# Properties

 Label 630.2.p Level 630 Weight 2 Character orbit p Rep. character $$\chi_{630}(307,\cdot)$$ Character field $$\Q(\zeta_{4})$$ Dimension 40 Newform subspaces 4 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.p (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$35$$ Character field: $$\Q(i)$$ Newform subspaces: $$4$$ Sturm bound: $$288$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$11$$, $$13$$

## Dimensions

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

Total New Old
Modular forms 320 40 280
Cusp forms 256 40 216
Eisenstein series 64 0 64

## Trace form

 $$40q + O(q^{10})$$ $$40q - 16q^{11} - 40q^{16} + 16q^{22} - 24q^{23} + 16q^{25} + 40q^{35} + 16q^{37} - 16q^{46} - 32q^{50} + 48q^{53} + 24q^{65} - 48q^{67} + 16q^{77} + 32q^{85} - 16q^{86} - 16q^{88} + 32q^{91} - 24q^{92} - 64q^{95} - 16q^{98} + O(q^{100})$$

## 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.p.a $$8$$ $$5.031$$ $$\Q(\zeta_{16})$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+\zeta_{16}^{2}q^{2}+\zeta_{16}^{4}q^{4}+(-\zeta_{16}-2\zeta_{16}^{5}+\cdots)q^{5}+\cdots$$
630.2.p.b $$8$$ $$5.031$$ 8.0.1698758656.6 None $$0$$ $$0$$ $$0$$ $$4$$ $$q-\beta _{3}q^{2}-\beta _{5}q^{4}+(-\beta _{1}-\beta _{4}-\beta _{5}+\cdots)q^{5}+\cdots$$
630.2.p.c $$8$$ $$5.031$$ 8.0.1698758656.6 None $$0$$ $$0$$ $$0$$ $$8$$ $$q+\beta _{1}q^{2}+\beta _{5}q^{4}+(-\beta _{2}-\beta _{3}-\beta _{5}+\cdots)q^{5}+\cdots$$
630.2.p.d $$16$$ $$5.031$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-\beta _{2}q^{2}-\beta _{9}q^{4}-\beta _{12}q^{5}-\beta _{1}q^{7}+\cdots$$

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

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{4}$$)
$3$ 1
$5$ ($$1 - 48 T^{4} + 625 T^{8}$$)($$1 + 2 T^{2} + 16 T^{3} + 2 T^{4} + 80 T^{5} + 50 T^{6} + 625 T^{8}$$)($$1 + 2 T^{2} - 16 T^{3} + 2 T^{4} - 80 T^{5} + 50 T^{6} + 625 T^{8}$$)($$( 1 - 6 T^{2} + 18 T^{4} - 150 T^{6} + 625 T^{8} )^{2}$$)
$7$ ($$1 + 8 T + 32 T^{2} + 88 T^{3} + 226 T^{4} + 616 T^{5} + 1568 T^{6} + 2744 T^{7} + 2401 T^{8}$$)($$1 - 4 T + 6 T^{2} + 28 T^{3} - 126 T^{4} + 196 T^{5} + 294 T^{6} - 1372 T^{7} + 2401 T^{8}$$)($$1 - 8 T + 34 T^{2} - 112 T^{3} + 322 T^{4} - 784 T^{5} + 1666 T^{6} - 2744 T^{7} + 2401 T^{8}$$)($$( 1 + 2 T + 2 T^{2} - 26 T^{3} - 62 T^{4} - 182 T^{5} + 98 T^{6} + 686 T^{7} + 2401 T^{8} )^{2}$$)
