Properties

Label 35.3.h
Level $35$
Weight $3$
Character orbit 35.h
Rep. character $\chi_{35}(26,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $12$
Newform subspaces $1$
Sturm bound $12$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 35.h (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(12\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(35, [\chi])\).

Total New Old
Modular forms 20 12 8
Cusp forms 12 12 0
Eisenstein series 8 0 8

Trace form

\( 12q - 2q^{2} - 6q^{3} - 10q^{4} - 2q^{7} - 4q^{8} + 14q^{9} + O(q^{10}) \) \( 12q - 2q^{2} - 6q^{3} - 10q^{4} - 2q^{7} - 4q^{8} + 14q^{9} - 14q^{11} + 18q^{12} - 2q^{14} - 20q^{15} - 22q^{16} + 48q^{17} + 64q^{18} - 30q^{19} - 84q^{21} - 88q^{22} - 14q^{23} - 36q^{24} + 30q^{25} + 66q^{26} + 202q^{28} + 64q^{29} + 20q^{30} + 132q^{31} - 54q^{32} - 192q^{33} + 30q^{35} + 156q^{36} + 44q^{37} - 300q^{38} - 24q^{39} - 138q^{42} - 4q^{43} + 6q^{44} - 180q^{45} - 214q^{46} + 204q^{47} - 24q^{49} - 20q^{50} - 132q^{51} + 252q^{52} + 196q^{53} + 168q^{54} - 460q^{56} - 48q^{57} + 158q^{58} + 72q^{59} + 150q^{60} + 72q^{61} + 536q^{63} - 140q^{64} + 30q^{65} + 744q^{66} - 138q^{67} - 348q^{68} + 240q^{70} - 8q^{71} - 196q^{72} - 528q^{73} + 50q^{74} - 30q^{75} - 176q^{77} - 312q^{78} - 12q^{79} - 240q^{80} - 310q^{81} - 378q^{82} - 276q^{84} - 40q^{86} + 138q^{87} + 604q^{88} + 204q^{89} - 480q^{91} + 732q^{92} + 84q^{93} - 42q^{94} + 60q^{95} + 540q^{96} + 898q^{98} - 44q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(35, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
35.3.h.a \(12\) \(0.954\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-2\) \(-6\) \(0\) \(-2\) \(q+(\beta _{1}+\beta _{2})q^{2}+(-1+\beta _{7})q^{3}+(-2+\cdots)q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T - 5 T^{2} - 6 T^{3} + 20 T^{4} + 10 T^{5} + 25 T^{6} + 58 T^{7} - 436 T^{8} - 470 T^{9} + 1107 T^{10} - 102 T^{11} - 5463 T^{12} - 408 T^{13} + 17712 T^{14} - 30080 T^{15} - 111616 T^{16} + 59392 T^{17} + 102400 T^{18} + 163840 T^{19} + 1310720 T^{20} - 1572864 T^{21} - 5242880 T^{22} + 8388608 T^{23} + 16777216 T^{24} \)
$3$ \( 1 + 6 T + 38 T^{2} + 156 T^{3} + 660 T^{4} + 2070 T^{5} + 5744 T^{6} + 10314 T^{7} - 68 T^{8} - 114228 T^{9} - 708426 T^{10} - 2725974 T^{11} - 9235782 T^{12} - 24533766 T^{13} - 57382506 T^{14} - 83272212 T^{15} - 446148 T^{16} + 609031386 T^{17} + 3052597104 T^{18} + 9900745830 T^{19} + 28410835860 T^{20} + 60437596284 T^{21} + 132497807238 T^{22} + 188286357654 T^{23} + 282429536481 T^{24} \)
