# Properties

 Label 210.3.e Level 210 Weight 3 Character orbit e Rep. character $$\chi_{210}(71,\cdot)$$ Character field $$\Q$$ Dimension 16 Newform subspaces 1 Sturm bound 144 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 210.e (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$3$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$144$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(210, [\chi])$$.

Total New Old
Modular forms 104 16 88
Cusp forms 88 16 72
Eisenstein series 16 0 16

## Trace form

 $$16q - 8q^{3} - 32q^{4} + 16q^{6} - 4q^{9} + O(q^{10})$$ $$16q - 8q^{3} - 32q^{4} + 16q^{6} - 4q^{9} + 16q^{12} - 20q^{15} + 64q^{16} - 32q^{18} + 48q^{19} + 28q^{21} - 96q^{22} - 32q^{24} - 80q^{25} + 64q^{27} - 88q^{33} + 160q^{34} + 8q^{36} + 80q^{37} + 156q^{39} - 336q^{43} - 80q^{45} + 32q^{46} - 32q^{48} + 112q^{49} + 84q^{51} - 32q^{54} - 80q^{55} - 264q^{57} + 96q^{58} + 40q^{60} + 112q^{61} + 112q^{63} - 128q^{64} + 240q^{67} + 8q^{69} + 64q^{72} + 48q^{73} + 40q^{75} - 96q^{76} + 208q^{78} + 8q^{79} - 124q^{81} - 608q^{82} - 56q^{84} + 120q^{85} - 120q^{87} + 192q^{88} + 160q^{90} - 56q^{91} + 104q^{93} + 32q^{94} + 64q^{96} - 192q^{97} - 52q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(210, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
210.3.e.a $$16$$ $$5.722$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$-8$$ $$0$$ $$0$$ $$q-\beta _{4}q^{2}-\beta _{1}q^{3}-2q^{4}-\beta _{9}q^{5}+(1+\cdots)q^{6}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(210, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(210, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} )^{8}$$
$3$ $$1 + 8 T + 34 T^{2} + 80 T^{3} + 97 T^{4} + 80 T^{5} + 498 T^{6} + 3288 T^{7} + 11844 T^{8} + 29592 T^{9} + 40338 T^{10} + 58320 T^{11} + 636417 T^{12} + 4723920 T^{13} + 18068994 T^{14} + 38263752 T^{15} + 43046721 T^{16}$$
$5$ $$( 1 + 5 T^{2} )^{8}$$
$7$ $$( 1 - 7 T^{2} )^{8}$$
$11$ $$1 - 588 T^{2} + 192110 T^{4} - 45729776 T^{6} + 8909879425 T^{8} - 1517060018584 T^{10} + 233462822472086 T^{12} - 32740893640977644 T^{14} + 4166709531307795396 T^{16} -$$$$47\!\cdots\!04$$$$T^{18} +$$$$50\!\cdots\!66$$$$T^{20} -$$$$47\!\cdots\!64$$$$T^{22} +$$$$40\!\cdots\!25$$$$T^{24} -$$$$30\!\cdots\!76$$$$T^{26} +$$$$18\!\cdots\!10$$$$T^{28} -$$$$84\!\cdots\!28$$$$T^{30} +$$$$21\!\cdots\!21$$$$T^{32}$$
$13$ $$( 1 + 494 T^{2} - 456 T^{3} + 146293 T^{4} - 55296 T^{5} + 35813146 T^{6} - 25079880 T^{7} + 6821269256 T^{8} - 4238499720 T^{9} + 1022859262906 T^{10} - 266903230464 T^{11} + 119335694367253 T^{12} - 62863472283144 T^{13} + 11509254050505614 T^{14} + 665416609183179841 T^{16} )^{2}$$