$11$ ($$( 1 + 14 T^{2} + 121 T^{4} )^{4}$$)($$( 1 + 4 T + 12 T^{2} - 4 T^{3} + 18 T^{4} - 44 T^{5} + 1452 T^{6} + 5324 T^{7} + 14641 T^{8} )^{2}$$)($$( 1 + 4 T + 12 T^{2} - 4 T^{3} + 18 T^{4} - 44 T^{5} + 1452 T^{6} + 5324 T^{7} + 14641 T^{8} )^{2}$$)($$( 1 + 20 T^{2} + 121 T^{4} )^{8}$$)
$13$ ($$( 1 - 240 T^{4} + 28561 T^{8} )^{2}$$)($$( 1 + 8 T + 32 T^{2} + 136 T^{3} + 562 T^{4} + 1768 T^{5} + 5408 T^{6} + 17576 T^{7} + 28561 T^{8} )^{2}$$)($$( 1 - 8 T + 32 T^{2} - 136 T^{3} + 562 T^{4} - 1768 T^{5} + 5408 T^{6} - 17576 T^{7} + 28561 T^{8} )^{2}$$)($$( 1 - 60 T^{4} - 36442 T^{8} - 1713660 T^{12} + 815730721 T^{16} )^{2}$$)
$17$ ($$1 - 252 T^{4} + 17030 T^{8} - 21047292 T^{12} + 6975757441 T^{16}$$)($$1 + 12 T + 72 T^{2} + 388 T^{3} + 1696 T^{4} + 4732 T^{5} + 9944 T^{6} + 2420 T^{7} - 98754 T^{8} + 41140 T^{9} + 2873816 T^{10} + 23248316 T^{11} + 141651616 T^{12} + 550904516 T^{13} + 1737904968 T^{14} + 4924064076 T^{15} + 6975757441 T^{16}$$)($$1 - 12 T + 72 T^{2} - 388 T^{3} + 1696 T^{4} - 4732 T^{5} + 9944 T^{6} - 2420 T^{7} - 98754 T^{8} - 41140 T^{9} + 2873816 T^{10} - 23248316 T^{11} + 141651616 T^{12} - 550904516 T^{13} + 1737904968 T^{14} - 4924064076 T^{15} + 6975757441 T^{16}$$)($$( 1 + 384 T^{4} + 84350 T^{8} + 32072064 T^{12} + 6975757441 T^{16} )^{2}$$)
$19$ ($$( 1 + 24 T^{2} + 288 T^{4} + 8664 T^{6} + 130321 T^{8} )^{2}$$)($$( 1 + 4 T + 62 T^{2} + 212 T^{3} + 1666 T^{4} + 4028 T^{5} + 22382 T^{6} + 27436 T^{7} + 130321 T^{8} )^{2}$$)($$( 1 - 4 T + 62 T^{2} - 212 T^{3} + 1666 T^{4} - 4028 T^{5} + 22382 T^{6} - 27436 T^{7} + 130321 T^{8} )^{2}$$)($$( 1 + 18 T^{2} + 762 T^{4} + 6498 T^{6} + 130321 T^{8} )^{4}$$)
$23$ ($$( 1 - 4 T + 8 T^{2} - 84 T^{3} + 878 T^{4} - 1932 T^{5} + 4232 T^{6} - 48668 T^{7} + 279841 T^{8} )^{2}$$)($$1 + 16 T + 128 T^{2} + 656 T^{3} + 3300 T^{4} + 21520 T^{5} + 137088 T^{6} + 691344 T^{7} + 3238918 T^{8} + 15900912 T^{9} + 72519552 T^{10} + 261833840 T^{11} + 923475300 T^{12} + 4222241008 T^{13} + 18948593792 T^{14} + 54477207152 T^{15} + 78310985281 T^{16}$$)($$1 + 16 T + 128 T^{2} + 656 T^{3} + 3300 T^{4} + 21520 T^{5} + 137088 T^{6} + 691344 T^{7} + 3238918 T^{8} + 15900912 T^{9} + 72519552 T^{10} + 261833840 T^{11} + 923475300 T^{12} + 4222241008 T^{13} + 18948593792 T^{14} + 54477207152 T^{15} + 78310985281 T^{16}$$)($$( 1 - 1756 T^{4} + 1320070 T^{8} - 491400796 T^{12} + 78310985281 T^{16} )^{2}$$)
