Properties

Label 81.9
Level 81
Weight 9
Dimension 1508
Nonzero newspaces 4
Newform subspaces 11
Sturm bound 4374
Trace bound 1

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Defining parameters

Level: \( N \) = \( 81 = 3^{4} \)
Weight: \( k \) = \( 9 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 11 \)
Sturm bound: \(4374\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(\Gamma_1(81))\).

Total New Old
Modular forms 1998 1564 434
Cusp forms 1890 1508 382
Eisenstein series 108 56 52

Trace form

\( 1508 q - 12 q^{2} - 18 q^{3} + 236 q^{4} + 429 q^{5} - 18 q^{6} - 2789 q^{7} - 9 q^{8} - 18 q^{9} + O(q^{10}) \) \( 1508 q - 12 q^{2} - 18 q^{3} + 236 q^{4} + 429 q^{5} - 18 q^{6} - 2789 q^{7} - 9 q^{8} - 18 q^{9} + 11235 q^{10} - 28686 q^{11} - 18 q^{12} + 52045 q^{13} + 120957 q^{14} - 18 q^{15} - 234376 q^{16} - 9 q^{17} - 274194 q^{18} + 732367 q^{19} - 815307 q^{20} - 1360953 q^{21} - 1124256 q^{22} + 863673 q^{23} + 3725550 q^{24} + 2283272 q^{25} - 27 q^{26} - 1933299 q^{27} - 4908329 q^{28} - 1909749 q^{29} - 1292562 q^{30} + 1720033 q^{31} + 14472126 q^{32} + 6069339 q^{33} - 3557322 q^{34} - 12450843 q^{35} - 18224658 q^{36} + 4384945 q^{37} + 36466050 q^{38} - 18 q^{39} - 7270149 q^{40} - 23149344 q^{41} + 39542067 q^{42} - 89648 q^{43} - 34682265 q^{44} - 37896786 q^{45} - 31245801 q^{46} - 8464917 q^{47} + 15217767 q^{48} + 23682702 q^{49} + 145747380 q^{50} + 54355878 q^{51} + 14270485 q^{52} - 27 q^{53} - 75728412 q^{54} - 62139633 q^{55} - 221070243 q^{56} - 65747826 q^{57} - 98272785 q^{58} - 68739708 q^{59} + 125543673 q^{60} + 99186013 q^{61} + 447419205 q^{62} + 101958462 q^{63} + 73639193 q^{64} - 65778843 q^{65} - 552090186 q^{66} - 61291730 q^{67} - 292730634 q^{68} + 273910662 q^{69} - 188308695 q^{70} + 125718795 q^{71} + 548536302 q^{72} + 33257476 q^{73} - 87410787 q^{74} - 246093768 q^{75} + 369351490 q^{76} - 230951739 q^{77} - 478793403 q^{78} + 78357691 q^{79} + 148365990 q^{81} - 136309578 q^{82} - 14587329 q^{83} + 1377777303 q^{84} - 213183261 q^{85} + 363808068 q^{86} + 61361982 q^{87} + 193071684 q^{88} - 1029546702 q^{89} - 624764277 q^{90} + 709272512 q^{91} + 597873567 q^{92} - 53952426 q^{93} - 287856975 q^{94} - 1195036359 q^{95} + 626058639 q^{96} - 915507962 q^{97} - 1883568663 q^{98} - 967332078 q^{99} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(\Gamma_1(81))\)

We only show spaces with odd 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
81.9.b \(\chi_{81}(80, \cdot)\) 81.9.b.a 14 1
81.9.b.b 16
81.9.d \(\chi_{81}(26, \cdot)\) 81.9.d.a 2 2
81.9.d.b 4
81.9.d.c 4
81.9.d.d 4
81.9.d.e 4
81.9.d.f 12
81.9.d.g 32
81.9.f \(\chi_{81}(8, \cdot)\) 81.9.f.a 138 6
81.9.h \(\chi_{81}(2, \cdot)\) 81.9.h.a 1278 18

Decomposition of \(S_{9}^{\mathrm{old}}(\Gamma_1(81))\) into lower level spaces

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