# Properties

 Label 630.2.g Level 630 Weight 2 Character orbit g Rep. character $$\chi_{630}(379,\cdot)$$ Character field $$\Q$$ Dimension 16 Newform subspaces 7 Sturm bound 288 Trace bound 11

# Related objects

## Defining parameters

 Level: $$N$$ = $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 630.g (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$5$$ Character field: $$\Q$$ Newform subspaces: $$7$$ Sturm bound: $$288$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$11$$, $$29$$

## Dimensions

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

Total New Old
Modular forms 160 16 144
Cusp forms 128 16 112
Eisenstein series 32 0 32

## Trace form

 $$16q - 16q^{4} - 4q^{5} + O(q^{10})$$ $$16q - 16q^{4} - 4q^{5} + 4q^{14} + 16q^{16} + 16q^{19} + 4q^{20} - 12q^{25} - 8q^{26} + 8q^{29} + 32q^{31} + 4q^{35} + 8q^{41} + 8q^{46} - 16q^{49} - 12q^{50} + 8q^{55} - 4q^{56} - 48q^{59} - 48q^{61} - 16q^{64} + 20q^{65} + 4q^{70} + 64q^{71} + 16q^{74} - 16q^{76} + 40q^{79} - 4q^{80} - 16q^{85} - 8q^{86} - 40q^{89} - 8q^{91} - 4q^{95} + O(q^{100})$$

