# Properties

 Label 315.2.b Level 315 Weight 2 Character orbit b Rep. character $$\chi_{315}(251,\cdot)$$ Character field $$\Q$$ Dimension 8 Newform subspaces 2 Sturm bound 96 Trace bound 5

# Related objects

## Defining parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$21$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$96$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$47$$

## Dimensions

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

Total New Old
Modular forms 56 8 48
Cusp forms 40 8 32
Eisenstein series 16 0 16

## Trace form

 $$8q + 8q^{7} + O(q^{10})$$ $$8q + 8q^{7} - 8q^{16} - 40q^{22} + 8q^{25} + 16q^{37} + 16q^{43} + 8q^{46} - 40q^{49} + 8q^{58} + 32q^{64} - 32q^{67} + 24q^{70} - 32q^{79} - 8q^{88} + 48q^{91} + O(q^{100})$$

## 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.b.a $$4$$ $$2.515$$ $$\Q(\sqrt{-2}, \sqrt{3})$$ None $$0$$ $$0$$ $$-4$$ $$4$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}-q^{5}+(1+\beta _{1}+\beta _{3})q^{7}+\cdots$$
315.2.b.b $$4$$ $$2.515$$ $$\Q(\sqrt{-2}, \sqrt{3})$$ None $$0$$ $$0$$ $$4$$ $$4$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}+q^{5}+(1-\beta _{1}-\beta _{3})q^{7}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(315, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - 4 T^{2} + 9 T^{4} - 16 T^{6} + 16 T^{8}$$)($$1 - 4 T^{2} + 9 T^{4} - 16 T^{6} + 16 T^{8}$$)
$3$ ()()
$5$ ($$( 1 + T )^{4}$$)($$( 1 - T )^{4}$$)
$7$ ($$( 1 - 2 T + 7 T^{2} )^{2}$$)($$( 1 - 2 T + 7 T^{2} )^{2}$$)
$11$ ($$1 - 16 T^{2} + 114 T^{4} - 1936 T^{6} + 14641 T^{8}$$)($$1 - 16 T^{2} + 114 T^{4} - 1936 T^{6} + 14641 T^{8}$$)
$13$ ($$1 - 4 T^{2} - 90 T^{4} - 676 T^{6} + 28561 T^{8}$$)($$1 - 4 T^{2} - 90 T^{4} - 676 T^{6} + 28561 T^{8}$$)
$17$ ($$( 1 + 22 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 22 T^{2} + 289 T^{4} )^{2}$$)
$19$ ($$1 - 28 T^{2} + 486 T^{4} - 10108 T^{6} + 130321 T^{8}$$)($$1 - 28 T^{2} + 486 T^{4} - 10108 T^{6} + 130321 T^{8}$$)
$23$ ($$( 1 - 44 T^{2} + 529 T^{4} )^{2}$$)($$( 1 - 44 T^{2} + 529 T^{4} )^{2}$$)
$29$ ($$( 1 - 56 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 56 T^{2} + 841 T^{4} )^{2}$$)
$31$ ($$1 - 76 T^{2} + 2934 T^{4} - 73036 T^{6} + 923521 T^{8}$$)($$1 - 76 T^{2} + 2934 T^{4} - 73036 T^{6} + 923521 T^{8}$$)
$37$ ($$( 1 - 4 T + 66 T^{2} - 148 T^{3} + 1369 T^{4} )^{2}$$)($$( 1 - 4 T + 66 T^{2} - 148 T^{3} + 1369 T^{4} )^{2}$$)
$41$ ($$( 1 - 26 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 - 26 T^{2} + 1681 T^{4} )^{2}$$)
$43$ ($$( 1 - 4 T + 42 T^{2} - 172 T^{3} + 1849 T^{4} )^{2}$$)($$( 1 - 4 T + 42 T^{2} - 172 T^{3} + 1849 T^{4} )^{2}$$)
$47$ ($$( 1 - 12 T + 118 T^{2} - 564 T^{3} + 2209 T^{4} )^{2}$$)($$( 1 + 12 T + 118 T^{2} + 564 T^{3} + 2209 T^{4} )^{2}$$)
$53$ ($$1 - 88 T^{2} + 5826 T^{4} - 247192 T^{6} + 7890481 T^{8}$$)($$1 - 88 T^{2} + 5826 T^{4} - 247192 T^{6} + 7890481 T^{8}$$)
$59$ ($$( 1 - 12 T + 142 T^{2} - 708 T^{3} + 3481 T^{4} )^{2}$$)($$( 1 + 12 T + 142 T^{2} + 708 T^{3} + 3481 T^{4} )^{2}$$)
$61$ ($$1 - 52 T^{2} + 1206 T^{4} - 193492 T^{6} + 13845841 T^{8}$$)($$1 - 52 T^{2} + 1206 T^{4} - 193492 T^{6} + 13845841 T^{8}$$)
$67$ ($$( 1 + 8 T + 102 T^{2} + 536 T^{3} + 4489 T^{4} )^{2}$$)($$( 1 + 8 T + 102 T^{2} + 536 T^{3} + 4489 T^{4} )^{2}$$)
$71$ ($$1 + 32 T^{2} + 5538 T^{4} + 161312 T^{6} + 25411681 T^{8}$$)($$1 + 32 T^{2} + 5538 T^{4} + 161312 T^{6} + 25411681 T^{8}$$)
$73$ ($$1 - 244 T^{2} + 25110 T^{4} - 1300276 T^{6} + 28398241 T^{8}$$)($$1 - 244 T^{2} + 25110 T^{4} - 1300276 T^{6} + 28398241 T^{8}$$)
$79$ ($$( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} )^{2}$$)($$( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} )^{2}$$)
$83$ ($$( 1 + 12 T + 190 T^{2} + 996 T^{3} + 6889 T^{4} )^{2}$$)($$( 1 - 12 T + 190 T^{2} - 996 T^{3} + 6889 T^{4} )^{2}$$)
$89$ ($$( 1 - 12 T + 166 T^{2} - 1068 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 12 T + 166 T^{2} + 1068 T^{3} + 7921 T^{4} )^{2}$$)
$97$ ($$1 - 52 T^{2} + 15606 T^{4} - 489268 T^{6} + 88529281 T^{8}$$)($$1 - 52 T^{2} + 15606 T^{4} - 489268 T^{6} + 88529281 T^{8}$$)