$5$ \( ( 1 - 5 T^{2} + 25 T^{4} )^{3} \)
$7$ \( 1 + 2 T + 14 T^{2} + 348 T^{3} + 1588 T^{4} + 14434 T^{5} + 136318 T^{6} + 707266 T^{7} + 3812788 T^{8} + 40941852 T^{9} + 80707214 T^{10} + 564950498 T^{11} + 13841287201 T^{12} \)
$11$ \( 1 + 14 T - 179 T^{2} - 3378 T^{3} - 10510 T^{4} - 101510 T^{5} - 1940909 T^{6} + 49596994 T^{7} + 1461472640 T^{8} + 7377470974 T^{9} - 78163118391 T^{10} - 1070296414578 T^{11} - 9257941759776 T^{12} - 129505866163938 T^{13} - 1144386216362631 T^{14} + 13069639856170414 T^{15} + 313279639722515840 T^{16} + 1286418292311249394 T^{17} - 6091403882233179389 T^{18} - 38548405607034793910 T^{19} - \)\(48\!\cdots\!10\)\( T^{20} - \)\(18\!\cdots\!18\)\( T^{21} - \)\(12\!\cdots\!79\)\( T^{22} + \)\(11\!\cdots\!94\)\( T^{23} + \)\(98\!\cdots\!41\)\( T^{24} \)
$13$ \( 1 - 1482 T^{2} + 1042167 T^{4} - 464849386 T^{6} + 148108329555 T^{8} - 35857522447788 T^{10} + 6808838187253242 T^{12} - 1024126698631273068 T^{14} + \)\(12\!\cdots\!55\)\( T^{16} - \)\(10\!\cdots\!66\)\( T^{18} + \)\(69\!\cdots\!47\)\( T^{20} - \)\(28\!\cdots\!82\)\( T^{22} + \)\(54\!\cdots\!61\)\( T^{24} \)
$17$ \( 1 - 48 T + 2382 T^{2} - 77472 T^{3} + 2498625 T^{4} - 67135632 T^{5} + 1727446034 T^{6} - 40136784912 T^{7} + 880931070942 T^{8} - 18174432672288 T^{9} + 351810098240190 T^{10} - 6513275972033712 T^{11} + 112870694757995013 T^{12} - 1882336755917742768 T^{13} + 29383531215118908990 T^{14} - \)\(43\!\cdots\!72\)\( T^{15} + \)\(61\!\cdots\!22\)\( T^{16} - \)\(80\!\cdots\!88\)\( T^{17} + \)\(10\!\cdots\!74\)\( T^{18} - \)\(11\!\cdots\!28\)\( T^{19} + \)\(12\!\cdots\!25\)\( T^{20} - \)\(10\!\cdots\!48\)\( T^{21} + \)\(96\!\cdots\!82\)\( T^{22} - \)\(56\!\cdots\!72\)\( T^{23} + \)\(33\!\cdots\!21\)\( T^{24} \)
$19$ \( 1 + 30 T + 1401 T^{2} + 33030 T^{3} + 884826 T^{4} + 20001714 T^{5} + 344601839 T^{6} + 6453829386 T^{7} + 66537113172 T^{8} + 653188957326 T^{9} - 1006933743831 T^{10} - 384405046252650 T^{11} - 5242806750443784 T^{12} - 138770221697206650 T^{13} - 131224612429799751 T^{14} + 30729849956873074206 T^{15} + \)\(11\!\cdots\!52\)\( T^{16} + \)\(39\!\cdots\!86\)\( T^{17} + \)\(76\!\cdots\!79\)\( T^{18} + \)\(15\!\cdots\!94\)\( T^{19} + \)\(25\!\cdots\!06\)\( T^{20} + \)\(34\!\cdots\!30\)\( T^{21} + \)\(52\!\cdots\!01\)\( T^{22} + \)\(40\!\cdots\!30\)\( T^{23} + \)\(48\!\cdots\!21\)\( T^{24} \)
