Properties

Label 9.8
Level 9
Weight 8
Dimension 15
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 48
Trace bound 1

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Defining parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(48\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(9))\).

Total New Old
Modular forms 25 20 5
Cusp forms 17 15 2
Eisenstein series 8 5 3

Trace form

\( 15 q - 15 q^{2} + 24 q^{3} + 51 q^{4} - 570 q^{5} - 1233 q^{6} + 372 q^{7} + 7242 q^{8} + 990 q^{9} + O(q^{10}) \) \( 15 q - 15 q^{2} + 24 q^{3} + 51 q^{4} - 570 q^{5} - 1233 q^{6} + 372 q^{7} + 7242 q^{8} + 990 q^{9} - 8928 q^{10} - 7512 q^{11} + 8052 q^{12} + 6834 q^{13} - 15888 q^{14} - 1188 q^{15} + 6927 q^{16} + 2178 q^{17} + 42876 q^{18} + 82164 q^{19} - 4908 q^{20} - 187224 q^{21} - 271089 q^{22} - 36300 q^{23} + 215469 q^{24} + 106791 q^{25} + 567060 q^{26} + 322272 q^{27} + 202044 q^{28} - 451158 q^{29} - 1112112 q^{30} - 265596 q^{31} - 1241073 q^{32} - 148518 q^{33} + 962253 q^{34} + 2235576 q^{35} + 2501811 q^{36} - 353670 q^{37} - 1900965 q^{38} - 2057316 q^{39} - 1407384 q^{40} - 1101864 q^{41} + 538866 q^{42} + 1109424 q^{43} + 5081898 q^{44} + 2687580 q^{45} - 577656 q^{46} - 2214780 q^{47} - 7434525 q^{48} - 2510343 q^{49} - 2098311 q^{50} + 2525688 q^{51} + 4347570 q^{52} + 3263706 q^{53} + 5816529 q^{54} + 3300624 q^{55} - 1868946 q^{56} - 3071850 q^{57} - 5970924 q^{58} - 3362016 q^{59} + 371484 q^{60} - 4400250 q^{61} + 634020 q^{62} - 2238804 q^{63} + 1845762 q^{64} + 600600 q^{65} + 5754762 q^{66} + 11014032 q^{67} + 4754151 q^{68} - 3002292 q^{69} - 2191518 q^{70} - 3152232 q^{71} - 9638325 q^{72} - 13310430 q^{73} + 8235348 q^{74} + 19174632 q^{75} + 9458877 q^{76} + 3418152 q^{77} - 13339962 q^{78} + 4506564 q^{79} - 39576288 q^{80} - 28538730 q^{81} + 8965494 q^{82} + 13981800 q^{83} + 30090090 q^{84} - 5005044 q^{85} + 30163929 q^{86} + 10290708 q^{87} - 39169269 q^{88} - 16101954 q^{89} - 13660596 q^{90} + 17479176 q^{91} + 37736898 q^{92} + 27331212 q^{93} + 37575864 q^{94} + 13783872 q^{95} - 3404376 q^{96} - 24764568 q^{97} - 90916632 q^{98} - 49382676 q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(9))\)

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 the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
9.8.a \(\chi_{9}(1, \cdot)\) 9.8.a.a 1 1
9.8.a.b 2
9.8.c \(\chi_{9}(4, \cdot)\) 9.8.c.a 12 2

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(9))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(9)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)