# Properties

 Label 315.2.g Level 315 Weight 2 Character orbit g Rep. character $$\chi_{315}(314,\cdot)$$ Character field $$\Q$$ Dimension 16 Newform subspaces 1 Sturm bound 96 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.g (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$105$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$96$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(315, [\chi])$$.

Total New Old
Modular forms 56 16 40
Cusp forms 40 16 24
Eisenstein series 16 0 16

## Trace form

 $$16q + 16q^{4} + O(q^{10})$$ $$16q + 16q^{4} + 16q^{16} + 16q^{25} - 96q^{46} + 16q^{49} - 80q^{64} - 48q^{70} - 64q^{79} + 32q^{85} + 32q^{91} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(315, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
315.2.g.a $$16$$ $$2.515$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{2}+(1+\beta _{14})q^{4}-\beta _{4}q^{5}-\beta _{12}q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(315, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(315, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} + 3 T^{4} + 8 T^{6} + 16 T^{8} )^{4}$$
$3$ 
$5$ $$( 1 - 4 T^{2} + 30 T^{4} - 100 T^{6} + 625 T^{8} )^{2}$$
$7$ $$( 1 - 4 T^{2} + 6 T^{4} - 196 T^{6} + 2401 T^{8} )^{2}$$
$11$ $$( 1 - 20 T^{2} + 121 T^{4} )^{8}$$
$13$ $$( 1 + 20 T^{2} + 222 T^{4} + 3380 T^{6} + 28561 T^{8} )^{4}$$
$17$ $$( 1 - 4 T^{2} + 558 T^{4} - 1156 T^{6} + 83521 T^{8} )^{4}$$
$19$ $$( 1 - 4 T^{2} + 510 T^{4} - 1444 T^{6} + 130321 T^{8} )^{4}$$
$23$ $$( 1 + 56 T^{2} + 1818 T^{4} + 29624 T^{6} + 279841 T^{8} )^{4}$$
$29$ $$( 1 - 16 T^{2} + 1362 T^{4} - 13456 T^{6} + 707281 T^{8} )^{4}$$
$31$ $$( 1 - 52 T^{2} + 1422 T^{4} - 49972 T^{6} + 923521 T^{8} )^{4}$$
$37$ $$( 1 - 76 T^{2} + 3318 T^{4} - 104044 T^{6} + 1874161 T^{8} )^{4}$$
$41$ $$( 1 + 44 T^{2} + 3246 T^{4} + 73964 T^{6} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 - 100 T^{2} + 6102 T^{4} - 184900 T^{6} + 3418801 T^{8} )^{4}$$
$47$ $$( 1 - 124 T^{2} + 7398 T^{4} - 273916 T^{6} + 4879681 T^{8} )^{4}$$
$53$ $$( 1 + 104 T^{2} + 6378 T^{4} + 292136 T^{6} + 7890481 T^{8} )^{4}$$
$59$ $$( 1 + 92 T^{2} + 4374 T^{4} + 320252 T^{6} + 12117361 T^{8} )^{4}$$
$61$ $$( 1 - 52 T^{2} + 6582 T^{4} - 193492 T^{6} + 13845841 T^{8} )^{4}$$
$67$ $$( 1 - 124 T^{2} + 12438 T^{4} - 556636 T^{6} + 20151121 T^{8} )^{4}$$
$71$ $$( 1 - 184 T^{2} + 18162 T^{4} - 927544 T^{6} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 + 164 T^{2} + 13326 T^{4} + 873956 T^{6} + 28398241 T^{8} )^{4}$$
$79$ $$( 1 + 4 T + 79 T^{2} )^{16}$$
$83$ $$( 1 - 268 T^{2} + 30870 T^{4} - 1846252 T^{6} + 47458321 T^{8} )^{4}$$
$89$ $$( 1 + 140 T^{2} + 18798 T^{4} + 1108940 T^{6} + 62742241 T^{8} )^{4}$$
$97$ $$( 1 + 356 T^{2} + 50286 T^{4} + 3349604 T^{6} + 88529281 T^{8} )^{4}$$