# Properties

 Label 630.2.ce Level 630 Weight 2 Character orbit ce Rep. character $$\chi_{630}(53,\cdot)$$ Character field $$\Q(\zeta_{12})$$ Dimension 64 Newform subspaces 3 Sturm bound 288 Trace bound 7

# Related objects

## Defining parameters

 Level: $$N$$ = $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 630.ce (of order $$12$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$105$$ Character field: $$\Q(\zeta_{12})$$ Newform subspaces: $$3$$ Sturm bound: $$288$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$11$$

## Dimensions

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

Total New Old
Modular forms 640 64 576
Cusp forms 512 64 448
Eisenstein series 128 0 128

## Trace form

 $$64q - 8q^{7} + O(q^{10})$$ $$64q - 8q^{7} - 8q^{10} + 32q^{16} + 48q^{22} + 16q^{25} - 8q^{28} - 64q^{31} + 16q^{37} + 32q^{43} + 32q^{55} + 8q^{58} - 96q^{61} + 32q^{67} + 24q^{70} + 64q^{73} + 64q^{76} - 32q^{82} - 64q^{85} - 24q^{88} + 48q^{91} - 208q^{97} + O(q^{100})$$

## 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.ce.a $$16$$ $$5.031$$ 16.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+(\beta _{4}-\beta _{13})q^{2}+(\beta _{2}+\beta _{10})q^{4}+(-2\beta _{4}+\cdots)q^{5}+\cdots$$
630.2.ce.b $$16$$ $$5.031$$ 16.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$12$$ $$q-\beta _{11}q^{2}+\beta _{9}q^{4}+(-2\beta _{11}+\beta _{15})q^{5}+\cdots$$
630.2.ce.c $$32$$ $$5.031$$ None $$0$$ $$0$$ $$0$$ $$-12$$

