# Properties

 Label 35.3.g Level 35 Weight 3 Character orbit g Rep. character $$\chi_{35}(8,\cdot)$$ Character field $$\Q(\zeta_{4})$$ 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.g (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q(i)$$ 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

 $$12q - 4q^{2} - 4q^{3} - 8q^{5} - 24q^{6} + 24q^{8} + O(q^{10})$$ $$12q - 4q^{2} - 4q^{3} - 8q^{5} - 24q^{6} + 24q^{8} + 28q^{10} - 12q^{11} + 16q^{12} - 4q^{13} - 64q^{15} + 40q^{16} - 12q^{17} - 56q^{18} + 60q^{20} + 28q^{21} - 68q^{22} - 16q^{23} + 64q^{25} - 56q^{26} + 164q^{27} - 76q^{30} - 96q^{31} + 32q^{32} + 124q^{33} + 232q^{36} - 104q^{37} + 80q^{38} - 124q^{40} - 208q^{41} - 140q^{42} + 76q^{43} + 92q^{45} - 80q^{46} - 164q^{47} - 392q^{48} - 52q^{50} + 220q^{51} + 216q^{52} - 204q^{53} + 116q^{55} + 168q^{56} - 236q^{57} + 356q^{58} + 152q^{60} + 280q^{61} + 568q^{62} + 112q^{63} - 192q^{65} - 544q^{66} + 324q^{67} + 184q^{68} - 112q^{70} + 144q^{71} - 440q^{72} - 248q^{73} + 108q^{75} - 632q^{76} - 56q^{77} + 12q^{78} + 60q^{80} - 260q^{81} - 376q^{82} - 224q^{83} - 324q^{85} + 456q^{86} + 244q^{87} - 24q^{88} + 780q^{90} + 84q^{91} - 424q^{92} + 236q^{93} + 52q^{95} + 504q^{96} + 564q^{97} + 28q^{98} + O(q^{100})$$

## 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.g.a $$12$$ $$0.954$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$-4$$ $$-4$$ $$-8$$ $$0$$ $$q-\beta _{1}q^{2}+\beta _{7}q^{3}+(\beta _{1}+\beta _{3}+\beta _{4}+\beta _{10}+\cdots)q^{4}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 8 T^{2} + 8 T^{3} - 26 T^{4} - 104 T^{5} - 176 T^{6} - 208 T^{7} + 9 T^{8} + 468 T^{9} + 968 T^{10} + 2568 T^{11} + 5732 T^{12} + 10272 T^{13} + 15488 T^{14} + 29952 T^{15} + 2304 T^{16} - 212992 T^{17} - 720896 T^{18} - 1703936 T^{19} - 1703936 T^{20} + 2097152 T^{21} + 8388608 T^{22} + 16777216 T^{23} + 16777216 T^{24}$$
$3$ $$1 + 4 T + 8 T^{2} - 32 T^{3} - 185 T^{4} - 104 T^{5} + 1576 T^{6} + 6004 T^{7} - 7229 T^{8} - 84904 T^{9} - 176944 T^{10} + 302584 T^{11} + 2631886 T^{12} + 2723256 T^{13} - 14332464 T^{14} - 61895016 T^{15} - 47429469 T^{16} + 354530196 T^{17} + 837551016 T^{18} - 497428776 T^{19} - 7963643385 T^{20} - 12397455648 T^{21} + 27894275208 T^{22} + 125524238436 T^{23} + 282429536481 T^{24}$$
$5$ $$1 + 8 T - 328 T^{3} - 1231 T^{4} + 2600 T^{5} + 41200 T^{6} + 65000 T^{7} - 769375 T^{8} - 5125000 T^{9} + 78125000 T^{11} + 244140625 T^{12}$$
$7$ $$( 1 + 49 T^{4} )^{3}$$
$11$ $$( 1 + 6 T + 319 T^{2} + 1166 T^{3} + 55579 T^{4} + 201340 T^{5} + 7907818 T^{6} + 24362140 T^{7} + 813732139 T^{8} + 2065640126 T^{9} + 68380483039 T^{10} + 155624547606 T^{11} + 3138428376721 T^{12} )^{2}$$
