# Properties

 Label 315.2.z Level 315 Weight 2 Character orbit z Rep. character $$\chi_{315}(104,\cdot)$$ 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.z (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

 $$88q - 44q^{4} + O(q^{10})$$ $$88q - 44q^{4} - 6q^{11} - 12q^{14} - 24q^{15} - 44q^{16} + 6q^{21} - 2q^{25} - 48q^{29} + 36q^{30} - 36q^{36} + 54q^{39} - 24q^{46} - 8q^{49} - 42q^{50} - 18q^{51} + 24q^{56} + 96q^{60} + 64q^{64} - 6q^{70} + 12q^{74} - 34q^{79} - 60q^{84} + 8q^{85} + 156q^{86} + 20q^{91} - 24q^{95} - 72q^{99} + O(q^{100})$$

## 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.z.a $$8$$ $$2.515$$ 8.0.$$\cdots$$.2 $$\Q(\sqrt{-35})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{5}q^{3}+(2+2\beta _{4})q^{4}+(\beta _{1}-\beta _{6})q^{5}+\cdots$$
315.2.z.b $$80$$ $$2.515$$ None $$0$$ $$0$$ $$0$$ $$0$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - 2 T^{2} + 4 T^{4} )^{4}$$)
$3$ ($$( 1 - T^{2} + 9 T^{4} )^{2}$$)
$5$ ($$( 1 - 5 T^{2} + 25 T^{4} )^{2}$$)
$7$ ($$( 1 + 7 T^{2} + 49 T^{4} )^{2}$$)
$11$ ($$( 1 - 3 T + 11 T^{2} )^{4}( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} )^{2}$$)
$13$ ($$( 1 + 19 T^{2} + 169 T^{4} )^{2}( 1 - 19 T^{2} + 192 T^{4} - 3211 T^{6} + 28561 T^{8} )$$)
$17$ ($$( 1 + 29 T^{2} + 552 T^{4} + 8381 T^{6} + 83521 T^{8} )^{2}$$)
$19$ ($$( 1 - 19 T^{2} )^{8}$$)
$23$ ($$( 1 - 23 T^{2} + 529 T^{4} )^{4}$$)
$29$ ($$( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} )^{2}( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} )^{2}$$)
$31$ ($$( 1 + 31 T^{2} + 961 T^{4} )^{4}$$)
$37$ ($$( 1 - 37 T^{2} )^{8}$$)
$41$ ($$( 1 - 41 T^{2} + 1681 T^{4} )^{4}$$)
$43$ ($$( 1 + 43 T^{2} + 1849 T^{4} )^{4}$$)
$47$ ($$( 1 + 31 T^{2} + 2209 T^{4} )^{2}( 1 - 31 T^{2} - 1248 T^{4} - 68479 T^{6} + 4879681 T^{8} )$$)
$53$ ($$( 1 + 53 T^{2} )^{8}$$)
$59$ ($$( 1 - 59 T^{2} + 3481 T^{4} )^{4}$$)
$61$ ($$( 1 + 61 T^{2} + 3721 T^{4} )^{4}$$)
$67$ ($$( 1 + 67 T^{2} + 4489 T^{4} )^{4}$$)
$71$ ($$( 1 - 12 T + 73 T^{2} - 852 T^{3} + 5041 T^{4} )^{2}( 1 + 12 T + 73 T^{2} + 852 T^{3} + 5041 T^{4} )^{2}$$)
$73$ ($$( 1 - 34 T^{2} - 4173 T^{4} - 181186 T^{6} + 28398241 T^{8} )^{2}$$)
$79$ ($$( 1 + T + 79 T^{2} )^{4}( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} )^{2}$$)
$83$ ($$( 1 - 86 T^{2} + 6889 T^{4} )^{2}( 1 + 86 T^{2} + 507 T^{4} + 592454 T^{6} + 47458321 T^{8} )$$)
$89$ ($$( 1 + 89 T^{2} )^{8}$$)
$97$ ($$( 1 - 149 T^{2} + 9409 T^{4} )^{2}( 1 + 149 T^{2} + 12792 T^{4} + 1401941 T^{6} + 88529281 T^{8} )$$)