# Properties

 Label 630.2.m Level 630 Weight 2 Character orbit m Rep. character $$\chi_{630}(197,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 24 Newform subspaces 4 Sturm bound 288 Trace bound 14

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 630.m (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$15$$ Character field: $$\Q(i)$$ Newform subspaces: $$4$$ Sturm bound: $$288$$ Trace bound: $$14$$ Distinguishing $$T_p$$: $$11$$, $$17$$

## Dimensions

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

Total New Old
Modular forms 320 24 296
Cusp forms 256 24 232
Eisenstein series 64 0 64

## Trace form

 $$24q + O(q^{10})$$ $$24q + 24q^{13} - 24q^{16} - 16q^{22} + 16q^{25} + 32q^{31} + 8q^{37} + 8q^{40} - 32q^{43} - 32q^{46} - 24q^{52} - 32q^{55} + 8q^{58} + 96q^{67} - 16q^{70} + 72q^{73} - 8q^{82} + 48q^{85} - 16q^{88} - 56q^{97} + O(q^{100})$$

## 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.m.a $$4$$ $$5.031$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$-8$$ $$0$$ $$q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(-2-\zeta_{8}^{2})q^{5}+\cdots$$
630.2.m.b $$4$$ $$5.031$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$8$$ $$0$$ $$q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(2+\zeta_{8}^{2})q^{5}+\zeta_{8}^{3}q^{7}+\cdots$$
630.2.m.c $$8$$ $$5.031$$ 8.0.1698758656.6 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{5}q^{4}+(-\beta _{1}-\beta _{4}-\beta _{5}+\cdots)q^{5}+\cdots$$
630.2.m.d $$8$$ $$5.031$$ 8.0.1698758656.6 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{2}-\beta _{5}q^{4}+(-\beta _{2}-\beta _{3}-\beta _{5}+\cdots)q^{5}+\cdots$$