$29$ ($$( 1 - 92 T^{2} + 3670 T^{4} - 77372 T^{6} + 707281 T^{8} )^{2}$$)($$1 - 4 T^{2} + 200 T^{4} - 5548 T^{6} + 1325934 T^{8} - 4665868 T^{10} + 141456200 T^{12} - 2379293284 T^{14} + 500246412961 T^{16}$$)($$1 - 4 T^{2} + 200 T^{4} - 5548 T^{6} + 1325934 T^{8} - 4665868 T^{10} + 141456200 T^{12} - 2379293284 T^{14} + 500246412961 T^{16}$$)($$( 1 - 50 T^{2} + 1282 T^{4} - 42050 T^{6} + 707281 T^{8} )^{4}$$)
$31$ ($$( 1 - 108 T^{2} + 4806 T^{4} - 103788 T^{6} + 923521 T^{8} )^{2}$$)($$1 - 148 T^{2} + 11192 T^{4} - 565212 T^{6} + 20507310 T^{8} - 543168732 T^{10} + 10336047032 T^{12} - 131350544788 T^{14} + 852891037441 T^{16}$$)($$1 - 148 T^{2} + 11192 T^{4} - 565212 T^{6} + 20507310 T^{8} - 543168732 T^{10} + 10336047032 T^{12} - 131350544788 T^{14} + 852891037441 T^{16}$$)($$( 1 - 18 T^{2} - 6 T^{4} - 17298 T^{6} + 923521 T^{8} )^{4}$$)
$37$ ($$( 1 - 16 T + 128 T^{2} - 1040 T^{3} + 7666 T^{4} - 38480 T^{5} + 175232 T^{6} - 810448 T^{7} + 1874161 T^{8} )^{2}$$)($$1 + 28 T + 392 T^{2} + 4180 T^{3} + 38912 T^{4} + 311708 T^{5} + 2210520 T^{6} + 14742164 T^{7} + 92944158 T^{8} + 545460068 T^{9} + 3026201880 T^{10} + 15788945324 T^{11} + 72927352832 T^{12} + 289857740260 T^{13} + 1005764752328 T^{14} + 2658092559724 T^{15} + 3512479453921 T^{16}$$)($$1 + 28 T + 392 T^{2} + 4180 T^{3} + 38912 T^{4} + 311708 T^{5} + 2210520 T^{6} + 14742164 T^{7} + 92944158 T^{8} + 545460068 T^{9} + 3026201880 T^{10} + 15788945324 T^{11} + 72927352832 T^{12} + 289857740260 T^{13} + 1005764752328 T^{14} + 2658092559724 T^{15} + 3512479453921 T^{16}$$)($$( 1 - 10 T + 50 T^{2} - 290 T^{3} + 1618 T^{4} - 10730 T^{5} + 68450 T^{6} - 506530 T^{7} + 1874161 T^{8} )^{4}$$)
$41$ ($$( 1 - 148 T^{2} + 8806 T^{4} - 248788 T^{6} + 2825761 T^{8} )^{2}$$)($$1 - 176 T^{2} + 13468 T^{4} - 643920 T^{6} + 26131718 T^{8} - 1082429520 T^{10} + 38057349148 T^{12} - 836018346416 T^{14} + 7984925229121 T^{16}$$)($$1 - 176 T^{2} + 13468 T^{4} - 643920 T^{6} + 26131718 T^{8} - 1082429520 T^{10} + 38057349148 T^{12} - 836018346416 T^{14} + 7984925229121 T^{16}$$)($$( 1 - 52 T^{2} + 1414 T^{4} - 87412 T^{6} + 2825761 T^{8} )^{4}$$)
