# 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

## 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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 6 T + 128 T^{2}$$)($$1 - 104 T^{2} + 16384 T^{4}$$)
$3$ 1
$5$ ($$1 + 390 T + 78125 T^{2}$$)($$1 + 64090 T^{2} + 6103515625 T^{4}$$)
$7$ ($$1 + 64 T + 823543 T^{2}$$)($$( 1 - 260 T + 823543 T^{2} )^{2}$$)
$11$ ($$1 - 948 T + 19487171 T^{2}$$)($$1 + 2110342 T^{2} + 379749833583241 T^{4}$$)
$13$ ($$1 + 5098 T + 62748517 T^{2}$$)($$( 1 - 6890 T + 62748517 T^{2} )^{2}$$)
$17$ ($$1 + 28386 T + 410338673 T^{2}$$)($$1 + 259975906 T^{2} + 168377826559400929 T^{4}$$)
$19$ ($$1 + 8620 T + 893871739 T^{2}$$)($$( 1 - 33176 T + 893871739 T^{2} )^{2}$$)
$23$ ($$1 - 15288 T + 3404825447 T^{2}$$)($$1 + 5812848334 T^{2} + 11592836324538749809 T^{4}$$)
$29$ ($$1 + 36510 T + 17249876309 T^{2}$$)($$1 + 15420328618 T^{2} +$$$$29\!\cdots\!81$$$$T^{4}$$)
$31$ ($$1 + 276808 T + 27512614111 T^{2}$$)($$( 1 - 1508 T + 27512614111 T^{2} )^{2}$$)
$37$ ($$1 - 268526 T + 94931877133 T^{2}$$)($$( 1 + 380770 T + 94931877133 T^{2} )^{2}$$)
$41$ ($$1 - 629718 T + 194754273881 T^{2}$$)($$1 + 381757891762 T^{2} +$$$$37\!\cdots\!61$$$$T^{4}$$)
$43$ ($$1 - 685772 T + 271818611107 T^{2}$$)($$( 1 - 7640 T + 271818611107 T^{2} )^{2}$$)
$47$ ($$1 + 583296 T + 506623120463 T^{2}$$)($$1 + 693036689566 T^{2} +$$$$25\!\cdots\!69$$$$T^{4}$$)
$53$ ($$1 - 428058 T + 1174711139837 T^{2}$$)($$1 + 1288434979834 T^{2} +$$$$13\!\cdots\!69$$$$T^{4}$$)
$59$ ($$1 + 1306380 T + 2488651484819 T^{2}$$)($$1 - 2355536454362 T^{2} +$$$$61\!\cdots\!61$$$$T^{4}$$)
$61$ ($$1 - 300662 T + 3142742836021 T^{2}$$)($$( 1 + 988858 T + 3142742836021 T^{2} )^{2}$$)
$67$ ($$1 + 507244 T + 6060711605323 T^{2}$$)($$( 1 - 3857360 T + 6060711605323 T^{2} )^{2}$$)
$71$ ($$1 + 5560632 T + 9095120158391 T^{2}$$)($$1 + 332728892782 T^{2} +$$$$82\!\cdots\!81$$$$T^{4}$$)
$73$ ($$1 - 1369082 T + 11047398519097 T^{2}$$)($$( 1 + 2004730 T + 11047398519097 T^{2} )^{2}$$)
$79$ ($$1 + 6913720 T + 19203908986159 T^{2}$$)($$( 1 - 2699684 T + 19203908986159 T^{2} )^{2}$$)
$83$ ($$1 - 4376748 T + 27136050989627 T^{2}$$)($$1 + 46919519671414 T^{2} +$$$$73\!\cdots\!29$$$$T^{4}$$)
$89$ ($$1 - 8528310 T + 44231334895529 T^{2}$$)($$1 + 28535629791058 T^{2} +$$$$19\!\cdots\!41$$$$T^{4}$$)
$97$ ($$1 + 8826814 T + 80798284478113 T^{2}$$)($$( 1 + 12957490 T + 80798284478113 T^{2} )^{2}$$)