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)
$3$ 1
$5$ ($$( 1 + 4 T + 5 T^{2} )^{2}$$)($$( 1 - 4 T + 5 T^{2} )^{2}$$)($$1 + 2 T^{2} + 16 T^{3} + 2 T^{4} + 80 T^{5} + 50 T^{6} + 625 T^{8}$$)($$1 + 2 T^{2} - 16 T^{3} + 2 T^{4} - 80 T^{5} + 50 T^{6} + 625 T^{8}$$)
$7$ ($$1 + T^{4}$$)($$1 + T^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)
$11$ ($$1 - 32 T^{2} + 466 T^{4} - 3872 T^{6} + 14641 T^{8}$$)($$1 - 32 T^{2} + 466 T^{4} - 3872 T^{6} + 14641 T^{8}$$)($$1 - 12 T^{2} + 40 T^{4} - 260 T^{6} + 15086 T^{8} - 31460 T^{10} + 585640 T^{12} - 21258732 T^{14} + 214358881 T^{16}$$)($$1 - 12 T^{2} + 40 T^{4} - 260 T^{6} + 15086 T^{8} - 31460 T^{10} + 585640 T^{12} - 21258732 T^{14} + 214358881 T^{16}$$)
$13$ ($$( 1 - 8 T + 32 T^{2} - 104 T^{3} + 169 T^{4} )^{2}$$)($$( 1 - 8 T + 32 T^{2} - 104 T^{3} + 169 T^{4} )^{2}$$)($$1 + 4 T + 8 T^{2} + 28 T^{3} + 80 T^{4} + 372 T^{5} + 1240 T^{6} + 6412 T^{7} + 34174 T^{8} + 83356 T^{9} + 209560 T^{10} + 817284 T^{11} + 2284880 T^{12} + 10396204 T^{13} + 38614472 T^{14} + 250994068 T^{15} + 815730721 T^{16}$$)($$1 + 4 T + 8 T^{2} + 28 T^{3} + 80 T^{4} + 372 T^{5} + 1240 T^{6} + 6412 T^{7} + 34174 T^{8} + 83356 T^{9} + 209560 T^{10} + 817284 T^{11} + 2284880 T^{12} + 10396204 T^{13} + 38614472 T^{14} + 250994068 T^{15} + 815730721 T^{16}$$)
$17$ ($$1 - 8 T + 32 T^{2} - 168 T^{3} + 866 T^{4} - 2856 T^{5} + 9248 T^{6} - 39304 T^{7} + 83521 T^{8}$$)($$1 + 8 T + 32 T^{2} + 168 T^{3} + 866 T^{4} + 2856 T^{5} + 9248 T^{6} + 39304 T^{7} + 83521 T^{8}$$)($$1 + 64 T^{3} - 252 T^{4} - 960 T^{5} + 2048 T^{6} - 6016 T^{7} - 17786 T^{8} - 102272 T^{9} + 591872 T^{10} - 4716480 T^{11} - 21047292 T^{12} + 90870848 T^{13} + 6975757441 T^{16}$$)($$1 - 64 T^{3} - 252 T^{4} + 960 T^{5} + 2048 T^{6} + 6016 T^{7} - 17786 T^{8} + 102272 T^{9} + 591872 T^{10} + 4716480 T^{11} - 21047292 T^{12} - 90870848 T^{13} + 6975757441 T^{16}$$)
$19$ ($$( 1 - 30 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 30 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 19 T^{2} )^{8}$$)($$( 1 - 19 T^{2} )^{8}$$)
$23$ ($$1 - 8 T + 32 T^{2} - 120 T^{3} + 386 T^{4} - 2760 T^{5} + 16928 T^{6} - 97336 T^{7} + 279841 T^{8}$$)($$1 + 8 T + 32 T^{2} + 120 T^{3} + 386 T^{4} + 2760 T^{5} + 16928 T^{6} + 97336 T^{7} + 279841 T^{8}$$)($$1 - 8 T + 32 T^{2} + 8 T^{3} - 924 T^{4} + 5032 T^{5} - 10656 T^{6} - 28456 T^{7} + 420614 T^{8} - 654488 T^{9} - 5637024 T^{10} + 61224344 T^{11} - 258573084 T^{12} + 51490744 T^{13} + 4737148448 T^{14} - 27238603576 T^{15} + 78310985281 T^{16}$$)($$1 + 8 T + 32 T^{2} - 8 T^{3} - 924 T^{4} - 5032 T^{5} - 10656 T^{6} + 28456 T^{7} + 420614 T^{8} + 654488 T^{9} - 5637024 T^{10} - 61224344 T^{11} - 258573084 T^{12} - 51490744 T^{13} + 4737148448 T^{14} + 27238603576 T^{15} + 78310985281 T^{16}$$)
$29$ ($$( 1 + 4 T + 54 T^{2} + 116 T^{3} + 841 T^{4} )^{2}$$)($$( 1 - 4 T + 54 T^{2} - 116 T^{3} + 841 T^{4} )^{2}$$)($$( 1 + 12 T + 78 T^{2} + 252 T^{3} + 738 T^{4} + 7308 T^{5} + 65598 T^{6} + 292668 T^{7} + 707281 T^{8} )^{2}$$)($$( 1 - 12 T + 78 T^{2} - 252 T^{3} + 738 T^{4} - 7308 T^{5} + 65598 T^{6} - 292668 T^{7} + 707281 T^{8} )^{2}$$)