$43$ ($$( 1 - 8 T + 32 T^{2} - 344 T^{3} + 1849 T^{4} )^{4}$$)($$1 - 32 T^{3} - 6588 T^{4} - 1632 T^{5} + 512 T^{6} + 117440 T^{7} + 17663206 T^{8} + 5049920 T^{9} + 946688 T^{10} - 129755424 T^{11} - 22523060988 T^{12} - 4704270176 T^{13} + 11688200277601 T^{16}$$)($$1 - 32 T^{3} - 6588 T^{4} - 1632 T^{5} + 512 T^{6} + 117440 T^{7} + 17663206 T^{8} + 5049920 T^{9} + 946688 T^{10} - 129755424 T^{11} - 22523060988 T^{12} - 4704270176 T^{13} + 11688200277601 T^{16}$$)($$( 1 + 4 T + 8 T^{2} + 172 T^{3} + 1849 T^{4} )^{8}$$)
$47$ ($$1 + 6020 T^{4} + 17872774 T^{8} + 29375679620 T^{12} + 23811286661761 T^{16}$$)($$1 + 24 T + 288 T^{2} + 2728 T^{3} + 26308 T^{4} + 243208 T^{5} + 1981280 T^{6} + 14819768 T^{7} + 104930310 T^{8} + 696529096 T^{9} + 4376647520 T^{10} + 25250584184 T^{11} + 128374647748 T^{12} + 625653179096 T^{13} + 3104414014752 T^{14} + 12158954891112 T^{15} + 23811286661761 T^{16}$$)($$1 - 24 T + 288 T^{2} - 2728 T^{3} + 26308 T^{4} - 243208 T^{5} + 1981280 T^{6} - 14819768 T^{7} + 104930310 T^{8} - 696529096 T^{9} + 4376647520 T^{10} - 25250584184 T^{11} + 128374647748 T^{12} - 625653179096 T^{13} + 3104414014752 T^{14} - 12158954891112 T^{15} + 23811286661761 T^{16}$$)($$( 1 - 700 T^{4} - 5274362 T^{8} - 3415776700 T^{12} + 23811286661761 T^{16} )^{2}$$)
$53$ ($$( 1 - 16 T + 128 T^{2} - 784 T^{3} + 4786 T^{4} - 41552 T^{5} + 359552 T^{6} - 2382032 T^{7} + 7890481 T^{8} )^{2}$$)($$1 - 8 T + 32 T^{2} - 264 T^{3} + 5828 T^{4} - 46136 T^{5} + 217440 T^{6} - 2204664 T^{7} + 21970150 T^{8} - 116847192 T^{9} + 610788960 T^{10} - 6868589272 T^{11} + 45985723268 T^{12} - 110403610152 T^{13} + 709259556128 T^{14} - 9397689118696 T^{15} + 62259690411361 T^{16}$$)($$1 - 8 T + 32 T^{2} - 264 T^{3} + 5828 T^{4} - 46136 T^{5} + 217440 T^{6} - 2204664 T^{7} + 21970150 T^{8} - 116847192 T^{9} + 610788960 T^{10} - 6868589272 T^{11} + 45985723268 T^{12} - 110403610152 T^{13} + 709259556128 T^{14} - 9397689118696 T^{15} + 62259690411361 T^{16}$$)($$( 1 - 2012 T^{4} - 1721946 T^{8} - 15875647772 T^{12} + 62259690411361 T^{16} )^{2}$$)
$59$ ($$( 1 + 136 T^{2} + 10336 T^{4} + 473416 T^{6} + 12117361 T^{8} )^{2}$$)($$( 1 + 4 T + 70 T^{2} + 116 T^{3} + 802 T^{4} + 6844 T^{5} + 243670 T^{6} + 821516 T^{7} + 12117361 T^{8} )^{2}$$)($$( 1 - 4 T + 70 T^{2} - 116 T^{3} + 802 T^{4} - 6844 T^{5} + 243670 T^{6} - 821516 T^{7} + 12117361 T^{8} )^{2}$$)($$( 1 + 34 T^{2} + 5242 T^{4} + 118354 T^{6} + 12117361 T^{8} )^{4}$$)