$23$ \( 1 + 14 T - 1631 T^{2} - 3594 T^{3} + 1665797 T^{4} - 8196524 T^{5} - 890248094 T^{6} + 13524083944 T^{7} + 223091831501 T^{8} - 7643356730246 T^{9} + 95428737941649 T^{10} + 2006796517495686 T^{11} - 92105247923668542 T^{12} + 1061595357755217894 T^{13} + 26704873454328997809 T^{14} - \)\(11\!\cdots\!94\)\( T^{15} + \)\(17\!\cdots\!81\)\( T^{16} + \)\(56\!\cdots\!56\)\( T^{17} - \)\(19\!\cdots\!74\)\( T^{18} - \)\(95\!\cdots\!16\)\( T^{19} + \)\(10\!\cdots\!17\)\( T^{20} - \)\(11\!\cdots\!86\)\( T^{21} - \)\(27\!\cdots\!31\)\( T^{22} + \)\(12\!\cdots\!06\)\( T^{23} + \)\(48\!\cdots\!41\)\( T^{24} \)
$29$ \( ( 1 - 32 T + 3552 T^{2} - 122256 T^{3} + 5916460 T^{4} - 196641728 T^{5} + 6117094942 T^{6} - 165375693248 T^{7} + 4184599745260 T^{8} - 72720719932176 T^{9} + 1776875258837472 T^{10} - 13462631465606432 T^{11} + 353814783205469041 T^{12} )^{2} \)
$31$ \( 1 - 132 T + 12894 T^{2} - 935352 T^{3} + 58065489 T^{4} - 3050976288 T^{5} + 143999803298 T^{6} - 6064589395908 T^{7} + 236052211330302 T^{8} - 8471733336563436 T^{9} + 289433514790600350 T^{10} - 9384030507705444000 T^{11} + \)\(29\!\cdots\!29\)\( T^{12} - \)\(90\!\cdots\!00\)\( T^{13} + \)\(26\!\cdots\!50\)\( T^{14} - \)\(75\!\cdots\!16\)\( T^{15} + \)\(20\!\cdots\!82\)\( T^{16} - \)\(49\!\cdots\!08\)\( T^{17} + \)\(11\!\cdots\!78\)\( T^{18} - \)\(23\!\cdots\!48\)\( T^{19} + \)\(42\!\cdots\!09\)\( T^{20} - \)\(65\!\cdots\!32\)\( T^{21} + \)\(86\!\cdots\!94\)\( T^{22} - \)\(85\!\cdots\!52\)\( T^{23} + \)\(62\!\cdots\!21\)\( T^{24} \)
$37$ \( 1 - 44 T - 4291 T^{2} + 241124 T^{3} + 9668742 T^{4} - 689019636 T^{5} - 12024976949 T^{6} + 1280625867436 T^{7} + 4009330536896 T^{8} - 1538422889903564 T^{9} + 15464652696139049 T^{10} + 849812472525660964 T^{11} - 34333638656751212332 T^{12} + \)\(11\!\cdots\!16\)\( T^{13} + \)\(28\!\cdots\!89\)\( T^{14} - \)\(39\!\cdots\!76\)\( T^{15} + \)\(14\!\cdots\!16\)\( T^{16} + \)\(61\!\cdots\!64\)\( T^{17} - \)\(79\!\cdots\!69\)\( T^{18} - \)\(62\!\cdots\!04\)\( T^{19} + \)\(11\!\cdots\!22\)\( T^{20} + \)\(40\!\cdots\!96\)\( T^{21} - \)\(99\!\cdots\!91\)\( T^{22} - \)\(13\!\cdots\!36\)\( T^{23} + \)\(43\!\cdots\!61\)\( T^{24} \)
