Properties

Label 9.9
Level 9
Weight 9
Dimension 16
Nonzero newspaces 2
Newform subspaces 2
Sturm bound 54
Trace bound 1

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Defining parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 9 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 2 \)
Sturm bound: \(54\)
Trace bound: \(1\)

Dimensions

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

Total New Old
Modular forms 28 20 8
Cusp forms 20 16 4
Eisenstein series 8 4 4

Trace form

\( 16 q - 3 q^{2} - 93 q^{3} + 1243 q^{4} + 438 q^{5} - 2259 q^{6} + 4226 q^{7} + 17973 q^{9} + O(q^{10}) \) \( 16 q - 3 q^{2} - 93 q^{3} + 1243 q^{4} + 438 q^{5} - 2259 q^{6} + 4226 q^{7} + 17973 q^{9} - 8904 q^{10} - 28677 q^{11} - 55110 q^{12} - 50860 q^{13} + 120966 q^{14} - 75276 q^{15} + 38791 q^{16} + 243324 q^{18} - 176350 q^{19} + 539454 q^{20} + 354054 q^{21} + 242607 q^{22} - 1000452 q^{23} - 1125513 q^{24} - 1107977 q^{25} - 524826 q^{27} + 1862060 q^{28} + 3797682 q^{29} + 2372562 q^{30} - 557020 q^{31} - 8461881 q^{32} - 5761521 q^{33} + 526149 q^{34} + 12958047 q^{36} + 3967160 q^{37} + 10967691 q^{38} + 3383976 q^{39} - 3530346 q^{40} - 10239447 q^{41} - 35844228 q^{42} - 3635113 q^{43} + 27631638 q^{45} + 3973488 q^{46} + 31148628 q^{47} + 17088189 q^{48} - 10898031 q^{49} - 63849453 q^{50} - 47946573 q^{51} - 18058192 q^{52} + 36967995 q^{54} + 50349444 q^{55} + 116638674 q^{56} + 37276797 q^{57} + 19430130 q^{58} - 83493795 q^{59} - 109012050 q^{60} - 40066204 q^{61} + 38322432 q^{63} - 6597398 q^{64} + 69232992 q^{65} + 36133866 q^{66} - 36934183 q^{67} - 77746743 q^{68} - 62108640 q^{69} + 13956780 q^{70} + 88821171 q^{72} - 54535666 q^{73} - 10383450 q^{74} + 13792011 q^{75} - 20622319 q^{76} + 56158710 q^{77} + 27355140 q^{78} + 32745866 q^{79} - 16336971 q^{81} + 242093670 q^{82} - 198915996 q^{83} - 187060614 q^{84} + 67465494 q^{85} + 146190669 q^{86} + 114294006 q^{87} + 114516051 q^{88} - 222035850 q^{90} - 288317192 q^{91} - 295365804 q^{92} + 94187784 q^{93} - 50554356 q^{94} + 386813838 q^{95} + 768275928 q^{96} + 78145673 q^{97} - 511060752 q^{99} + O(q^{100}) \)

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

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
9.9.b \(\chi_{9}(8, \cdot)\) 9.9.b.a 2 1
9.9.d \(\chi_{9}(2, \cdot)\) 9.9.d.a 14 2

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

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