Properties

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

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Defining parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 2 \)
Newforms: \( 3 \)
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

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

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

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