# Properties

 Label 315.2.k Level 315 Weight 2 Character orbit k Rep. character $$\chi_{315}(16,\cdot)$$ Character field $$\Q(\zeta_{3})$$ 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.k (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$63$$ Character field: $$\Q(\zeta_{3})$$ 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

 $$64q - 32q^{4} + 8q^{5} - 4q^{6} - 2q^{7} + O(q^{10})$$ $$64q - 32q^{4} + 8q^{5} - 4q^{6} - 2q^{7} - 4q^{11} - 20q^{12} + 2q^{13} - 18q^{14} - 2q^{15} - 32q^{16} - 16q^{17} - 12q^{18} - 4q^{19} - 12q^{20} - 4q^{21} + 12q^{23} - 14q^{24} + 64q^{25} - 8q^{26} + 6q^{27} - 8q^{28} - 10q^{29} - 4q^{30} + 8q^{31} + 20q^{32} - 4q^{33} - 60q^{36} + 2q^{37} + 88q^{38} - 28q^{39} + 10q^{41} - 36q^{42} + 8q^{43} - 14q^{44} + 6q^{45} - 6q^{46} - 52q^{47} - 64q^{48} - 14q^{49} - 46q^{51} - 16q^{52} + 120q^{54} + 102q^{56} + 26q^{57} - 10q^{59} + 30q^{60} + 8q^{61} - 24q^{62} + 26q^{63} + 64q^{64} + 2q^{65} + 16q^{66} + 14q^{67} + 116q^{68} - 24q^{69} + 6q^{70} + 48q^{71} - 28q^{72} - 28q^{73} - 88q^{74} - 16q^{76} + 10q^{77} + 40q^{78} + 8q^{79} - 28q^{80} + 44q^{81} - 68q^{83} + 54q^{84} + 6q^{85} - 4q^{86} - 22q^{87} - 14q^{89} + 18q^{90} - 22q^{91} - 100q^{92} - 36q^{93} + 12q^{94} + 64q^{96} + 2q^{97} - 10q^{98} - 90q^{99} + O(q^{100})$$

## 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.k.a $$4$$ $$2.515$$ $$\Q(\sqrt{-3}, \sqrt{13})$$ None $$1$$ $$0$$ $$-4$$ $$-8$$ $$q+\beta _{1}q^{2}+(1+2\beta _{2})q^{3}+(-1+\beta _{1}+\cdots)q^{4}+\cdots$$
315.2.k.b $$24$$ $$2.515$$ None $$-1$$ $$1$$ $$-24$$ $$7$$
315.2.k.c $$36$$ $$2.515$$ None $$0$$ $$-1$$ $$36$$ $$-1$$

## 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

$p$ $F_p(T)$
$2$ ($$1 - T + 3 T^{3} - 5 T^{4} + 6 T^{5} - 8 T^{7} + 16 T^{8}$$)
$3$ ($$( 1 + 3 T^{2} )^{2}$$)
$5$ ($$( 1 + T )^{4}$$)
$7$ ($$( 1 + 4 T + 7 T^{2} )^{2}$$)
$11$ ($$( 1 + 11 T^{2} )^{4}$$)
$13$ ($$1 - 4 T - T^{2} + 36 T^{3} - 88 T^{4} + 468 T^{5} - 169 T^{6} - 8788 T^{7} + 28561 T^{8}$$)
$17$ ($$1 + 4 T - 9 T^{2} - 36 T^{3} + 64 T^{4} - 612 T^{5} - 2601 T^{6} + 19652 T^{7} + 83521 T^{8}$$)
$19$ ($$1 - 25 T^{2} + 264 T^{4} - 9025 T^{6} + 130321 T^{8}$$)
$23$ ($$( 1 - 4 T - 2 T^{2} - 92 T^{3} + 529 T^{4} )^{2}$$)
$29$ ($$1 + 2 T - 3 T^{2} - 102 T^{3} - 908 T^{4} - 2958 T^{5} - 2523 T^{6} + 48778 T^{7} + 707281 T^{8}$$)
$31$ ($$1 - 49 T^{2} + 1440 T^{4} - 47089 T^{6} + 923521 T^{8}$$)
$37$ ($$1 - 61 T^{2} + 2352 T^{4} - 83509 T^{6} + 1874161 T^{8}$$)
$41$ ($$( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} )^{2}$$)
$43$ ($$1 - 6 T - 7 T^{2} + 258 T^{3} - 1548 T^{4} + 11094 T^{5} - 12943 T^{6} - 477042 T^{7} + 3418801 T^{8}$$)
$47$ ($$1 - 2 T - 39 T^{2} + 102 T^{3} - 548 T^{4} + 4794 T^{5} - 86151 T^{6} - 207646 T^{7} + 4879681 T^{8}$$)
$53$ ($$1 + 8 T - 45 T^{2} + 24 T^{3} + 5680 T^{4} + 1272 T^{5} - 126405 T^{6} + 1191016 T^{7} + 7890481 T^{8}$$)
$59$ ($$1 - 16 T + 87 T^{2} - 816 T^{3} + 9976 T^{4} - 48144 T^{5} + 302847 T^{6} - 3286064 T^{7} + 12117361 T^{8}$$)
$61$ ($$1 - 6 T - 43 T^{2} + 258 T^{3} + 324 T^{4} + 15738 T^{5} - 160003 T^{6} - 1361886 T^{7} + 13845841 T^{8}$$)
$67$ ($$1 - 6 T - 55 T^{2} + 258 T^{3} + 1380 T^{4} + 17286 T^{5} - 246895 T^{6} - 1804578 T^{7} + 20151121 T^{8}$$)
$71$ ($$( 1 + 71 T^{2} )^{4}$$)
$73$ ($$1 - 133 T^{2} + 12360 T^{4} - 708757 T^{6} + 28398241 T^{8}$$)
$79$ ($$1 + 4 T - 29 T^{2} - 452 T^{3} - 5480 T^{4} - 35708 T^{5} - 180989 T^{6} + 1972156 T^{7} + 38950081 T^{8}$$)
$83$ ($$1 - 14 T + 33 T^{2} + 42 T^{3} + 3412 T^{4} + 3486 T^{5} + 227337 T^{6} - 8005018 T^{7} + 47458321 T^{8}$$)
$89$ ($$( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} )^{2}$$)
$97$ ($$1 + 20 T + 119 T^{2} + 1740 T^{3} + 30752 T^{4} + 168780 T^{5} + 1119671 T^{6} + 18253460 T^{7} + 88529281 T^{8}$$)