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

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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

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