## 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.g.a $$2$$ $$5.031$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+iq^{2}-q^{4}+(-2-i)q^{5}+iq^{7}+\cdots$$
630.2.g.b $$2$$ $$5.031$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q+iq^{2}-q^{4}+(-1+2i)q^{5}-iq^{7}+\cdots$$
630.2.g.c $$2$$ $$5.031$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q+iq^{2}-q^{4}+(-1-2i)q^{5}+iq^{7}+\cdots$$
630.2.g.d $$2$$ $$5.031$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+iq^{2}-q^{4}+(1+2i)q^{5}-iq^{7}-iq^{8}+\cdots$$
630.2.g.e $$2$$ $$5.031$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+iq^{2}-q^{4}+(1+2i)q^{5}+iq^{7}-iq^{8}+\cdots$$
630.2.g.f $$2$$ $$5.031$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+iq^{2}-q^{4}+(2-i)q^{5}-iq^{7}-iq^{8}+\cdots$$
630.2.g.g $$4$$ $$5.031$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q-\beta _{2}q^{2}-q^{4}+(-1-\beta _{1}+\beta _{2})q^{5}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(630, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(315, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$( 1 + T^{2} )^{2}$$)
$3$ 1
$5$ ($$1 + 4 T + 5 T^{2}$$)($$1 + 2 T + 5 T^{2}$$)($$1 + 2 T + 5 T^{2}$$)($$1 - 2 T + 5 T^{2}$$)($$1 - 2 T + 5 T^{2}$$)($$1 - 4 T + 5 T^{2}$$)($$1 + 4 T + 8 T^{2} + 20 T^{3} + 25 T^{4}$$)
$7$ ($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$1 + T^{2}$$)($$( 1 + T^{2} )^{2}$$)
$11$ ($$( 1 + 11 T^{2} )^{2}$$)($$( 1 + 6 T + 11 T^{2} )^{2}$$)($$( 1 + 2 T + 11 T^{2} )^{2}$$)($$( 1 - 2 T + 11 T^{2} )^{2}$$)($$( 1 - 6 T + 11 T^{2} )^{2}$$)($$( 1 + 11 T^{2} )^{2}$$)($$( 1 - 2 T^{2} + 121 T^{4} )^{2}$$)
$13$ ($$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$1 - 22 T^{2} + 169 T^{4}$$)($$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$)($$1 - 32 T^{2} + 498 T^{4} - 5408 T^{6} + 28561 T^{8}$$)
$17$ ($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$1 - 18 T^{2} + 289 T^{4}$$)($$( 1 - 2 T + 17 T^{2} )( 1 + 2 T + 17 T^{2} )$$)($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$)($$( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2}$$)
$19$ ($$( 1 - 8 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 - 6 T + 19 T^{2} )^{2}$$)($$( 1 - 2 T + 19 T^{2} )^{2}$$)($$( 1 + 4 T + 19 T^{2} )^{2}$$)($$( 1 - 8 T + 19 T^{2} )^{2}$$)($$( 1 + 8 T + 48 T^{2} + 152 T^{3} + 361 T^{4} )^{2}$$)
$23$ ($$1 + 18 T^{2} + 529 T^{4}$$)($$1 - 30 T^{2} + 529 T^{4}$$)($$1 + 18 T^{2} + 529 T^{4}$$)($$( 1 - 23 T^{2} )^{2}$$)($$1 - 30 T^{2} + 529 T^{4}$$)($$1 + 18 T^{2} + 529 T^{4}$$)($$1 - 36 T^{2} + 998 T^{4} - 19044 T^{6} + 279841 T^{8}$$)
$29$ ($$( 1 + 8 T + 29 T^{2} )^{2}$$)($$( 1 - 2 T + 29 T^{2} )^{2}$$)($$( 1 - 6 T + 29 T^{2} )^{2}$$)($$( 1 + 6 T + 29 T^{2} )^{2}$$)($$( 1 + 2 T + 29 T^{2} )^{2}$$)($$( 1 - 8 T + 29 T^{2} )^{2}$$)($$( 1 - 4 T + 38 T^{2} - 116 T^{3} + 841 T^{4} )^{2}$$)
$31$ ($$( 1 - 4 T + 31 T^{2} )^{2}$$)($$( 1 + 2 T + 31 T^{2} )^{2}$$)($$( 1 + 2 T + 31 T^{2} )^{2}$$)($$( 1 - 6 T + 31 T^{2} )^{2}$$)($$( 1 + 2 T + 31 T^{2} )^{2}$$)($$( 1 - 4 T + 31 T^{2} )^{2}$$)($$( 1 - 8 T + 54 T^{2} - 248 T^{3} + 961 T^{4} )^{2}$$)
$37$ ($$1 - 10 T^{2} + 1369 T^{4}$$)($$1 + 26 T^{2} + 1369 T^{4}$$)($$1 - 58 T^{2} + 1369 T^{4}$$)($$1 - 10 T^{2} + 1369 T^{4}$$)($$1 + 26 T^{2} + 1369 T^{4}$$)($$1 - 10 T^{2} + 1369 T^{4}$$)($$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$)