$17$ $$1 - 2012 T^{2} + 2016998 T^{4} - 1352297560 T^{6} + 695200323953 T^{8} - 299701490294440 T^{10} + 114417537477919350 T^{12} - 39314188029348430308 T^{14} +$$$$12\!\cdots\!52$$$$T^{16} -$$$$32\!\cdots\!68$$$$T^{18} +$$$$79\!\cdots\!50$$$$T^{20} -$$$$17\!\cdots\!40$$$$T^{22} +$$$$33\!\cdots\!93$$$$T^{24} -$$$$54\!\cdots\!60$$$$T^{26} +$$$$68\!\cdots\!58$$$$T^{28} -$$$$57\!\cdots\!92$$$$T^{30} +$$$$23\!\cdots\!61$$$$T^{32}$$
$19$ $$( 1 - 24 T + 1612 T^{2} - 39608 T^{3} + 1389160 T^{4} - 32391560 T^{5} + 808980068 T^{6} - 17015640488 T^{7} + 341271444494 T^{8} - 6142646216168 T^{9} + 105427091441828 T^{10} - 1523889477164360 T^{11} + 23592886434035560 T^{12} - 242839272338982008 T^{13} + 3567863649534651532 T^{14} - 19176160458789218904 T^{15} +$$$$28\!\cdots\!81$$$$T^{16} )^{2}$$
$23$ $$1 - 3856 T^{2} + 7749432 T^{4} - 10915385904 T^{6} + 12001409108892 T^{8} - 10839581553458448 T^{10} + 8275120566094665352 T^{12} -$$$$54\!\cdots\!60$$$$T^{14} +$$$$30\!\cdots\!06$$$$T^{16} -$$$$15\!\cdots\!60$$$$T^{18} +$$$$64\!\cdots\!12$$$$T^{20} -$$$$23\!\cdots\!08$$$$T^{22} +$$$$73\!\cdots\!12$$$$T^{24} -$$$$18\!\cdots\!04$$$$T^{26} +$$$$37\!\cdots\!12$$$$T^{28} -$$$$51\!\cdots\!36$$$$T^{30} +$$$$37\!\cdots\!21$$$$T^{32}$$
$29$ $$1 - 3972 T^{2} + 8120294 T^{4} - 13034144872 T^{6} + 18580208769649 T^{8} - 22683669812342744 T^{10} + 24087258397916480502 T^{12} -$$$$23\!\cdots\!44$$$$T^{14} +$$$$20\!\cdots\!28$$$$T^{16} -$$$$16\!\cdots\!64$$$$T^{18} +$$$$12\!\cdots\!22$$$$T^{20} -$$$$80\!\cdots\!04$$$$T^{22} +$$$$46\!\cdots\!29$$$$T^{24} -$$$$23\!\cdots\!72$$$$T^{26} +$$$$10\!\cdots\!14$$$$T^{28} -$$$$35\!\cdots\!92$$$$T^{30} +$$$$62\!\cdots\!41$$$$T^{32}$$
$31$ $$( 1 + 3684 T^{2} - 19472 T^{3} + 7802120 T^{4} - 58737456 T^{5} + 11372951532 T^{6} - 93476864320 T^{7} + 12493787155086 T^{8} - 89831266611520 T^{9} + 10503159571784172 T^{10} - 52129708412575536 T^{11} + 6654358221039174920 T^{12} - 15959802004090157072 T^{13} +$$$$29\!\cdots\!24$$$$T^{14} +$$$$72\!\cdots\!81$$$$T^{16} )^{2}$$
$37$ $$( 1 - 40 T + 6968 T^{2} - 291512 T^{3} + 25042268 T^{4} - 987943976 T^{5} + 58242343432 T^{6} - 2052793579384 T^{7} + 94406736665478 T^{8} - 2810274410176696 T^{9} + 109155528608860552 T^{10} - 2534793949835662184 T^{11} + 87960451829583332828 T^{12} -$$$$14\!\cdots\!88$$$$T^{13} +$$$$45\!\cdots\!08$$$$T^{14} -$$$$36\!\cdots\!60$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16} )^{2}$$
$41$ $$1 - 10832 T^{2} + 65454200 T^{4} - 277210214128 T^{6} + 920889789641756 T^{8} - 2528312319464856400 T^{10} +$$$$59\!\cdots\!28$$$$T^{12} -$$$$12\!\cdots\!00$$$$T^{14} +$$$$21\!\cdots\!66$$$$T^{16} -$$$$34\!\cdots\!00$$$$T^{18} +$$$$47\!\cdots\!88$$$$T^{20} -$$$$57\!\cdots\!00$$$$T^{22} +$$$$58\!\cdots\!96$$$$T^{24} -$$$$49\!\cdots\!28$$$$T^{26} +$$$$33\!\cdots\!00$$$$T^{28} -$$$$15\!\cdots\!72$$$$T^{30} +$$$$40\!\cdots\!81$$$$T^{32}$$