$31$ ($$( 1 - 12 T + 80 T^{2} - 372 T^{3} + 961 T^{4} )^{2}$$)($$( 1 - 12 T + 80 T^{2} - 372 T^{3} + 961 T^{4} )^{2}$$)($$( 1 + 4 T + 110 T^{2} + 356 T^{3} + 4930 T^{4} + 11036 T^{5} + 105710 T^{6} + 119164 T^{7} + 923521 T^{8} )^{2}$$)($$( 1 + 4 T + 110 T^{2} + 356 T^{3} + 4930 T^{4} + 11036 T^{5} + 105710 T^{6} + 119164 T^{7} + 923521 T^{8} )^{2}$$)
$37$ ($$1 - 8 T + 32 T^{2} - 72 T^{3} - 622 T^{4} - 2664 T^{5} + 43808 T^{6} - 405224 T^{7} + 1874161 T^{8}$$)($$1 - 8 T + 32 T^{2} - 72 T^{3} - 622 T^{4} - 2664 T^{5} + 43808 T^{6} - 405224 T^{7} + 1874161 T^{8}$$)($$1 + 4 T + 8 T^{2} + 220 T^{3} - 48 T^{4} - 8684 T^{5} - 10152 T^{6} - 208404 T^{7} - 3196994 T^{8} - 7710948 T^{9} - 13898088 T^{10} - 439870652 T^{11} - 89959728 T^{12} + 15255670540 T^{13} + 20525811272 T^{14} + 379727508532 T^{15} + 3512479453921 T^{16}$$)($$1 + 4 T + 8 T^{2} + 220 T^{3} - 48 T^{4} - 8684 T^{5} - 10152 T^{6} - 208404 T^{7} - 3196994 T^{8} - 7710948 T^{9} - 13898088 T^{10} - 439870652 T^{11} - 89959728 T^{12} + 15255670540 T^{13} + 20525811272 T^{14} + 379727508532 T^{15} + 3512479453921 T^{16}$$)
$41$ ($$1 - 76 T^{2} + 3654 T^{4} - 127756 T^{6} + 2825761 T^{8}$$)($$1 - 76 T^{2} + 3654 T^{4} - 127756 T^{6} + 2825761 T^{8}$$)($$1 - 76 T^{2} + 5480 T^{4} - 315940 T^{6} + 12648846 T^{8} - 531095140 T^{10} + 15485170280 T^{12} - 361007922316 T^{14} + 7984925229121 T^{16}$$)($$1 - 76 T^{2} + 5480 T^{4} - 315940 T^{6} + 12648846 T^{8} - 531095140 T^{10} + 15485170280 T^{12} - 361007922316 T^{14} + 7984925229121 T^{16}$$)
$43$ ($$1 + 4 T + 8 T^{2} - 220 T^{3} - 3554 T^{4} - 9460 T^{5} + 14792 T^{6} + 318028 T^{7} + 3418801 T^{8}$$)($$1 + 4 T + 8 T^{2} - 220 T^{3} - 3554 T^{4} - 9460 T^{5} + 14792 T^{6} + 318028 T^{7} + 3418801 T^{8}$$)($$1 + 12 T + 72 T^{2} + 300 T^{3} - 1008 T^{4} - 17988 T^{5} - 98280 T^{6} - 509316 T^{7} - 2377282 T^{8} - 21900588 T^{9} - 181719720 T^{10} - 1430171916 T^{11} - 3446151408 T^{12} + 44102532900 T^{13} + 455138139528 T^{14} + 3261823333284 T^{15} + 11688200277601 T^{16}$$)($$1 + 12 T + 72 T^{2} + 300 T^{3} - 1008 T^{4} - 17988 T^{5} - 98280 T^{6} - 509316 T^{7} - 2377282 T^{8} - 21900588 T^{9} - 181719720 T^{10} - 1430171916 T^{11} - 3446151408 T^{12} + 44102532900 T^{13} + 455138139528 T^{14} + 3261823333284 T^{15} + 11688200277601 T^{16}$$)
$47$ ($$1 - 20 T + 200 T^{2} - 1220 T^{3} + 7246 T^{4} - 57340 T^{5} + 441800 T^{6} - 2076460 T^{7} + 4879681 T^{8}$$)($$1 + 20 T + 200 T^{2} + 1220 T^{3} + 7246 T^{4} + 57340 T^{5} + 441800 T^{6} + 2076460 T^{7} + 4879681 T^{8}$$)($$1 + 12 T + 72 T^{2} + 220 T^{3} - 2352 T^{4} - 10116 T^{5} + 72152 T^{6} + 1599596 T^{7} + 18271774 T^{8} + 75181012 T^{9} + 159383768 T^{10} - 1050273468 T^{11} - 11477009712 T^{12} + 50455901540 T^{13} + 776103503688 T^{14} + 6079477445556 T^{15} + 23811286661761 T^{16}$$)($$1 - 12 T + 72 T^{2} - 220 T^{3} - 2352 T^{4} + 10116 T^{5} + 72152 T^{6} - 1599596 T^{7} + 18271774 T^{8} - 75181012 T^{9} + 159383768 T^{10} + 1050273468 T^{11} - 11477009712 T^{12} - 50455901540 T^{13} + 776103503688 T^{14} - 6079477445556 T^{15} + 23811286661761 T^{16}$$)