$41$ \( 1 - 6978 T^{2} + 29395173 T^{4} - 89601849670 T^{6} + 218675773246986 T^{8} - 448976563577597898 T^{10} + \)\(80\!\cdots\!77\)\( T^{12} - \)\(12\!\cdots\!78\)\( T^{14} + \)\(17\!\cdots\!06\)\( T^{16} - \)\(20\!\cdots\!70\)\( T^{18} + \)\(18\!\cdots\!93\)\( T^{20} - \)\(12\!\cdots\!78\)\( T^{22} + \)\(50\!\cdots\!61\)\( T^{24} \)
$43$ \( ( 1 + 2 T + 7538 T^{2} + 1596 T^{3} + 27008008 T^{4} - 11306606 T^{5} + 60967237342 T^{6} - 20905914494 T^{7} + 92335004758408 T^{8} + 10088895426204 T^{9} + 88105653692556338 T^{10} + 43222964626568498 T^{11} + 39959630797262576401 T^{12} )^{2} \)
$47$ \( 1 - 204 T + 29397 T^{2} - 3167100 T^{3} + 290418402 T^{4} - 23370921444 T^{5} + 1705985151887 T^{6} - 114387377070276 T^{7} + 7121727778488432 T^{8} - 413844362647971516 T^{9} + 22564123030969504557 T^{10} - \)\(11\!\cdots\!96\)\( T^{11} + \)\(56\!\cdots\!04\)\( T^{12} - \)\(25\!\cdots\!64\)\( T^{13} + \)\(11\!\cdots\!17\)\( T^{14} - \)\(44\!\cdots\!64\)\( T^{15} + \)\(16\!\cdots\!52\)\( T^{16} - \)\(60\!\cdots\!24\)\( T^{17} + \)\(19\!\cdots\!67\)\( T^{18} - \)\(59\!\cdots\!36\)\( T^{19} + \)\(16\!\cdots\!42\)\( T^{20} - \)\(39\!\cdots\!00\)\( T^{21} + \)\(81\!\cdots\!97\)\( T^{22} - \)\(12\!\cdots\!36\)\( T^{23} + \)\(13\!\cdots\!81\)\( T^{24} \)
$53$ \( 1 - 196 T + 13213 T^{2} - 531828 T^{3} + 46624490 T^{4} - 3460227932 T^{5} + 114727083151 T^{6} - 3187972234364 T^{7} + 109323786787880 T^{8} + 2196815143901836 T^{9} - 301463170054474635 T^{10} + 37355670780781789692 T^{11} - \)\(32\!\cdots\!76\)\( T^{12} + \)\(10\!\cdots\!28\)\( T^{13} - \)\(23\!\cdots\!35\)\( T^{14} + \)\(48\!\cdots\!44\)\( T^{15} + \)\(68\!\cdots\!80\)\( T^{16} - \)\(55\!\cdots\!36\)\( T^{17} + \)\(56\!\cdots\!91\)\( T^{18} - \)\(47\!\cdots\!08\)\( T^{19} + \)\(18\!\cdots\!90\)\( T^{20} - \)\(57\!\cdots\!92\)\( T^{21} + \)\(40\!\cdots\!13\)\( T^{22} - \)\(16\!\cdots\!64\)\( T^{23} + \)\(24\!\cdots\!81\)\( T^{24} \)
$59$ \( 1 - 72 T + 17682 T^{2} - 1148688 T^{3} + 164787345 T^{4} - 10566780840 T^{5} + 1093761678926 T^{6} - 70159293997992 T^{7} + 5748831154849182 T^{8} - 368559907967058624 T^{9} + 25324575120331389906 T^{10} - \)\(15\!\cdots\!28\)\( T^{11} + \)\(95\!\cdots\!93\)\( T^{12} - \)\(54\!\cdots\!68\)\( T^{13} + \)\(30\!\cdots\!66\)\( T^{14} - \)\(15\!\cdots\!84\)\( T^{15} + \)\(84\!\cdots\!22\)\( T^{16} - \)\(35\!\cdots\!92\)\( T^{17} + \)\(19\!\cdots\!06\)\( T^{18} - \)\(65\!\cdots\!40\)\( T^{19} + \)\(35\!\cdots\!45\)\( T^{20} - \)\(86\!\cdots\!48\)\( T^{21} + \)\(46\!\cdots\!82\)\( T^{22} - \)\(65\!\cdots\!32\)\( T^{23} + \)\(31\!\cdots\!61\)\( T^{24} \)