$13$ $$1 + 4 T + 8 T^{2} - 2424 T^{3} + 42679 T^{4} + 821112 T^{5} + 5880904 T^{6} - 59122612 T^{7} - 455341821 T^{8} + 28627711344 T^{9} + 347588147888 T^{10} + 1969414999296 T^{11} - 34074574484818 T^{12} + 332831134881024 T^{13} + 9927465091829168 T^{14} + 138180494764621296 T^{15} - 371436311945782941 T^{16} - 8150554124493589588 T^{17} +$$$$13\!\cdots\!24$$$$T^{18} +$$$$32\!\cdots\!68$$$$T^{19} +$$$$28\!\cdots\!39$$$$T^{20} -$$$$27\!\cdots\!96$$$$T^{21} +$$$$15\!\cdots\!08$$$$T^{22} +$$$$12\!\cdots\!76$$$$T^{23} +$$$$54\!\cdots\!61$$$$T^{24}$$
$17$ $$1 + 12 T + 72 T^{2} + 2784 T^{3} + 76619 T^{4} + 611296 T^{5} + 5694312 T^{6} + 329614108 T^{7} - 1813984861 T^{8} - 125752615304 T^{9} - 710229888304 T^{10} - 21298315351968 T^{11} - 439455529642318 T^{12} - 6155213136718752 T^{13} - 59319110501038384 T^{14} - 3035362428830755976 T^{15} - 12653918391982100701 T^{16} +$$$$66\!\cdots\!92$$$$T^{17} +$$$$33\!\cdots\!32$$$$T^{18} +$$$$10\!\cdots\!84$$$$T^{19} +$$$$37\!\cdots\!39$$$$T^{20} +$$$$39\!\cdots\!56$$$$T^{21} +$$$$29\!\cdots\!72$$$$T^{22} +$$$$14\!\cdots\!68$$$$T^{23} +$$$$33\!\cdots\!21$$$$T^{24}$$
$19$ $$1 - 2208 T^{2} + 2306338 T^{4} - 1539873472 T^{6} + 753579200867 T^{8} - 300967197557024 T^{10} + 110083881158431556 T^{12} - 39222346152828924704 T^{14} +$$$$12\!\cdots\!47$$$$T^{16} -$$$$34\!\cdots\!92$$$$T^{18} +$$$$66\!\cdots\!78$$$$T^{20} -$$$$82\!\cdots\!08$$$$T^{22} +$$$$48\!\cdots\!21$$$$T^{24}$$
$23$ $$1 + 16 T + 128 T^{2} + 12576 T^{3} + 414846 T^{4} - 3136832 T^{5} - 24211712 T^{6} - 1482627024 T^{7} + 34808010943 T^{8} + 851590367168 T^{9} + 5488492394624 T^{10} + 645053822277920 T^{11} + 70544571658994180 T^{12} + 341233471985019680 T^{13} + 1535905200203974784 T^{14} +$$$$12\!\cdots\!52$$$$T^{15} +$$$$27\!\cdots\!83$$$$T^{16} -$$$$61\!\cdots\!76$$$$T^{17} -$$$$53\!\cdots\!52$$$$T^{18} -$$$$36\!\cdots\!88$$$$T^{19} +$$$$25\!\cdots\!06$$$$T^{20} +$$$$40\!\cdots\!44$$$$T^{21} +$$$$21\!\cdots\!28$$$$T^{22} +$$$$14\!\cdots\!64$$$$T^{23} +$$$$48\!\cdots\!41$$$$T^{24}$$
$29$ $$1 - 7018 T^{2} + 22990943 T^{4} - 47164435002 T^{6} + 68979050432827 T^{8} - 78004905252931004 T^{10} + 71933262446730943946 T^{12} -$$$$55\!\cdots\!24$$$$T^{14} +$$$$34\!\cdots\!47$$$$T^{16} -$$$$16\!\cdots\!82$$$$T^{18} +$$$$57\!\cdots\!03$$$$T^{20} -$$$$12\!\cdots\!18$$$$T^{22} +$$$$12\!\cdots\!81$$$$T^{24}$$
$31$ $$( 1 + 48 T + 3144 T^{2} + 79440 T^{3} + 3260215 T^{4} + 42545920 T^{5} + 2350062944 T^{6} + 40886629120 T^{7} + 3010877017015 T^{8} + 70503292418640 T^{9} + 2681489421714504 T^{10} + 39342157775078448 T^{11} + 787662783788549761 T^{12} )^{2}$$
$37$ $$1 + 104 T + 5408 T^{2} + 174344 T^{3} + 2697254 T^{4} + 10557800 T^{5} + 1709176736 T^{6} + 138455876040 T^{7} + 4386451218735 T^{8} + 70184122125200 T^{9} + 3161821544956736 T^{10} + 334287451090079440 T^{11} + 17427941461580290900 T^{12} +$$$$45\!\cdots\!60$$$$T^{13} +$$$$59\!\cdots\!96$$$$T^{14} +$$$$18\!\cdots\!00$$$$T^{15} +$$$$15\!\cdots\!35$$$$T^{16} +$$$$66\!\cdots\!60$$$$T^{17} +$$$$11\!\cdots\!16$$$$T^{18} +$$$$95\!\cdots\!00$$$$T^{19} +$$$$33\!\cdots\!14$$$$T^{20} +$$$$29\!\cdots\!76$$$$T^{21} +$$$$12\!\cdots\!08$$$$T^{22} +$$$$32\!\cdots\!76$$$$T^{23} +$$$$43\!\cdots\!61$$$$T^{24}$$