$61$ ($$( 1 - 96 T^{2} + 8688 T^{4} - 357216 T^{6} + 13845841 T^{8} )^{2}$$)($$1 - 264 T^{2} + 37532 T^{4} - 3615096 T^{6} + 255247078 T^{8} - 13451772216 T^{10} + 519662104412 T^{12} - 13601378831304 T^{14} + 191707312997281 T^{16}$$)($$1 - 264 T^{2} + 37532 T^{4} - 3615096 T^{6} + 255247078 T^{8} - 13451772216 T^{10} + 519662104412 T^{12} - 13601378831304 T^{14} + 191707312997281 T^{16}$$)($$( 1 - 192 T^{2} + 16494 T^{4} - 714432 T^{6} + 13845841 T^{8} )^{4}$$)
$67$ ($$( 1 - 8 T + 32 T^{2} + 552 T^{3} - 8974 T^{4} + 36984 T^{5} + 143648 T^{6} - 2406104 T^{7} + 20151121 T^{8} )^{2}$$)($$1 - 64 T^{3} + 9348 T^{4} + 10560 T^{5} + 2048 T^{6} + 428416 T^{7} + 51599014 T^{8} + 28703872 T^{9} + 9193472 T^{10} + 3176057280 T^{11} + 188372679108 T^{12} - 86408006848 T^{13} + 406067677556641 T^{16}$$)($$1 - 64 T^{3} + 9348 T^{4} + 10560 T^{5} + 2048 T^{6} + 428416 T^{7} + 51599014 T^{8} + 28703872 T^{9} + 9193472 T^{10} + 3176057280 T^{11} + 188372679108 T^{12} - 86408006848 T^{13} + 406067677556641 T^{16}$$)($$( 1 + 8 T + 32 T^{2} + 536 T^{3} + 4489 T^{4} )^{8}$$)
$71$ ($$( 1 - 4 T + 144 T^{2} - 284 T^{3} + 5041 T^{4} )^{4}$$)($$( 1 + 4 T + 158 T^{2} - 220 T^{3} + 10018 T^{4} - 15620 T^{5} + 796478 T^{6} + 1431644 T^{7} + 25411681 T^{8} )^{2}$$)($$( 1 + 4 T + 158 T^{2} - 220 T^{3} + 10018 T^{4} - 15620 T^{5} + 796478 T^{6} + 1431644 T^{7} + 25411681 T^{8} )^{2}$$)($$( 1 + 218 T^{2} + 20938 T^{4} + 1098938 T^{6} + 25411681 T^{8} )^{4}$$)
$73$ ($$1 + 1220 T^{4} + 55730374 T^{8} + 34645854020 T^{12} + 806460091894081 T^{16}$$)($$1 + 28 T + 392 T^{2} + 4756 T^{3} + 46736 T^{4} + 302508 T^{5} + 1459480 T^{6} + 3207460 T^{7} - 21483554 T^{8} + 234144580 T^{9} + 7777568920 T^{10} + 117680754636 T^{11} + 1327220191376 T^{12} + 9859528496308 T^{13} + 59323016705288 T^{14} + 309327158534716 T^{15} + 806460091894081 T^{16}$$)($$1 - 28 T + 392 T^{2} - 4756 T^{3} + 46736 T^{4} - 302508 T^{5} + 1459480 T^{6} - 3207460 T^{7} - 21483554 T^{8} - 234144580 T^{9} + 7777568920 T^{10} - 117680754636 T^{11} + 1327220191376 T^{12} - 9859528496308 T^{13} + 59323016705288 T^{14} - 309327158534716 T^{15} + 806460091894081 T^{16}$$)($$( 1 + 2432 T^{4} + 57090238 T^{8} + 69064522112 T^{12} + 806460091894081 T^{16} )^{2}$$)
