Properties

 Label 315.2.bo Level 315 Weight 2 Character orbit bo Rep. character $$\chi_{315}(4,\cdot)$$ 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.bo (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

 $$88q + 42q^{4} - 2q^{5} - 2q^{9} + O(q^{10})$$ $$88q + 42q^{4} - 2q^{5} - 2q^{9} + 2q^{10} - 8q^{14} + 4q^{15} - 34q^{16} - 4q^{19} - 12q^{20} - 8q^{21} + 18q^{24} - 2q^{25} - 32q^{26} - 14q^{29} - 34q^{30} + 2q^{31} - 8q^{34} + 24q^{35} - 12q^{36} - 20q^{39} + 16q^{40} - 34q^{41} - 16q^{44} - 18q^{45} + 10q^{46} - 14q^{49} + 34q^{50} + 2q^{51} - 30q^{55} + 18q^{56} + 50q^{59} + 54q^{60} + 8q^{61} - 56q^{64} - 8q^{65} - 74q^{66} - 32q^{69} - 24q^{70} + 12q^{71} - 76q^{74} + 6q^{75} + 12q^{76} - 4q^{79} - 7q^{80} - 70q^{81} - 52q^{84} - 7q^{85} - 88q^{86} + 40q^{89} + 5q^{90} - 36q^{91} - 2q^{94} - 37q^{95} + 84q^{96} - 30q^{99} + O(q^{100})$$

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.bo.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.bo.b $$84$$ $$2.515$$ None $$0$$ $$0$$ $$-6$$ $$0$$

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8}$$)
$3$ ($$( 1 - 3 T^{2} )^{2}$$)
$5$ ($$( 1 - 2 T + 5 T^{2} )^{2}$$)
$7$ ($$1 + 11 T^{2} + 49 T^{4}$$)
$11$ ($$( 1 - 6 T + 11 T^{2} )^{4}$$)
$13$ ($$( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} )( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} )$$)
$17$ ($$( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} )$$)
$19$ ($$( 1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4} )^{2}$$)
$23$ ($$( 1 - 37 T^{2} + 529 T^{4} )^{2}$$)
$29$ ($$( 1 + 2 T - 25 T^{2} + 58 T^{3} + 841 T^{4} )^{2}$$)
$31$ ($$( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$)
$37$ ($$1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8}$$)
$41$ ($$( 1 + 2 T - 37 T^{2} + 82 T^{3} + 1681 T^{4} )^{2}$$)
$43$ ($$1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8}$$)
$47$ ($$1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8}$$)
$53$ ($$1 - 38 T^{2} - 1365 T^{4} - 106742 T^{6} + 7890481 T^{8}$$)
$59$ ($$( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} )^{2}$$)
$61$ ($$( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} )^{2}$$)
$67$ ($$1 + 85 T^{2} + 2736 T^{4} + 381565 T^{6} + 20151121 T^{8}$$)
$71$ ($$( 1 - 4 T + 71 T^{2} )^{4}$$)
$73$ ($$( 1 + 73 T^{2} + 5329 T^{4} )^{2}$$)
$79$ ($$( 1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4} )^{2}$$)
$83$ ($$1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8}$$)
$89$ ($$( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} )^{2}$$)
$97$ ($$1 + 190 T^{2} + 26691 T^{4} + 1787710 T^{6} + 88529281 T^{8}$$)