$61$ \( 1 - 72 T + 19176 T^{2} - 1256256 T^{3} + 196053300 T^{4} - 13507864560 T^{5} + 1475851108916 T^{6} - 102662314431264 T^{7} + 8714109428620692 T^{8} - 590990079276027360 T^{9} + 42479838876985802160 T^{10} - \)\(27\!\cdots\!84\)\( T^{11} + \)\(17\!\cdots\!82\)\( T^{12} - \)\(10\!\cdots\!64\)\( T^{13} + \)\(58\!\cdots\!60\)\( T^{14} - \)\(30\!\cdots\!60\)\( T^{15} + \)\(16\!\cdots\!52\)\( T^{16} - \)\(73\!\cdots\!64\)\( T^{17} + \)\(39\!\cdots\!36\)\( T^{18} - \)\(13\!\cdots\!60\)\( T^{19} + \)\(72\!\cdots\!00\)\( T^{20} - \)\(17\!\cdots\!36\)\( T^{21} + \)\(97\!\cdots\!76\)\( T^{22} - \)\(13\!\cdots\!12\)\( T^{23} + \)\(70\!\cdots\!41\)\( T^{24} \)
$67$ \( 1 + 138 T - 9270 T^{2} - 1437548 T^{3} + 115886436 T^{4} + 10557292578 T^{5} - 1139954744456 T^{6} - 58817128757610 T^{7} + 8471104062197196 T^{8} + 229786064617296868 T^{9} - 50182738990417249110 T^{10} - \)\(37\!\cdots\!50\)\( T^{11} + \)\(25\!\cdots\!50\)\( T^{12} - \)\(16\!\cdots\!50\)\( T^{13} - \)\(10\!\cdots\!10\)\( T^{14} + \)\(20\!\cdots\!92\)\( T^{15} + \)\(34\!\cdots\!36\)\( T^{16} - \)\(10\!\cdots\!90\)\( T^{17} - \)\(93\!\cdots\!16\)\( T^{18} + \)\(38\!\cdots\!62\)\( T^{19} + \)\(19\!\cdots\!16\)\( T^{20} - \)\(10\!\cdots\!32\)\( T^{21} - \)\(30\!\cdots\!70\)\( T^{22} + \)\(20\!\cdots\!82\)\( T^{23} + \)\(66\!\cdots\!21\)\( T^{24} \)
$71$ \( ( 1 + 4 T + 14826 T^{2} - 442692 T^{3} + 122154931 T^{4} - 3956292368 T^{5} + 781022121076 T^{6} - 19943669827088 T^{7} + 3104162139149011 T^{8} - 56708970889555332 T^{9} + 9573941854249652586 T^{10} + 13020974204039524804 T^{11} + \)\(16\!\cdots\!41\)\( T^{12} )^{2} \)
$73$ \( 1 + 528 T + 147318 T^{2} + 28717920 T^{3} + 4362863697 T^{4} + 544924287120 T^{5} + 57462036213962 T^{6} + 5186233581298704 T^{7} + 403863565075955790 T^{8} + 27374688378441994752 T^{9} + \)\(16\!\cdots\!86\)\( T^{10} + \)\(95\!\cdots\!24\)\( T^{11} + \)\(62\!\cdots\!73\)\( T^{12} + \)\(51\!\cdots\!96\)\( T^{13} + \)\(46\!\cdots\!26\)\( T^{14} + \)\(41\!\cdots\!28\)\( T^{15} + \)\(32\!\cdots\!90\)\( T^{16} + \)\(22\!\cdots\!96\)\( T^{17} + \)\(13\!\cdots\!02\)\( T^{18} + \)\(66\!\cdots\!80\)\( T^{19} + \)\(28\!\cdots\!17\)\( T^{20} + \)\(99\!\cdots\!80\)\( T^{21} + \)\(27\!\cdots\!18\)\( T^{22} + \)\(51\!\cdots\!12\)\( T^{23} + \)\(52\!\cdots\!41\)\( T^{24} \)
