# Properties

 Label 630.2.t Level 630 Weight 2 Character orbit t Rep. character $$\chi_{630}(311,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 64 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.t (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$63$$ 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 64 240
Cusp forms 272 64 208
Eisenstein series 32 0 32

## Trace form

 $$64q + 32q^{4} - 4q^{7} + 6q^{9} + O(q^{10})$$ $$64q + 32q^{4} - 4q^{7} + 6q^{9} + 12q^{13} + 6q^{14} + 4q^{15} - 32q^{16} - 8q^{18} + 22q^{21} + 6q^{24} + 64q^{25} + 24q^{26} + 36q^{27} + 4q^{28} - 6q^{29} + 4q^{30} - 12q^{31} + 24q^{33} + 4q^{37} + 36q^{39} - 6q^{41} - 32q^{42} + 4q^{43} - 12q^{44} + 6q^{45} - 6q^{46} + 36q^{47} - 8q^{49} + 8q^{51} - 72q^{53} - 36q^{54} - 12q^{57} - 60q^{59} + 8q^{60} + 102q^{61} - 80q^{63} - 64q^{64} - 12q^{65} - 48q^{66} - 28q^{67} - 6q^{70} + 8q^{72} - 12q^{77} + 16q^{78} + 8q^{79} - 38q^{81} + 14q^{84} + 12q^{85} + 96q^{87} + 6q^{89} + 24q^{91} - 36q^{92} + 8q^{93} + 6q^{96} + 12q^{97} - 48q^{98} - 48q^{99} + O(q^{100})$$

## 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.t.a $$4$$ $$5.031$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$-4$$ $$2$$ $$q-\zeta_{12}q^{2}+(-\zeta_{12}+2\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots$$
630.2.t.b $$28$$ $$5.031$$ None $$0$$ $$-2$$ $$-28$$ $$-4$$
630.2.t.c $$32$$ $$5.031$$ None $$0$$ $$2$$ $$32$$ $$-2$$

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

$$S_{2}^{\mathrm{old}}(630, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(315, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 - T^{2} + T^{4}$$)
$3$ ($$1 + 3 T^{2} + 9 T^{4}$$)
$5$ ($$( 1 + T )^{4}$$)
$7$ ($$( 1 - T + 7 T^{2} )^{2}$$)
$11$ ($$1 - 20 T^{2} + 234 T^{4} - 2420 T^{6} + 14641 T^{8}$$)
$13$ ($$( 1 - 6 T + 25 T^{2} - 78 T^{3} + 169 T^{4} )^{2}$$)
$17$ ($$( 1 - 17 T^{2} + 289 T^{4} )^{2}$$)
$19$ ($$1 - 6 T + 44 T^{2} - 192 T^{3} + 891 T^{4} - 3648 T^{5} + 15884 T^{6} - 41154 T^{7} + 130321 T^{8}$$)
$23$ ($$1 + 4 T^{2} - 666 T^{4} + 2116 T^{6} + 279841 T^{8}$$)
$29$ ($$1 + 12 T + 109 T^{2} + 732 T^{3} + 4272 T^{4} + 21228 T^{5} + 91669 T^{6} + 292668 T^{7} + 707281 T^{8}$$)
$31$ ($$1 + 18 T + 188 T^{2} + 1440 T^{3} + 8787 T^{4} + 44640 T^{5} + 180668 T^{6} + 536238 T^{7} + 923521 T^{8}$$)
$37$ ($$1 + 2 T - 44 T^{2} - 52 T^{3} + 787 T^{4} - 1924 T^{5} - 60236 T^{6} + 101306 T^{7} + 1874161 T^{8}$$)
$41$ ($$( 1 + 9 T + 40 T^{2} + 369 T^{3} + 1681 T^{4} )^{2}$$)
$43$ ($$1 - 4 T - 47 T^{2} + 92 T^{3} + 1432 T^{4} + 3956 T^{5} - 86903 T^{6} - 318028 T^{7} + 3418801 T^{8}$$)
$47$ ($$( 1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4} )^{2}$$)
$53$ ($$1 - 6 T + 40 T^{2} - 168 T^{3} - 1389 T^{4} - 8904 T^{5} + 112360 T^{6} - 893262 T^{7} + 7890481 T^{8}$$)
$59$ ($$1 + 6 T - 64 T^{2} - 108 T^{3} + 4395 T^{4} - 6372 T^{5} - 222784 T^{6} + 1232274 T^{7} + 12117361 T^{8}$$)
$61$ ($$1 - 24 T + 326 T^{2} - 3216 T^{3} + 25947 T^{4} - 196176 T^{5} + 1213046 T^{6} - 5447544 T^{7} + 13845841 T^{8}$$)
$67$ ($$( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} )^{2}$$)
$71$ ($$1 - 260 T^{2} + 26874 T^{4} - 1310660 T^{6} + 25411681 T^{8}$$)
$73$ ($$( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} )^{2}$$)
$79$ ($$1 - 2 T + 88 T^{2} + 484 T^{3} + 499 T^{4} + 38236 T^{5} + 549208 T^{6} - 986078 T^{7} + 38950081 T^{8}$$)
$83$ ($$1 - 12 T - 31 T^{2} - 108 T^{3} + 11784 T^{4} - 8964 T^{5} - 213559 T^{6} - 6861444 T^{7} + 47458321 T^{8}$$)
$89$ ($$1 - 12 T + 38 T^{2} + 864 T^{3} - 9501 T^{4} + 76896 T^{5} + 300998 T^{6} - 8459628 T^{7} + 62742241 T^{8}$$)
$97$ ($$1 + 12 T + 110 T^{2} + 744 T^{3} - 909 T^{4} + 72168 T^{5} + 1034990 T^{6} + 10952076 T^{7} + 88529281 T^{8}$$)