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

$$S_{2}^{\mathrm{old}}(630, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(315, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - T^{4} + T^{8} )^{2}$$)($$( 1 - T^{4} + T^{8} )^{2}$$)
$3$ ()()
$5$ ($$( 1 - 8 T^{2} + 39 T^{4} - 200 T^{6} + 625 T^{8} )^{2}$$)($$1 + 49 T^{4} + 1776 T^{8} + 30625 T^{12} + 390625 T^{16}$$)
$7$ ($$( 1 + 2 T + 2 T^{2} + 14 T^{3} + 49 T^{4} )^{4}$$)($$( 1 - 6 T + 18 T^{2} - 36 T^{3} + 71 T^{4} - 252 T^{5} + 882 T^{6} - 2058 T^{7} + 2401 T^{8} )^{2}$$)
$11$ ($$( 1 + 4 T^{2} - 191 T^{4} - 140 T^{6} + 26272 T^{8} - 16940 T^{10} - 2796431 T^{12} + 7086244 T^{14} + 214358881 T^{16} )^{2}$$)($$( 1 + 32 T^{2} + 537 T^{4} + 7840 T^{6} + 98624 T^{8} + 948640 T^{10} + 7862217 T^{12} + 56689952 T^{14} + 214358881 T^{16} )^{2}$$)
$13$ ($$( 1 + 6 T + 18 T^{2} - 12 T^{3} - 217 T^{4} - 156 T^{5} + 3042 T^{6} + 13182 T^{7} + 28561 T^{8} )^{4}$$)($$( 1 - 6 T + 13 T^{2} )^{8}( 1 + 4 T + 13 T^{2} )^{8}$$)
$17$ ($$1 + 772 T^{4} + 302410 T^{8} + 97682704 T^{12} + 29432756371 T^{16} + 8158557120784 T^{20} + 2109538807732810 T^{24} + 449784367141375492 T^{28} + 48661191875666868481 T^{32}$$)($$1 + 100 T^{4} - 74358 T^{8} - 8268400 T^{12} - 694695277 T^{16} - 690585036400 T^{20} - 518703371797878 T^{24} + 58262223722976100 T^{28} + 48661191875666868481 T^{32}$$)
$19$ ($$( 1 + 11 T^{2} + 361 T^{4} )^{4}( 1 + 26 T^{2} + 361 T^{4} )^{4}$$)($$( 1 + 36 T^{2} + 646 T^{4} - 2592 T^{6} - 175677 T^{8} - 935712 T^{10} + 84187366 T^{12} + 1693651716 T^{14} + 16983563041 T^{16} )^{2}$$)
$23$ ($$1 + 850 T^{4} + 4657 T^{8} + 134436850 T^{12} + 234802464868 T^{16} + 37620942540850 T^{20} + 364694258453617 T^{24} + 18627430767217272850 T^{28} +$$$$61\!\cdots\!61$$$$T^{32}$$)($$1 - 284 T^{4} - 59190 T^{8} + 119233424 T^{12} - 81324937261 T^{16} + 33366400605584 T^{20} - 4635227218782390 T^{24} - 6223753338693771164 T^{28} +$$$$61\!\cdots\!61$$$$T^{32}$$)
$29$ ($$( 1 + 50 T^{2} + 841 T^{4} )^{8}$$)($$( 1 + 80 T^{2} + 3007 T^{4} + 67280 T^{6} + 707281 T^{8} )^{4}$$)
$31$ ($$( 1 - 7 T + 31 T^{2} )^{8}( 1 + 11 T + 31 T^{2} )^{8}$$)($$( 1 - 3 T - 22 T^{2} - 93 T^{3} + 961 T^{4} )^{8}$$)
$37$ ($$( 1 - 18 T + 162 T^{2} - 828 T^{3} + 1165 T^{4} + 18576 T^{5} - 180306 T^{6} + 1048086 T^{7} - 5780316 T^{8} + 38779182 T^{9} - 246838914 T^{10} + 940930128 T^{11} + 2183397565 T^{12} - 57416796396 T^{13} + 415647678258 T^{14} - 1708773788394 T^{15} + 3512479453921 T^{16} )^{2}$$)($$( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} )^{4}( 1 + 12 T + 107 T^{2} + 444 T^{3} + 1369 T^{4} )^{4}$$)
$41$ ($$( 1 - 100 T^{2} + 4887 T^{4} - 168100 T^{6} + 2825761 T^{8} )^{4}$$)($$( 1 - 74 T^{2} + 1681 T^{4} )^{8}$$)
$43$ ($$( 1 - 6 T + 18 T^{2} - 258 T^{3} + 1849 T^{4} )^{8}$$)($$( 1 + 4 T + 8 T^{2} - 172 T^{3} - 3698 T^{4} - 7396 T^{5} + 14792 T^{6} + 318028 T^{7} + 3418801 T^{8} )^{4}$$)
$47$ ($$( 1 + 4249 T^{4} + 13174320 T^{8} + 20733764569 T^{12} + 23811286661761 T^{16} )^{2}$$)($$1 - 2908 T^{4} + 5256970 T^{8} + 19076096144 T^{12} - 54839371513709 T^{16} + 93085263908050064 T^{20} +$$$$12\!\cdots\!70$$$$T^{24} -$$$$33\!\cdots\!28$$$$T^{28} +$$$$56\!\cdots\!21$$$$T^{32}$$)
