# Properties

 Label 630.2.z Level 630 Weight 2 Character orbit z Rep. character $$\chi_{630}(169,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 72 Newform subspaces 3 Sturm bound 288 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 630.z (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$45$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$3$$ Sturm bound: $$288$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(630, [\chi])$$.

Total New Old
Modular forms 304 72 232
Cusp forms 272 72 200
Eisenstein series 32 0 32

## Trace form

 $$72q + 36q^{4} + 4q^{5} + 16q^{9} + O(q^{10})$$ $$72q + 36q^{4} + 4q^{5} + 16q^{9} - 12q^{11} + 8q^{14} + 20q^{15} - 36q^{16} - 4q^{20} + 4q^{21} - 12q^{25} + 16q^{29} - 16q^{30} + 24q^{31} + 20q^{36} - 4q^{39} - 16q^{41} - 24q^{44} - 24q^{45} + 36q^{49} + 4q^{50} - 112q^{51} - 72q^{54} + 24q^{55} - 8q^{56} - 16q^{59} + 4q^{60} - 72q^{64} + 48q^{65} + 40q^{66} - 16q^{69} + 88q^{71} + 56q^{74} + 12q^{79} - 8q^{80} + 32q^{81} + 8q^{84} + 24q^{85} - 64q^{86} + 160q^{89} + 60q^{90} + 24q^{91} - 12q^{95} - 108q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(630, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
630.2.z.a $$4$$ $$5.031$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+\zeta_{12}q^{2}+(-2\zeta_{12}+\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots$$
630.2.z.b $$24$$ $$5.031$$ None $$0$$ $$0$$ $$-2$$ $$0$$
630.2.z.c $$44$$ $$5.031$$ None $$0$$ $$0$$ $$2$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(630, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(630, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(315, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - T^{2} + T^{4}$$)
$3$ ($$( 1 - 3 T^{2} )^{2}$$)
$5$ ($$1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4}$$)
$7$ ($$1 - T^{2} + T^{4}$$)
$11$ ($$( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} )^{2}$$)
$13$ ($$1 + 25 T^{2} + 456 T^{4} + 4225 T^{6} + 28561 T^{8}$$)
$17$ ($$( 1 - 33 T^{2} + 289 T^{4} )^{2}$$)
$19$ ($$( 1 - 2 T + 19 T^{2} )^{4}$$)
$23$ ($$( 1 + 23 T^{2} + 529 T^{4} )^{2}$$)
$29$ ($$( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} )^{2}$$)
$31$ ($$( 1 - 31 T^{2} + 961 T^{4} )^{2}$$)
$37$ ($$( 1 - 10 T^{2} + 1369 T^{4} )^{2}$$)
$41$ ($$( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} )^{2}$$)
$43$ ($$1 + 70 T^{2} + 3051 T^{4} + 129430 T^{6} + 3418801 T^{8}$$)
$47$ ($$1 + 93 T^{2} + 6440 T^{4} + 205437 T^{6} + 4879681 T^{8}$$)
$53$ ($$( 1 - 102 T^{2} + 2809 T^{4} )^{2}$$)
$59$ ($$( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} )^{2}$$)
$61$ ($$( 1 - 13 T + 61 T^{2} )^{2}( 1 - T + 61 T^{2} )^{2}$$)
$67$ ($$1 - 10 T^{2} - 4389 T^{4} - 44890 T^{6} + 20151121 T^{8}$$)
$71$ ($$( 1 - 11 T + 71 T^{2} )^{4}$$)
$73$ ($$( 1 - 25 T^{2} + 5329 T^{4} )^{2}$$)
$79$ ($$( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} )^{2}$$)
$83$ ($$1 + 45 T^{2} - 4864 T^{4} + 310005 T^{6} + 47458321 T^{8}$$)
$89$ ($$( 1 - 10 T + 89 T^{2} )^{4}$$)
$97$ ($$1 + 25 T^{2} - 8784 T^{4} + 235225 T^{6} + 88529281 T^{8}$$)