$79$ ($$( 1 - 208 T^{2} + 22498 T^{4} - 1298128 T^{6} + 38950081 T^{8} )^{2}$$)($$1 - 272 T^{2} + 39644 T^{4} - 4069360 T^{6} + 348070342 T^{8} - 25396875760 T^{10} + 1544137011164 T^{12} - 66119787901712 T^{14} + 1517108809906561 T^{16}$$)($$1 - 272 T^{2} + 39644 T^{4} - 4069360 T^{6} + 348070342 T^{8} - 25396875760 T^{10} + 1544137011164 T^{12} - 66119787901712 T^{14} + 1517108809906561 T^{16}$$)($$( 1 - 232 T^{2} + 25774 T^{4} - 1447912 T^{6} + 38950081 T^{8} )^{4}$$)
$83$ ($$1 - 4224 T^{4} + 99283874 T^{8} - 200463947904 T^{12} + 2252292232139041 T^{16}$$)($$1 - 16 T + 128 T^{2} - 976 T^{3} + 16964 T^{4} - 235664 T^{5} + 2075520 T^{6} - 16972368 T^{7} + 135669670 T^{8} - 1408706544 T^{9} + 14298257280 T^{10} - 134749611568 T^{11} + 805082957444 T^{12} - 3844503667568 T^{13} + 41848367791232 T^{14} - 434176815834032 T^{15} + 2252292232139041 T^{16}$$)($$1 + 16 T + 128 T^{2} + 976 T^{3} + 16964 T^{4} + 235664 T^{5} + 2075520 T^{6} + 16972368 T^{7} + 135669670 T^{8} + 1408706544 T^{9} + 14298257280 T^{10} + 134749611568 T^{11} + 805082957444 T^{12} + 3844503667568 T^{13} + 41848367791232 T^{14} + 434176815834032 T^{15} + 2252292232139041 T^{16}$$)($$( 1 - 26076 T^{4} + 264528230 T^{8} - 1237523178396 T^{12} + 2252292232139041 T^{16} )^{2}$$)
$89$ ($$( 1 - 60 T^{2} + 10470 T^{4} - 475260 T^{6} + 62742241 T^{8} )^{2}$$)($$( 1 - 32 T + 628 T^{2} - 8320 T^{3} + 88630 T^{4} - 740480 T^{5} + 4974388 T^{6} - 22559008 T^{7} + 62742241 T^{8} )^{2}$$)($$( 1 + 32 T + 628 T^{2} + 8320 T^{3} + 88630 T^{4} + 740480 T^{5} + 4974388 T^{6} + 22559008 T^{7} + 62742241 T^{8} )^{2}$$)($$( 1 + 72 T^{2} + 16974 T^{4} + 570312 T^{6} + 62742241 T^{8} )^{4}$$)
$97$ ($$1 - 11900 T^{4} + 108781062 T^{8} - 1053498443900 T^{12} + 7837433594376961 T^{16}$$)($$1 + 28 T + 392 T^{2} + 4900 T^{3} + 61488 T^{4} + 704956 T^{5} + 7640472 T^{6} + 89011524 T^{7} + 981394654 T^{8} + 8634117828 T^{9} + 71889201048 T^{10} + 643394307388 T^{11} + 5443488430128 T^{12} + 42077967259300 T^{13} + 326525025932168 T^{14} + 2262351965387164 T^{15} + 7837433594376961 T^{16}$$)($$1 - 28 T + 392 T^{2} - 4900 T^{3} + 61488 T^{4} - 704956 T^{5} + 7640472 T^{6} - 89011524 T^{7} + 981394654 T^{8} - 8634117828 T^{9} + 71889201048 T^{10} - 643394307388 T^{11} + 5443488430128 T^{12} - 42077967259300 T^{13} + 326525025932168 T^{14} - 2262351965387164 T^{15} + 7837433594376961 T^{16}$$)($$( 1 + 3328 T^{4} - 48139266 T^{8} + 294625447168 T^{12} + 7837433594376961 T^{16} )^{2}$$)