# Properties

 Label 315.2.bh Level 315 Weight 2 Character orbit bh Rep. character $$\chi_{315}(169,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 72 Newform subspaces 3 Sturm bound 96 Trace bound 4

# Related objects

## Defining parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.bh (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$45$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$3$$ Sturm bound: $$96$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

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

Total New Old
Modular forms 104 72 32
Cusp forms 88 72 16
Eisenstein series 16 0 16

## Trace form

 $$72q + 36q^{4} - 2q^{5} - 8q^{9} + O(q^{10})$$ $$72q + 36q^{4} - 2q^{5} - 8q^{9} + 18q^{11} - 8q^{14} - 20q^{15} - 36q^{16} - 6q^{20} - 2q^{21} - 12q^{24} + 6q^{25} - 32q^{26} - 20q^{29} + 20q^{30} - 12q^{31} + 12q^{34} - 36q^{36} - 14q^{39} + 8q^{41} + 32q^{44} + 18q^{45} + 24q^{46} + 36q^{49} - 50q^{50} + 74q^{51} + 36q^{54} - 12q^{55} + 24q^{56} - 4q^{59} - 48q^{60} - 96q^{64} - 20q^{65} - 128q^{66} - 20q^{69} - 168q^{71} - 28q^{74} + 60q^{75} - 12q^{76} - 6q^{79} + 104q^{80} - 40q^{81} - 16q^{84} - 24q^{85} + 44q^{86} - 80q^{89} - 16q^{90} - 12q^{91} - 48q^{94} + 20q^{95} + 180q^{96} + 144q^{99} + O(q^{100})$$

## 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.bh.a $$4$$ $$2.515$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+\zeta_{12}q^{2}+(2\zeta_{12}-\zeta_{12}^{3})q^{3}-\zeta_{12}^{2}q^{4}+\cdots$$
315.2.bh.b $$4$$ $$2.515$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+2\zeta_{12}q^{2}+(\zeta_{12}-2\zeta_{12}^{3})q^{3}+2\zeta_{12}^{2}q^{4}+\cdots$$
315.2.bh.c $$64$$ $$2.515$$ None $$0$$ $$0$$ $$-10$$ $$0$$

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

$$S_{2}^{\mathrm{old}}(315, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8}$$)($$( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )$$)
$3$ ($$( 1 - 3 T^{2} )^{2}$$)($$1 + 3 T^{2} + 9 T^{4}$$)
$5$ ($$1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4}$$)($$1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4}$$)
$7$ ($$1 - T^{2} + T^{4}$$)($$1 - T^{2} + T^{4}$$)
$11$ ($$( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} )^{2}$$)($$( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} )^{2}$$)
$13$ ($$1 + 25 T^{2} + 456 T^{4} + 4225 T^{6} + 28561 T^{8}$$)($$( 1 - T^{2} + 169 T^{4} )( 1 + 23 T^{2} + 169 T^{4} )$$)
$17$ ($$( 1 + 15 T^{2} + 289 T^{4} )^{2}$$)($$( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2}$$)
$19$ ($$( 1 + 6 T + 19 T^{2} )^{4}$$)($$( 1 + 6 T + 19 T^{2} )^{4}$$)
$23$ ($$( 1 + 23 T^{2} + 529 T^{4} )^{2}$$)($$1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8}$$)
$29$ ($$( 1 - 10 T + 71 T^{2} - 290 T^{3} + 841 T^{4} )^{2}$$)($$( 1 + 5 T - 4 T^{2} + 145 T^{3} + 841 T^{4} )^{2}$$)
$31$ ($$( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} )^{2}$$)($$( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$)
$37$ ($$( 1 - 10 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$)
$41$ ($$( 1 + 2 T - 37 T^{2} + 82 T^{3} + 1681 T^{4} )^{2}$$)($$( 1 + 8 T + 23 T^{2} + 328 T^{3} + 1681 T^{4} )^{2}$$)
$43$ ($$1 + 70 T^{2} + 3051 T^{4} + 129430 T^{6} + 3418801 T^{8}$$)($$1 + 82 T^{2} + 4875 T^{4} + 151618 T^{6} + 3418801 T^{8}$$)
$47$ ($$1 + 13 T^{2} - 2040 T^{4} + 28717 T^{6} + 4879681 T^{8}$$)($$1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8}$$)
$53$ ($$( 1 - 70 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 53 T^{2} )^{4}$$)
$59$ ($$( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 + 14 T + 137 T^{2} + 826 T^{3} + 3481 T^{4} )^{2}$$)
$61$ ($$( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} )^{2}$$)($$( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} )^{2}$$)
$67$ ($$1 + 118 T^{2} + 9435 T^{4} + 529702 T^{6} + 20151121 T^{8}$$)($$1 + 34 T^{2} - 3333 T^{4} + 152626 T^{6} + 20151121 T^{8}$$)
$71$ ($$( 1 + 5 T + 71 T^{2} )^{4}$$)($$( 1 - 7 T + 71 T^{2} )^{4}$$)
$73$ ($$( 1 - 137 T^{2} + 5329 T^{4} )^{2}$$)($$( 1 - 65 T^{2} + 5329 T^{4} )^{2}$$)
$79$ ($$( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} )^{2}$$)
$83$ ($$1 + 141 T^{2} + 12992 T^{4} + 971349 T^{6} + 47458321 T^{8}$$)($$1 - 123 T^{2} + 8240 T^{4} - 847347 T^{6} + 47458321 T^{8}$$)
$89$ ($$( 1 - 18 T + 89 T^{2} )^{4}$$)($$( 1 - 12 T + 89 T^{2} )^{4}$$)
$97$ ($$( 1 + 2 T^{2} + 9409 T^{4} )( 1 + 167 T^{2} + 9409 T^{4} )$$)($$1 + 145 T^{2} + 11616 T^{4} + 1364305 T^{6} + 88529281 T^{8}$$)