# Properties

 Label 315.2.r Level 315 Weight 2 Character orbit r Rep. character $$\chi_{315}(184,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 88 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.r (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$315$$ 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 104 104 0
Cusp forms 88 88 0
Eisenstein series 16 16 0

## Trace form

 $$88q - 84q^{4} + q^{5} - 8q^{9} + O(q^{10})$$ $$88q - 84q^{4} + q^{5} - 8q^{9} + 2q^{10} + 16q^{14} + 4q^{15} + 68q^{16} - 4q^{19} - 12q^{20} - 8q^{21} + q^{25} - 32q^{26} - 14q^{29} + 5q^{30} - 4q^{31} - 8q^{34} + 24q^{35} - 12q^{36} - 32q^{39} - 8q^{40} - 34q^{41} - 16q^{44} + 27q^{45} + 10q^{46} + 4q^{49} + 34q^{50} - 52q^{51} + 24q^{54} - 30q^{55} + 6q^{56} - 100q^{59} - 21q^{60} - 16q^{61} - 56q^{64} + 16q^{65} + 40q^{66} - 32q^{69} - 9q^{70} + 12q^{71} + 38q^{74} + 42q^{75} + 12q^{76} + 8q^{79} - 7q^{80} + 20q^{81} + 74q^{84} - 7q^{85} + 44q^{86} + 40q^{89} + 5q^{90} - 36q^{91} + 4q^{94} + 74q^{95} + 48q^{96} - 30q^{99} + O(q^{100})$$

## 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.r.a $$4$$ $$2.515$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q+\zeta_{12}^{3}q^{2}+(\zeta_{12}+\zeta_{12}^{3})q^{3}+q^{4}+\cdots$$
315.2.r.b $$84$$ $$2.515$$ None $$0$$ $$0$$ $$3$$ $$0$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - 3 T^{2} + 4 T^{4} )^{2}$$)
$3$ ($$1 + 3 T^{2} + 9 T^{4}$$)
$5$ ($$1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4}$$)
$7$ ($$1 + 2 T^{2} + 49 T^{4}$$)
$11$ ($$( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{2}$$)
$13$ ($$( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} )( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} )$$)
$17$ ($$( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} )$$)
$19$ ($$( 1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4} )^{2}$$)
$23$ ($$1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8}$$)
$29$ ($$( 1 + 2 T - 25 T^{2} + 58 T^{3} + 841 T^{4} )^{2}$$)
$31$ ($$( 1 + 4 T + 31 T^{2} )^{4}$$)
$37$ ($$1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8}$$)
$41$ ($$( 1 + 2 T - 37 T^{2} + 82 T^{3} + 1681 T^{4} )^{2}$$)
$43$ ($$1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8}$$)
$47$ ($$( 1 - 85 T^{2} + 2209 T^{4} )^{2}$$)
$53$ ($$1 - 38 T^{2} - 1365 T^{4} - 106742 T^{6} + 7890481 T^{8}$$)
$59$ ($$( 1 + 4 T + 59 T^{2} )^{4}$$)
$61$ ($$( 1 - 7 T + 61 T^{2} )^{4}$$)
$67$ ($$( 1 - 85 T^{2} + 4489 T^{4} )^{2}$$)
$71$ ($$( 1 - 4 T + 71 T^{2} )^{4}$$)
$73$ ($$( 1 + 73 T^{2} + 5329 T^{4} )^{2}$$)
$79$ ($$( 1 - 14 T + 79 T^{2} )^{4}$$)
$83$ ($$1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8}$$)
$89$ ($$( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} )^{2}$$)
$97$ ($$1 + 190 T^{2} + 26691 T^{4} + 1787710 T^{6} + 88529281 T^{8}$$)