$43$ $$( 1 + 168 T + 21576 T^{2} + 1867768 T^{3} + 137965244 T^{4} + 8243885160 T^{5} + 447366405624 T^{6} + 21255986843000 T^{7} + 961142451546054 T^{8} + 39302319672707000 T^{9} + 1529456714913736824 T^{10} + 52112591030623452840 T^{11} +$$$$16\!\cdots\!44$$$$T^{12} +$$$$40\!\cdots\!32$$$$T^{13} +$$$$86\!\cdots\!76$$$$T^{14} +$$$$12\!\cdots\!32$$$$T^{15} +$$$$13\!\cdots\!01$$$$T^{16} )^{2}$$
$47$ $$1 - 18860 T^{2} + 165198086 T^{4} - 880812409480 T^{6} + 3099424527294161 T^{8} - 7043887723434328360 T^{10} +$$$$79\!\cdots\!02$$$$T^{12} +$$$$68\!\cdots\!80$$$$T^{14} -$$$$40\!\cdots\!04$$$$T^{16} +$$$$33\!\cdots\!80$$$$T^{18} +$$$$18\!\cdots\!22$$$$T^{20} -$$$$81\!\cdots\!60$$$$T^{22} +$$$$17\!\cdots\!81$$$$T^{24} -$$$$24\!\cdots\!80$$$$T^{26} +$$$$22\!\cdots\!66$$$$T^{28} -$$$$12\!\cdots\!60$$$$T^{30} +$$$$32\!\cdots\!41$$$$T^{32}$$
$53$ $$1 - 15704 T^{2} + 138921152 T^{4} - 871766999944 T^{6} + 4342207935569660 T^{8} - 18113936181752364376 T^{10} +$$$$65\!\cdots\!52$$$$T^{12} -$$$$21\!\cdots\!64$$$$T^{14} +$$$$62\!\cdots\!50$$$$T^{16} -$$$$16\!\cdots\!84$$$$T^{18} +$$$$40\!\cdots\!72$$$$T^{20} -$$$$88\!\cdots\!16$$$$T^{22} +$$$$16\!\cdots\!60$$$$T^{24} -$$$$26\!\cdots\!44$$$$T^{26} +$$$$33\!\cdots\!12$$$$T^{28} -$$$$29\!\cdots\!44$$$$T^{30} +$$$$15\!\cdots\!41$$$$T^{32}$$
$59$ $$1 - 26768 T^{2} + 368873336 T^{4} - 3501619664560 T^{6} + 25676127920457884 T^{8} -$$$$15\!\cdots\!96$$$$T^{10} +$$$$77\!\cdots\!60$$$$T^{12} -$$$$33\!\cdots\!08$$$$T^{14} +$$$$12\!\cdots\!58$$$$T^{16} -$$$$40\!\cdots\!88$$$$T^{18} +$$$$11\!\cdots\!60$$$$T^{20} -$$$$27\!\cdots\!76$$$$T^{22} +$$$$55\!\cdots\!44$$$$T^{24} -$$$$91\!\cdots\!60$$$$T^{26} +$$$$11\!\cdots\!96$$$$T^{28} -$$$$10\!\cdots\!28$$$$T^{30} +$$$$46\!\cdots\!81$$$$T^{32}$$
$61$ $$( 1 - 56 T + 11072 T^{2} - 269288 T^{3} + 50054524 T^{4} - 711106488 T^{5} + 247917300928 T^{6} - 7112512171688 T^{7} + 1188015110923334 T^{8} - 26465657790851048 T^{9} + 3432623529798240448 T^{10} - 36636472472295954168 T^{11} +$$$$95\!\cdots\!44$$$$T^{12} -$$$$19\!\cdots\!88$$$$T^{13} +$$$$29\!\cdots\!12$$$$T^{14} -$$$$55\!\cdots\!96$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16} )^{2}$$
$67$ $$( 1 - 120 T + 21032 T^{2} - 1363784 T^{3} + 159985756 T^{4} - 7095679000 T^{5} + 808761061016 T^{6} - 25648223522216 T^{7} + 3474267779595526 T^{8} - 115134875391227624 T^{9} + 16297442000621798936 T^{10} -$$$$64\!\cdots\!00$$$$T^{11} +$$$$64\!\cdots\!96$$$$T^{12} -$$$$24\!\cdots\!16$$$$T^{13} +$$$$17\!\cdots\!52$$$$T^{14} -$$$$44\!\cdots\!80$$$$T^{15} +$$$$16\!\cdots\!81$$$$T^{16} )^{2}$$