$79$ \( 1 + 12 T - 26574 T^{2} - 366848 T^{3} + 390887217 T^{4} + 5621569728 T^{5} - 3680496208466 T^{6} - 54811745357412 T^{7} + 24735046819850526 T^{8} + 328584508359816652 T^{9} - \)\(13\!\cdots\!54\)\( T^{10} - \)\(88\!\cdots\!28\)\( T^{11} + \)\(71\!\cdots\!73\)\( T^{12} - \)\(55\!\cdots\!48\)\( T^{13} - \)\(50\!\cdots\!74\)\( T^{14} + \)\(79\!\cdots\!92\)\( T^{15} + \)\(37\!\cdots\!86\)\( T^{16} - \)\(51\!\cdots\!12\)\( T^{17} - \)\(21\!\cdots\!06\)\( T^{18} + \)\(20\!\cdots\!68\)\( T^{19} + \)\(89\!\cdots\!57\)\( T^{20} - \)\(52\!\cdots\!28\)\( T^{21} - \)\(23\!\cdots\!74\)\( T^{22} + \)\(67\!\cdots\!92\)\( T^{23} + \)\(34\!\cdots\!81\)\( T^{24} \)
$83$ \( 1 - 43200 T^{2} + 987057204 T^{4} - 15370870045228 T^{6} + 180607989502531968 T^{8} - \)\(16\!\cdots\!36\)\( T^{10} + \)\(12\!\cdots\!66\)\( T^{12} - \)\(79\!\cdots\!56\)\( T^{14} + \)\(40\!\cdots\!88\)\( T^{16} - \)\(16\!\cdots\!08\)\( T^{18} + \)\(50\!\cdots\!24\)\( T^{20} - \)\(10\!\cdots\!00\)\( T^{22} + \)\(11\!\cdots\!21\)\( T^{24} \)
$89$ \( 1 - 204 T + 49104 T^{2} - 7187328 T^{3} + 1045283496 T^{4} - 127643879244 T^{5} + 15255527568428 T^{6} - 1684635845896692 T^{7} + 183174646224057864 T^{8} - 18563328978636389184 T^{9} + \)\(18\!\cdots\!76\)\( T^{10} - \)\(17\!\cdots\!52\)\( T^{11} + \)\(16\!\cdots\!74\)\( T^{12} - \)\(13\!\cdots\!92\)\( T^{13} + \)\(11\!\cdots\!16\)\( T^{14} - \)\(92\!\cdots\!24\)\( T^{15} + \)\(72\!\cdots\!84\)\( T^{16} - \)\(52\!\cdots\!92\)\( T^{17} + \)\(37\!\cdots\!88\)\( T^{18} - \)\(24\!\cdots\!04\)\( T^{19} + \)\(16\!\cdots\!56\)\( T^{20} - \)\(88\!\cdots\!68\)\( T^{21} + \)\(47\!\cdots\!04\)\( T^{22} - \)\(15\!\cdots\!84\)\( T^{23} + \)\(61\!\cdots\!41\)\( T^{24} \)
$97$ \( 1 - 64284 T^{2} + 2051576418 T^{4} - 44019752488108 T^{6} + 711686325387159663 T^{8} - \)\(91\!\cdots\!92\)\( T^{10} + \)\(94\!\cdots\!68\)\( T^{12} - \)\(80\!\cdots\!52\)\( T^{14} + \)\(55\!\cdots\!43\)\( T^{16} - \)\(30\!\cdots\!28\)\( T^{18} + \)\(12\!\cdots\!78\)\( T^{20} - \)\(34\!\cdots\!84\)\( T^{22} + \)\(48\!\cdots\!81\)\( T^{24} \)
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