$53$ ($$1 + 16 T + 128 T^{2} + 1296 T^{3} + 12338 T^{4} + 68688 T^{5} + 359552 T^{6} + 2382032 T^{7} + 7890481 T^{8}$$)($$1 - 16 T + 128 T^{2} - 1296 T^{3} + 12338 T^{4} - 68688 T^{5} + 359552 T^{6} - 2382032 T^{7} + 7890481 T^{8}$$)($$1 - 32 T + 512 T^{2} - 5792 T^{3} + 60388 T^{4} - 619872 T^{5} + 5690880 T^{6} - 44817632 T^{7} + 328258854 T^{8} - 2375334496 T^{9} + 15985681920 T^{10} - 92284683744 T^{11} + 476490366628 T^{12} - 2422188295456 T^{13} + 11348152898048 T^{14} - 37590756474784 T^{15} + 62259690411361 T^{16}$$)($$1 + 32 T + 512 T^{2} + 5792 T^{3} + 60388 T^{4} + 619872 T^{5} + 5690880 T^{6} + 44817632 T^{7} + 328258854 T^{8} + 2375334496 T^{9} + 15985681920 T^{10} + 92284683744 T^{11} + 476490366628 T^{12} + 2422188295456 T^{13} + 11348152898048 T^{14} + 37590756474784 T^{15} + 62259690411361 T^{16}$$)
$59$ ($$( 1 + 110 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 110 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 + 8 T + 148 T^{2} + 520 T^{3} + 8710 T^{4} + 30680 T^{5} + 515188 T^{6} + 1643032 T^{7} + 12117361 T^{8} )^{2}$$)($$( 1 - 8 T + 148 T^{2} - 520 T^{3} + 8710 T^{4} - 30680 T^{5} + 515188 T^{6} - 1643032 T^{7} + 12117361 T^{8} )^{2}$$)
$61$ ($$( 1 + 24 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 + 24 T^{2} + 3721 T^{4} )^{2}$$)($$( 1 + 226 T^{2} + 16 T^{3} + 20162 T^{4} + 976 T^{5} + 840946 T^{6} + 13845841 T^{8} )^{2}$$)($$( 1 + 226 T^{2} + 16 T^{3} + 20162 T^{4} + 976 T^{5} + 840946 T^{6} + 13845841 T^{8} )^{2}$$)
$67$ ($$1 - 28 T + 392 T^{2} - 4508 T^{3} + 43006 T^{4} - 302036 T^{5} + 1759688 T^{6} - 8421364 T^{7} + 20151121 T^{8}$$)($$1 - 28 T + 392 T^{2} - 4508 T^{3} + 43006 T^{4} - 302036 T^{5} + 1759688 T^{6} - 8421364 T^{7} + 20151121 T^{8}$$)($$1 - 20 T + 200 T^{2} - 1332 T^{3} + 9104 T^{4} - 102596 T^{5} + 1118232 T^{6} - 8938244 T^{7} + 69729150 T^{8} - 598862348 T^{9} + 5019743448 T^{10} - 30857080748 T^{11} + 183455805584 T^{12} - 1798366642524 T^{13} + 18091676433800 T^{14} - 121214232106460 T^{15} + 406067677556641 T^{16}$$)($$1 - 20 T + 200 T^{2} - 1332 T^{3} + 9104 T^{4} - 102596 T^{5} + 1118232 T^{6} - 8938244 T^{7} + 69729150 T^{8} - 598862348 T^{9} + 5019743448 T^{10} - 30857080748 T^{11} + 183455805584 T^{12} - 1798366642524 T^{13} + 18091676433800 T^{14} - 121214232106460 T^{15} + 406067677556641 T^{16}$$)
$71$ ($$1 - 236 T^{2} + 23494 T^{4} - 1189676 T^{6} + 25411681 T^{8}$$)($$1 - 236 T^{2} + 23494 T^{4} - 1189676 T^{6} + 25411681 T^{8}$$)($$1 - 216 T^{2} + 31388 T^{4} - 3350120 T^{6} + 263319558 T^{8} - 16887954920 T^{10} + 797621843228 T^{12} - 27669661326936 T^{14} + 645753531245761 T^{16}$$)($$1 - 216 T^{2} + 31388 T^{4} - 3350120 T^{6} + 263319558 T^{8} - 16887954920 T^{10} + 797621843228 T^{12} - 27669661326936 T^{14} + 645753531245761 T^{16}$$)
