# 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 $0$

# Related objects

## 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

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