$53$ ($$1 + 1042 T^{4} + 11001745 T^{8} - 26776214606 T^{12} + 27499788568516 T^{16} - 211277212600565486 T^{20} +$$$$68\!\cdots\!45$$$$T^{24} +$$$$51\!\cdots\!22$$$$T^{28} +$$$$38\!\cdots\!21$$$$T^{32}$$)($$1 - 302 T^{4} + 8381841 T^{8} + 7269622898 T^{12} + 5755328025284 T^{16} + 57360821353833938 T^{20} +$$$$52\!\cdots\!01$$$$T^{24} -$$$$14\!\cdots\!82$$$$T^{28} +$$$$38\!\cdots\!21$$$$T^{32}$$)
$59$ ($$( 1 - 64 T^{2} - 1394 T^{4} + 94208 T^{6} + 3764563 T^{8} + 327938048 T^{10} - 16891601234 T^{12} - 2699554153024 T^{14} + 146830437604321 T^{16} )^{2}$$)($$( 1 - 88 T^{2} + 3697 T^{4} + 256520 T^{6} - 22696016 T^{8} + 892946120 T^{10} + 44797883617 T^{12} - 3711886960408 T^{14} + 146830437604321 T^{16} )^{2}$$)
$61$ ($$( 1 + 6 T - 56 T^{2} - 180 T^{3} + 2547 T^{4} - 10980 T^{5} - 208376 T^{6} + 1361886 T^{7} + 13845841 T^{8} )^{4}$$)($$( 1 + 6 T - 84 T^{2} - 12 T^{3} + 8483 T^{4} - 732 T^{5} - 312564 T^{6} + 1361886 T^{7} + 13845841 T^{8} )^{4}$$)
$67$ ($$( 1 - 14 T + 98 T^{2} + 504 T^{3} - 8017 T^{4} + 33768 T^{5} + 439922 T^{6} - 4210682 T^{7} + 20151121 T^{8} )^{4}$$)($$( 1 + 16 T + 128 T^{2} + 1696 T^{3} + 16462 T^{4} + 126928 T^{5} + 1361920 T^{6} + 10477584 T^{7} + 66591603 T^{8} + 701998128 T^{9} + 6113658880 T^{10} + 38175246064 T^{11} + 331727753902 T^{12} + 2289812181472 T^{13} + 11578672917632 T^{14} + 96971385685168 T^{15} + 406067677556641 T^{16} )^{2}$$)
$71$ ($$( 1 - 124 T^{2} + 13302 T^{4} - 625084 T^{6} + 25411681 T^{8} )^{4}$$)($$( 1 - 224 T^{2} + 21922 T^{4} - 1129184 T^{6} + 25411681 T^{8} )^{4}$$)
$73$ ($$( 1 + 16 T + 128 T^{2} + 1184 T^{3} + 10130 T^{4} + 107888 T^{5} + 1130496 T^{6} + 8780592 T^{7} + 70277539 T^{8} + 640983216 T^{9} + 6024413184 T^{10} + 41970266096 T^{11} + 287674181330 T^{12} + 2454516766112 T^{13} + 19370780964992 T^{14} + 176758376305552 T^{15} + 806460091894081 T^{16} )^{2}$$)($$( 1 - 24 T + 288 T^{2} - 1008 T^{3} - 15662 T^{4} + 266712 T^{5} - 1382400 T^{6} - 3986376 T^{7} + 112103139 T^{8} - 291005448 T^{9} - 7366809600 T^{10} + 103755502104 T^{11} - 444773250542 T^{12} - 2089656165744 T^{13} + 43584257171232 T^{14} - 265137564458328 T^{15} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$( 1 + 76 T^{2} + 4486 T^{4} - 850592 T^{6} - 73333997 T^{8} - 5308544672 T^{10} + 174730063366 T^{12} + 18474646619596 T^{14} + 1517108809906561 T^{16} )^{2}$$)($$( 1 + 210 T^{2} + 22177 T^{4} + 1982610 T^{6} + 166439748 T^{8} + 12373469010 T^{10} + 863795946337 T^{12} + 51048365659410 T^{14} + 1517108809906561 T^{16} )^{2}$$)
$83$ ($$( 1 + 3100 T^{4} + 19121958 T^{8} + 147120795100 T^{12} + 2252292232139041 T^{16} )^{2}$$)($$( 1 - 25906 T^{4} + 262356115 T^{8} - 1229455263826 T^{12} + 2252292232139041 T^{16} )^{2}$$)
$89$ ($$( 1 - 80 T^{2} - 1521 T^{4} - 633680 T^{6} + 62742241 T^{8} )^{4}$$)($$( 1 + 44 T^{2} - 12806 T^{4} - 48400 T^{6} + 139249267 T^{8} - 383376400 T^{10} - 803477138246 T^{12} + 21867176802284 T^{14} + 3936588805702081 T^{16} )^{2}$$)
$97$ ($$( 1 + 12 T + 72 T^{2} + 444 T^{3} - 862 T^{4} + 43068 T^{5} + 677448 T^{6} + 10952076 T^{7} + 88529281 T^{8} )^{4}$$)($$( 1 + 30 T + 450 T^{2} + 6120 T^{3} + 71783 T^{4} + 593640 T^{5} + 4234050 T^{6} + 27380190 T^{7} + 88529281 T^{8} )^{4}$$)