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Properties

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

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

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3
9.8.a.a 9.a 1.a $1$ $2.811$ \(\Q\) None \(-6\) \(0\) \(-390\) \(-64\) $-$ $\mathrm{SU}(2)$ \(q-6q^{2}-92q^{4}-390q^{5}-2^{6}q^{7}+\cdots\)
9.8.a.b 9.a 1.a $2$ $2.811$ \(\Q(\sqrt{10}) \) None \(0\) \(0\) \(0\) \(520\) $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+232q^{4}-2^{4}\beta q^{5}+260q^{7}+\cdots\)

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

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