$71$ $$1 - 58328 T^{2} + 1630496768 T^{4} - 29214173879944 T^{6} + 378588472445657852 T^{8} -$$$$37\!\cdots\!40$$$$T^{10} +$$$$30\!\cdots\!12$$$$T^{12} -$$$$20\!\cdots\!48$$$$T^{14} +$$$$11\!\cdots\!90$$$$T^{16} -$$$$51\!\cdots\!88$$$$T^{18} +$$$$19\!\cdots\!32$$$$T^{20} -$$$$62\!\cdots\!40$$$$T^{22} +$$$$15\!\cdots\!92$$$$T^{24} -$$$$30\!\cdots\!44$$$$T^{26} +$$$$43\!\cdots\!08$$$$T^{28} -$$$$39\!\cdots\!08$$$$T^{30} +$$$$17\!\cdots\!41$$$$T^{32}$$
$73$ $$( 1 - 24 T + 15156 T^{2} + 137320 T^{3} + 133090184 T^{4} + 1190310168 T^{5} + 1034805205980 T^{6} + 7023266129048 T^{7} + 5908298409616782 T^{8} + 37426985201696792 T^{9} + 29386647627474681180 T^{10} +$$$$18\!\cdots\!52$$$$T^{11} +$$$$10\!\cdots\!04$$$$T^{12} +$$$$59\!\cdots\!80$$$$T^{13} +$$$$34\!\cdots\!76$$$$T^{14} -$$$$29\!\cdots\!16$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16} )^{2}$$
$79$ $$( 1 - 4 T + 21374 T^{2} - 1216 T^{3} + 230456449 T^{4} + 2685965400 T^{5} + 1772108711446 T^{6} + 44455008585260 T^{7} + 11672772046344164 T^{8} + 277443708580607660 T^{9} + 69023777851627327126 T^{10} +$$$$65\!\cdots\!00$$$$T^{11} +$$$$34\!\cdots\!89$$$$T^{12} -$$$$11\!\cdots\!16$$$$T^{13} +$$$$12\!\cdots\!34$$$$T^{14} -$$$$14\!\cdots\!24$$$$T^{15} +$$$$23\!\cdots\!21$$$$T^{16} )^{2}$$
$83$ $$1 - 26752 T^{2} + 559157880 T^{4} - 7483229844864 T^{6} + 85634770716597660 T^{8} -$$$$74\!\cdots\!16$$$$T^{10} +$$$$60\!\cdots\!76$$$$T^{12} -$$$$41\!\cdots\!76$$$$T^{14} +$$$$29\!\cdots\!06$$$$T^{16} -$$$$19\!\cdots\!96$$$$T^{18} +$$$$13\!\cdots\!16$$$$T^{20} -$$$$80\!\cdots\!76$$$$T^{22} +$$$$43\!\cdots\!60$$$$T^{24} -$$$$18\!\cdots\!64$$$$T^{26} +$$$$63\!\cdots\!80$$$$T^{28} -$$$$14\!\cdots\!32$$$$T^{30} +$$$$25\!\cdots\!61$$$$T^{32}$$
$89$ $$1 - 71136 T^{2} + 2535583160 T^{4} - 60654761408672 T^{6} + 1094469962087760028 T^{8} -$$$$15\!\cdots\!12$$$$T^{10} +$$$$18\!\cdots\!28$$$$T^{12} -$$$$19\!\cdots\!80$$$$T^{14} +$$$$16\!\cdots\!02$$$$T^{16} -$$$$11\!\cdots\!80$$$$T^{18} +$$$$74\!\cdots\!68$$$$T^{20} -$$$$39\!\cdots\!52$$$$T^{22} +$$$$16\!\cdots\!08$$$$T^{24} -$$$$58\!\cdots\!72$$$$T^{26} +$$$$15\!\cdots\!60$$$$T^{28} -$$$$27\!\cdots\!16$$$$T^{30} +$$$$24\!\cdots\!21$$$$T^{32}$$
$97$ $$( 1 + 96 T + 62878 T^{2} + 5357144 T^{3} + 1785824677 T^{4} + 133941251648 T^{5} + 30441979716138 T^{6} + 1968996759462584 T^{7} + 345736614004039176 T^{8} + 18526290509783452856 T^{9} +$$$$26\!\cdots\!78$$$$T^{10} +$$$$11\!\cdots\!92$$$$T^{11} +$$$$13\!\cdots\!97$$$$T^{12} +$$$$39\!\cdots\!56$$$$T^{13} +$$$$43\!\cdots\!98$$$$T^{14} +$$$$62\!\cdots\!24$$$$T^{15} +$$$$61\!\cdots\!21$$$$T^{16} )^{2}$$