$73$ ($$1 - 8542 T^{4} + 28398241 T^{8}$$)($$1 - 8542 T^{4} + 28398241 T^{8}$$)($$1 - 36 T + 648 T^{2} - 8620 T^{3} + 98192 T^{4} - 980116 T^{5} + 8807960 T^{6} - 73999132 T^{7} + 618591198 T^{8} - 5401936636 T^{9} + 46937618840 T^{10} - 381281785972 T^{11} + 2788480080272 T^{12} - 17869877131660 T^{13} + 98064578635272 T^{14} - 397706346687492 T^{15} + 806460091894081 T^{16}$$)($$1 - 36 T + 648 T^{2} - 8620 T^{3} + 98192 T^{4} - 980116 T^{5} + 8807960 T^{6} - 73999132 T^{7} + 618591198 T^{8} - 5401936636 T^{9} + 46937618840 T^{10} - 381281785972 T^{11} + 2788480080272 T^{12} - 17869877131660 T^{13} + 98064578635272 T^{14} - 397706346687492 T^{15} + 806460091894081 T^{16}$$)
$79$ ($$( 1 - 126 T^{2} + 6241 T^{4} )^{2}$$)($$( 1 - 126 T^{2} + 6241 T^{4} )^{2}$$)($$1 - 280 T^{2} + 32732 T^{4} - 1914792 T^{6} + 96372294 T^{8} - 11950216872 T^{10} + 1274914051292 T^{12} - 68064487545880 T^{14} + 1517108809906561 T^{16}$$)($$1 - 280 T^{2} + 32732 T^{4} - 1914792 T^{6} + 96372294 T^{8} - 11950216872 T^{10} + 1274914051292 T^{12} - 68064487545880 T^{14} + 1517108809906561 T^{16}$$)
$83$ ($$1 - 8 T + 32 T^{2} - 600 T^{3} + 11186 T^{4} - 49800 T^{5} + 220448 T^{6} - 4574296 T^{7} + 47458321 T^{8}$$)($$1 + 8 T + 32 T^{2} + 600 T^{3} + 11186 T^{4} + 49800 T^{5} + 220448 T^{6} + 4574296 T^{7} + 47458321 T^{8}$$)($$1 - 56 T + 1568 T^{2} - 30056 T^{3} + 449924 T^{4} - 5649064 T^{5} + 62548320 T^{6} - 632947128 T^{7} + 5962291750 T^{8} - 52534611624 T^{9} + 430895376480 T^{10} - 3230061357368 T^{11} + 21352637617604 T^{12} - 118391805566008 T^{13} + 512642505442592 T^{14} - 1519618855419112 T^{15} + 2252292232139041 T^{16}$$)($$1 + 56 T + 1568 T^{2} + 30056 T^{3} + 449924 T^{4} + 5649064 T^{5} + 62548320 T^{6} + 632947128 T^{7} + 5962291750 T^{8} + 52534611624 T^{9} + 430895376480 T^{10} + 3230061357368 T^{11} + 21352637617604 T^{12} + 118391805566008 T^{13} + 512642505442592 T^{14} + 1519618855419112 T^{15} + 2252292232139041 T^{16}$$)
$89$ ($$( 1 - 12 T + 206 T^{2} - 1068 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 12 T + 206 T^{2} + 1068 T^{3} + 7921 T^{4} )^{2}$$)($$( 1 + 36 T + 734 T^{2} + 10004 T^{3} + 106562 T^{4} + 890356 T^{5} + 5814014 T^{6} + 25378884 T^{7} + 62742241 T^{8} )^{2}$$)($$( 1 - 36 T + 734 T^{2} - 10004 T^{3} + 106562 T^{4} - 890356 T^{5} + 5814014 T^{6} - 25378884 T^{7} + 62742241 T^{8} )^{2}$$)
$97$ ($$1 + 24 T + 288 T^{2} + 3960 T^{3} + 49826 T^{4} + 384120 T^{5} + 2709792 T^{6} + 21904152 T^{7} + 88529281 T^{8}$$)($$1 + 24 T + 288 T^{2} + 3960 T^{3} + 49826 T^{4} + 384120 T^{5} + 2709792 T^{6} + 21904152 T^{7} + 88529281 T^{8}$$)($$1 + 4 T + 8 T^{2} + 1852 T^{3} + 7856 T^{4} - 48156 T^{5} + 1459480 T^{6} + 10407580 T^{7} - 90680354 T^{8} + 1009535260 T^{9} + 13732247320 T^{10} - 43950680988 T^{11} + 695486031536 T^{12} + 15903754155964 T^{13} + 6663776039432 T^{14} + 323193137912452 T^{15} + 7837433594376961 T^{16}$$)($$1 + 4 T + 8 T^{2} + 1852 T^{3} + 7856 T^{4} - 48156 T^{5} + 1459480 T^{6} + 10407580 T^{7} - 90680354 T^{8} + 1009535260 T^{9} + 13732247320 T^{10} - 43950680988 T^{11} + 695486031536 T^{12} + 15903754155964 T^{13} + 6663776039432 T^{14} + 323193137912452 T^{15} + 7837433594376961 T^{16}$$)