Properties

Label 9.8.a
Level 9
Weight 8
Character orbit a
Rep. character \(\chi_{9}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 2
Sturm bound 8
Trace bound 1

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(8\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(9))\).

Total New Old
Modular forms 9 4 5
Cusp forms 5 3 2
Eisenstein series 4 1 3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)Dim.
\(+\)\(2\)
\(-\)\(1\)

Trace form

\( 3q - 6q^{2} + 372q^{4} - 390q^{5} + 456q^{7} + 1320q^{8} + O(q^{10}) \) \( 3q - 6q^{2} + 372q^{4} - 390q^{5} + 456q^{7} + 1320q^{8} - 9180q^{10} + 948q^{11} + 8682q^{13} + 384q^{14} + 19344q^{16} - 28386q^{17} + 57732q^{19} + 35880q^{20} - 236088q^{22} + 15288q^{23} + 102045q^{25} + 30588q^{26} + 126528q^{28} - 36510q^{29} - 273792q^{31} - 192096q^{32} + 1068876q^{34} + 24960q^{35} - 493014q^{37} + 51720q^{38} - 1712880q^{40} + 629718q^{41} + 701052q^{43} - 87216q^{44} + 1106352q^{46} - 583296q^{47} - 2331333q^{49} - 443850q^{50} + 3665976q^{52} + 428058q^{53} + 3316680q^{55} - 84480q^{56} - 5022540q^{58} - 1306380q^{59} - 1677054q^{61} + 1660848q^{62} - 5332416q^{64} + 1988220q^{65} + 7207476q^{67} + 2611512q^{68} - 3144960q^{70} - 5560632q^{71} - 2640378q^{73} - 1611156q^{74} + 16186704q^{76} - 60672q^{77} - 1514352q^{79} - 1503840q^{80} - 437508q^{82} + 4376748q^{83} - 3306420q^{85} - 4114632q^{86} - 22710240q^{88} + 8528310q^{89} + 3909072q^{91} - 1406496q^{92} + 24973056q^{94} + 3361800q^{95} - 34741794q^{97} + 4916682q^{98} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(9))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
9.8.a.a \(1\) \(2.811\) \(\Q\) None \(-6\) \(0\) \(-390\) \(-64\) \(-\) \(q-6q^{2}-92q^{4}-390q^{5}-2^{6}q^{7}+\cdots\)
9.8.a.b \(2\) \(2.811\) \(\Q(\sqrt{10}) \) None \(0\) \(0\) \(0\) \(520\) \(+\) \(q+\beta q^{2}+232q^{4}-2^{4}\beta q^{5}+260q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)