# 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 $1$

# Related objects

## 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

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

## 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

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