Properties

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

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Defining parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 9.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
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

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

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(9))\) into irreducible Hecke orbits

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

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