# Properties

 Label 315.2.bb Level 315 Weight 2 Character orbit bb Rep. character $$\chi_{315}(89,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 32 Newform subspaces 2 Sturm bound 96 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.bb (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$105$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$2$$ Sturm bound: $$96$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

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

Total New Old
Modular forms 112 32 80
Cusp forms 80 32 48
Eisenstein series 32 0 32

## Trace form

 $$32q - 16q^{4} + O(q^{10})$$ $$32q - 16q^{4} - 12q^{10} - 16q^{16} - 24q^{19} + 20q^{25} - 24q^{31} + 96q^{40} - 24q^{46} + 8q^{49} - 24q^{61} - 16q^{64} - 48q^{70} - 32q^{79} - 128q^{85} + 88q^{91} + 48q^{94} + O(q^{100})$$

## 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.bb.a $$8$$ $$2.515$$ 8.0.$$\cdots$$.5 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+2\beta _{4}q^{4}+(\beta _{1}-\beta _{2}+\beta _{5})q^{5}+(\beta _{3}+\cdots)q^{7}+\cdots$$
315.2.bb.b $$24$$ $$2.515$$ None $$0$$ $$0$$ $$0$$ $$0$$

## 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

$p$ $F_p(T)$
$2$ ($$( 1 - 2 T^{2} + 4 T^{4} )^{4}$$)
$3$ 1
$5$ ($$1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8}$$)
$7$ ($$( 1 + 7 T^{2} + 49 T^{4} )^{2}$$)
$11$ ($$( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2}( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{2}$$)
$13$ ($$( 1 + 19 T^{2} + 169 T^{4} )^{4}$$)
$17$ ($$( 1 + 20 T^{2} + 111 T^{4} + 5780 T^{6} + 83521 T^{8} )^{2}$$)
$19$ ($$( 1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4} )^{4}$$)
$23$ ($$( 1 - 4 T^{2} - 513 T^{4} - 2116 T^{6} + 279841 T^{8} )^{2}$$)
$29$ ($$( 1 - 56 T^{2} + 841 T^{4} )^{4}$$)
$31$ ($$( 1 - 9 T + 58 T^{2} - 279 T^{3} + 961 T^{4} )^{4}$$)
$37$ ($$( 1 + 53 T^{2} + 1440 T^{4} + 72557 T^{6} + 1874161 T^{8} )^{2}$$)
$41$ ($$( 1 + 58 T^{2} + 1681 T^{4} )^{4}$$)
$43$ ($$( 1 - 65 T^{2} + 1849 T^{4} )^{4}$$)
$47$ ($$( 1 + 80 T^{2} + 4191 T^{4} + 176720 T^{6} + 4879681 T^{8} )^{2}$$)
$53$ ($$( 1 + 62 T^{2} + 1035 T^{4} + 174158 T^{6} + 7890481 T^{8} )^{2}$$)
$59$ ($$( 1 - 64 T^{2} + 615 T^{4} - 222784 T^{6} + 12117361 T^{8} )^{2}$$)
$61$ ($$( 1 + 18 T + 169 T^{2} + 1098 T^{3} + 3721 T^{4} )^{4}$$)
$67$ ($$( 1 - 55 T^{2} - 1464 T^{4} - 246895 T^{6} + 20151121 T^{8} )^{2}$$)
$71$ ($$( 1 - 14 T^{2} + 5041 T^{4} )^{4}$$)
$73$ ($$( 1 + 29 T^{2} - 4488 T^{4} + 154541 T^{6} + 28398241 T^{8} )^{2}$$)
$79$ ($$( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} )^{4}$$)
$83$ ($$( 1 + 58 T^{2} + 6889 T^{4} )^{4}$$)
$89$ ($$( 1 + 116 T^{2} + 5535 T^{4} + 918836 T^{6} + 62742241 T^{8} )^{2}$$)
$97$ ($$( 1 + 166 T^{2} + 9409 T^{4} )^{4}$$)