$41$ ($$( 1 - 12 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 + 2 T + 41 T^{2} )^{2}$$)($$( 1 + 6 T + 41 T^{2} )^{2}$$)($$( 1 - 6 T + 41 T^{2} )^{2}$$)($$( 1 + 12 T + 41 T^{2} )^{2}$$)($$( 1 - 12 T + 94 T^{2} - 492 T^{3} + 1681 T^{4} )^{2}$$)
$43$ ($$1 - 22 T^{2} + 1849 T^{4}$$)($$1 - 82 T^{2} + 1849 T^{4}$$)($$1 - 70 T^{2} + 1849 T^{4}$$)($$1 - 22 T^{2} + 1849 T^{4}$$)($$1 - 82 T^{2} + 1849 T^{4}$$)($$1 - 22 T^{2} + 1849 T^{4}$$)($$1 - 92 T^{2} + 4278 T^{4} - 170108 T^{6} + 3418801 T^{8}$$)
$47$ ($$1 - 78 T^{2} + 2209 T^{4}$$)($$1 - 90 T^{2} + 2209 T^{4}$$)($$1 - 30 T^{2} + 2209 T^{4}$$)($$1 - 78 T^{2} + 2209 T^{4}$$)($$1 - 90 T^{2} + 2209 T^{4}$$)($$1 - 78 T^{2} + 2209 T^{4}$$)($$1 - 108 T^{2} + 5798 T^{4} - 238572 T^{6} + 4879681 T^{8}$$)
$53$ ($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 102 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 70 T^{2} + 2809 T^{4}$$)($$1 - 92 T^{2} + 4278 T^{4} - 258428 T^{6} + 7890481 T^{8}$$)
$59$ ($$( 1 + 8 T + 59 T^{2} )^{2}$$)($$( 1 + 4 T + 59 T^{2} )^{2}$$)($$( 1 + 8 T + 59 T^{2} )^{2}$$)($$( 1 + 8 T + 59 T^{2} )^{2}$$)($$( 1 - 4 T + 59 T^{2} )^{2}$$)($$( 1 - 8 T + 59 T^{2} )^{2}$$)($$( 1 + 8 T + 128 T^{2} + 472 T^{3} + 3481 T^{4} )^{2}$$)
$61$ ($$( 1 + 6 T + 61 T^{2} )^{2}$$)($$( 1 + 12 T + 61 T^{2} )^{2}$$)($$( 1 + 10 T + 61 T^{2} )^{2}$$)($$( 1 - 10 T + 61 T^{2} )^{2}$$)($$( 1 + 12 T + 61 T^{2} )^{2}$$)($$( 1 + 6 T + 61 T^{2} )^{2}$$)($$( 1 - 12 T + 152 T^{2} - 732 T^{3} + 3721 T^{4} )^{2}$$)
$67$ ($$1 - 70 T^{2} + 4489 T^{4}$$)($$1 - 34 T^{2} + 4489 T^{4}$$)($$1 - 70 T^{2} + 4489 T^{4}$$)($$1 + 10 T^{2} + 4489 T^{4}$$)($$1 - 34 T^{2} + 4489 T^{4}$$)($$1 - 70 T^{2} + 4489 T^{4}$$)($$( 1 - 70 T^{2} + 4489 T^{4} )^{2}$$)
$71$ ($$( 1 + 71 T^{2} )^{2}$$)($$( 1 + 12 T + 71 T^{2} )^{2}$$)($$( 1 - 6 T + 71 T^{2} )^{2}$$)($$( 1 - 14 T + 71 T^{2} )^{2}$$)($$( 1 - 12 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 - 12 T + 154 T^{2} - 852 T^{3} + 5041 T^{4} )^{2}$$)
$73$ ($$1 - 130 T^{2} + 5329 T^{4}$$)($$1 - 142 T^{2} + 5329 T^{4}$$)($$1 + 50 T^{2} + 5329 T^{4}$$)($$1 - 46 T^{2} + 5329 T^{4}$$)($$1 - 142 T^{2} + 5329 T^{4}$$)($$1 - 130 T^{2} + 5329 T^{4}$$)($$1 - 236 T^{2} + 24198 T^{4} - 1257644 T^{6} + 28398241 T^{8}$$)
$79$ ($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 - 16 T + 79 T^{2} )^{2}$$)($$( 1 - 12 T + 79 T^{2} )^{2}$$)($$( 1 + 4 T + 79 T^{2} )^{2}$$)($$( 1 - 16 T + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 + 4 T + 138 T^{2} + 316 T^{3} + 6241 T^{4} )^{2}$$)
$83$ ($$( 1 - 83 T^{2} )^{2}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$1 - 102 T^{2} + 6889 T^{4}$$)($$1 + 90 T^{2} + 6889 T^{4}$$)($$1 - 22 T^{2} + 6889 T^{4}$$)($$( 1 - 83 T^{2} )^{2}$$)($$( 1 - 160 T^{2} + 6889 T^{4} )^{2}$$)
$89$ ($$( 1 - 4 T + 89 T^{2} )^{2}$$)($$( 1 - 14 T + 89 T^{2} )^{2}$$)($$( 1 + 10 T + 89 T^{2} )^{2}$$)($$( 1 - 10 T + 89 T^{2} )^{2}$$)($$( 1 + 14 T + 89 T^{2} )^{2}$$)($$( 1 + 4 T + 89 T^{2} )^{2}$$)($$( 1 + 10 T + 89 T^{2} )^{4}$$)
$97$ ($$1 - 50 T^{2} + 9409 T^{4}$$)($$( 1 - 8 T + 97 T^{2} )( 1 + 8 T + 97 T^{2} )$$)($$1 - 94 T^{2} + 9409 T^{4}$$)($$1 - 94 T^{2} + 9409 T^{4}$$)($$( 1 - 8 T + 97 T^{2} )( 1 + 8 T + 97 T^{2} )$$)($$1 - 50 T^{2} + 9409 T^{4}$$)($$1 - 124 T^{2} + 8838 T^{4} - 1166716 T^{6} + 88529281 T^{8}$$)