56268 q^{75} + 213502 q^{76} - 220947 q^{77} - 741447 q^{78} + 438589 q^{79} - 688722 q^{80} - 454986 q^{81} + 200838 q^{82} + 29895 q^{83} - 824391 q^{84} + 188397 q^{85} - 30876 q^{86} + 405918 q^{87} - 332196 q^{88} + 1200708 q^{89} + 2137815 q^{90} + 44702 q^{91} + 727437 q^{92} + 30024 q^{93} - 1180083 q^{94} - 1983621 q^{95} - 1414989 q^{96} - 546425 q^{97} - 3162735 q^{98} - 1736586 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(81))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
81.6.a \(\chi_{81}(1, \cdot)\) 81.6.a.a 2 1
81.6.a.b 2
81.6.a.c 4
81.6.a.d 4
81.6.a.e 6
81.6.c \(\chi_{81}(28, \cdot)\) 81.6.c.a 2 2
81.6.c.b 2
81.6.c.c 2
81.6.c.d 4
81.6.c.e 4
81.6.c.f 4
81.6.c.g 4
81.6.c.h 4
81.6.c.i 12
81.6.e \(\chi_{81}(10, \cdot)\) 81.6.e.a 84 6
81.6.g \(\chi_{81}(4, \cdot)\) 81.6.g.a 792 18

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(81))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(81)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 2}\)