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Properties

Label 81.8
Level 81
Weight 8
Dimension 1316
Nonzero newspaces 4
Newform subspaces 18
Sturm bound 3888
Trace bound 1

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Defining parameters

Level: \( N \) = \( 81 = 3^{4} \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 18 \)
Sturm bound: \(3888\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(81))\).

Total New Old
Modular forms 1755 1372 383
Cusp forms 1647 1316 331
Eisenstein series 108 56 52

Trace form

\( 1316 q - 12 q^{2} - 18 q^{3} - 148 q^{4} + 201 q^{5} - 18 q^{6} + 229 q^{7} + 4593 q^{8} - 18 q^{9} + O(q^{10}) \) \( 1316 q - 12 q^{2} - 18 q^{3} - 148 q^{4} + 201 q^{5} - 18 q^{6} + 229 q^{7} + 4593 q^{8} - 18 q^{9} - 4269 q^{10} - 9417 q^{11} - 18 q^{12} - 3377 q^{13} - 16665 q^{14} - 18 q^{15} + 16160 q^{16} + 58941 q^{17} + 108774 q^{18} - 136175 q^{19} - 403845 q^{20} + 185985 q^{21} + 140592 q^{22} + 355017 q^{23} - 100242 q^{24} - 177238 q^{25} - 1392315 q^{26} - 416979 q^{27} - 168665 q^{28} + 162339 q^{29} + 1355166 q^{30} + 372697 q^{31} + 2136006 q^{32} + 154233 q^{33} + 239670 q^{34} - 1284339 q^{35} - 2486034 q^{36} + 295267 q^{37} + 2295570 q^{38} - 18 q^{39} - 1302891 q^{40} + 1907103 q^{41} - 4341303 q^{42} - 1842899 q^{43} - 2846127 q^{44} + 1459008 q^{45} + 8979063 q^{46} + 11416839 q^{47} + 9962649 q^{48} - 245196 q^{49} - 3438534 q^{50} - 4677327 q^{51} - 8174321 q^{52} - 13671117 q^{53} - 17669880 q^{54} - 8034825 q^{55} - 8809887 q^{56} + 3123720 q^{57} + 10004385 q^{58} + 25818213 q^{59} + 28050615 q^{60} + 11662843 q^{61} + 22534941 q^{62} - 1006956 q^{63} - 12361543 q^{64} - 50738727 q^{65} - 11249946 q^{66} - 20656847 q^{67} + 33243264 q^{68} + 31645980 q^{69} + 14376999 q^{70} - 2625813 q^{71} - 87478290 q^{72} + 740524 q^{73} - 13362765 q^{74} - 18281268 q^{75} + 13653838 q^{76} + 15550743 q^{77} + 48218193 q^{78} - 42151955 q^{79} + 27943374 q^{80} + 35458470 q^{81} - 20001522 q^{82} + 13604307 q^{83} + 4745169 q^{84} + 33886521 q^{85} + 5931552 q^{86} - 92159586 q^{87} + 69730620 q^{88} + 40335318 q^{89} - 99084321 q^{90} - 49367662 q^{91} - 182315715 q^{92} - 51873804 q^{93} - 75451731 q^{94} + 71483307 q^{95} + 241882443 q^{96} + 18083317 q^{97} + 245015973 q^{98} + 10370646 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(81))\)

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
81.8.a \(\chi_{81}(1, \cdot)\) 81.8.a.a 4 1
81.8.a.b 4
81.8.a.c 6
81.8.a.d 6
81.8.a.e 6
81.8.c \(\chi_{81}(28, \cdot)\) 81.8.c.a 2 2
81.8.c.b 2
81.8.c.c 2
81.8.c.d 4
81.8.c.e 4
81.8.c.f 4
81.8.c.g 4
81.8.c.h 4
81.8.c.i 8
81.8.c.j 8
81.8.c.k 12
81.8.e \(\chi_{81}(10, \cdot)\) 81.8.e.a 120 6
81.8.g \(\chi_{81}(4, \cdot)\) 81.8.g.a 1116 18

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(81))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(81)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(81))\)\(^{\oplus 1}\)