# Properties

 Label 315.2.be Level 315 Weight 2 Character orbit be Rep. character $$\chi_{315}(236,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 64 Newform subspaces 3 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.be (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$63$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$3$$ 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 64 40
Cusp forms 88 64 24
Eisenstein series 16 0 16

## Trace form

 $$64q + 32q^{4} + 2q^{7} - 4q^{9} + O(q^{10})$$ $$64q + 32q^{4} + 2q^{7} - 4q^{9} - 6q^{13} - 6q^{14} - 2q^{15} - 32q^{16} + 28q^{18} - 24q^{21} - 18q^{24} + 64q^{25} - 24q^{26} - 18q^{27} - 8q^{28} + 18q^{29} - 4q^{30} + 24q^{31} - 24q^{33} + 36q^{36} - 2q^{37} - 120q^{38} - 36q^{39} + 6q^{41} - 8q^{43} - 42q^{44} - 18q^{45} + 6q^{46} - 36q^{47} + 60q^{48} + 10q^{49} + 42q^{51} + 48q^{53} - 36q^{54} + 102q^{56} + 6q^{57} + 30q^{59} - 30q^{60} - 60q^{61} + 14q^{63} - 64q^{64} + 6q^{65} + 48q^{66} + 14q^{67} - 60q^{68} + 6q^{70} - 76q^{72} - 54q^{77} - 24q^{78} - 4q^{79} + 44q^{81} + 60q^{83} - 54q^{84} - 6q^{85} - 102q^{87} - 42q^{89} - 54q^{90} - 6q^{91} + 12q^{92} - 60q^{93} + 60q^{96} - 6q^{97} - 54q^{98} - 2q^{99} + O(q^{100})$$

## 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.be.a $$2$$ $$2.515$$ $$\Q(\sqrt{-3})$$ None $$-3$$ $$0$$ $$2$$ $$-5$$ $$q+(-1-\zeta_{6})q^{2}+(1-2\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots$$
315.2.be.b $$30$$ $$2.515$$ None $$3$$ $$-1$$ $$30$$ $$6$$
315.2.be.c $$32$$ $$2.515$$ None $$0$$ $$1$$ $$-32$$ $$1$$

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

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$)
$3$ ($$1 + 3 T^{2}$$)
$5$ ($$( 1 - T )^{2}$$)
$7$ ($$1 + 5 T + 7 T^{2}$$)
$11$ ($$1 - 10 T^{2} + 121 T^{4}$$)
$13$ ($$( 1 + 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)
$17$ ($$1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4}$$)
$19$ ($$1 + 19 T^{2} + 361 T^{4}$$)
$23$ ($$1 - 19 T^{2} + 529 T^{4}$$)
$29$ ($$1 + 29 T^{2} + 841 T^{4}$$)
$31$ ($$1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4}$$)
$37$ ($$1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4}$$)
$41$ ($$1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4}$$)
$43$ ($$1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4}$$)
$47$ ($$1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4}$$)
$53$ ($$1 + 6 T + 65 T^{2} + 318 T^{3} + 2809 T^{4}$$)
$59$ ($$1 - 59 T^{2} + 3481 T^{4}$$)
$61$ ($$1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4}$$)
$67$ ($$( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$)
$71$ ($$1 - 94 T^{2} + 5041 T^{4}$$)
$73$ ($$1 + 18 T + 181 T^{2} + 1314 T^{3} + 5329 T^{4}$$)
$79$ ($$1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4}$$)
$83$ ($$1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4}$$)
$89$ ($$1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4}$$)
$97$ ($$1 + 12 T + 145 T^{2} + 1164 T^{3} + 9409 T^{4}$$)