$41$ $$( 1 + 104 T + 11808 T^{2} + 782920 T^{3} + 51161335 T^{4} + 2462767696 T^{5} + 114809897552 T^{6} + 4139912496976 T^{7} + 144569705150935 T^{8} + 3718951612363720 T^{9} + 94285997105460768 T^{10} + 1395956568255849704 T^{11} + 22563490300366186081 T^{12} )^{2}$$
$43$ $$1 - 76 T + 2888 T^{2} - 67684 T^{3} + 11551602 T^{4} - 935583796 T^{5} + 40033903848 T^{6} - 1100447483004 T^{7} + 64203765803999 T^{8} - 5036169803944728 T^{9} + 247013912808013264 T^{10} - 7962007386466391112 T^{11} +$$$$24\!\cdots\!96$$$$T^{12} -$$$$14\!\cdots\!88$$$$T^{13} +$$$$84\!\cdots\!64$$$$T^{14} -$$$$31\!\cdots\!72$$$$T^{15} +$$$$75\!\cdots\!99$$$$T^{16} -$$$$23\!\cdots\!96$$$$T^{17} +$$$$15\!\cdots\!48$$$$T^{18} -$$$$69\!\cdots\!04$$$$T^{19} +$$$$15\!\cdots\!02$$$$T^{20} -$$$$17\!\cdots\!16$$$$T^{21} +$$$$13\!\cdots\!88$$$$T^{22} -$$$$65\!\cdots\!24$$$$T^{23} +$$$$15\!\cdots\!01$$$$T^{24}$$
$47$ $$1 + 164 T + 13448 T^{2} + 868672 T^{3} + 63690059 T^{4} + 4678476896 T^{5} + 288061819304 T^{6} + 15844760579700 T^{7} + 894357153623779 T^{8} + 50754877518397512 T^{9} + 2657783539457568848 T^{10} +$$$$12\!\cdots\!80$$$$T^{11} +$$$$60\!\cdots\!22$$$$T^{12} +$$$$28\!\cdots\!20$$$$T^{13} +$$$$12\!\cdots\!88$$$$T^{14} +$$$$54\!\cdots\!48$$$$T^{15} +$$$$21\!\cdots\!19$$$$T^{16} +$$$$83\!\cdots\!00$$$$T^{17} +$$$$33\!\cdots\!64$$$$T^{18} +$$$$12\!\cdots\!24$$$$T^{19} +$$$$36\!\cdots\!39$$$$T^{20} +$$$$10\!\cdots\!08$$$$T^{21} +$$$$37\!\cdots\!48$$$$T^{22} +$$$$10\!\cdots\!76$$$$T^{23} +$$$$13\!\cdots\!81$$$$T^{24}$$
$53$ $$1 + 204 T + 20808 T^{2} + 1399092 T^{3} + 75768866 T^{4} + 3705029700 T^{5} + 157956707304 T^{6} + 4490093784828 T^{7} - 72880136612449 T^{8} - 22948283226553704 T^{9} - 1987483652559492912 T^{10} -$$$$11\!\cdots\!40$$$$T^{11} -$$$$61\!\cdots\!96$$$$T^{12} -$$$$33\!\cdots\!60$$$$T^{13} -$$$$15\!\cdots\!72$$$$T^{14} -$$$$50\!\cdots\!16$$$$T^{15} -$$$$45\!\cdots\!89$$$$T^{16} +$$$$78\!\cdots\!72$$$$T^{17} +$$$$77\!\cdots\!64$$$$T^{18} +$$$$51\!\cdots\!00$$$$T^{19} +$$$$29\!\cdots\!86$$$$T^{20} +$$$$15\!\cdots\!88$$$$T^{21} +$$$$63\!\cdots\!08$$$$T^{22} +$$$$17\!\cdots\!36$$$$T^{23} +$$$$24\!\cdots\!81$$$$T^{24}$$
$59$ $$1 - 27144 T^{2} + 365962322 T^{4} - 3231050981288 T^{6} + 20796912147061923 T^{8} -$$$$10\!\cdots\!08$$$$T^{10} +$$$$40\!\cdots\!08$$$$T^{12} -$$$$12\!\cdots\!88$$$$T^{14} +$$$$30\!\cdots\!83$$$$T^{16} -$$$$57\!\cdots\!28$$$$T^{18} +$$$$78\!\cdots\!02$$$$T^{20} -$$$$70\!\cdots\!44$$$$T^{22} +$$$$31\!\cdots\!61$$$$T^{24}$$
$61$ $$( 1 - 140 T + 18986 T^{2} - 1612508 T^{3} + 146964673 T^{4} - 10078896872 T^{5} + 706860431120 T^{6} - 37503575260712 T^{7} + 2034849494974993 T^{8} - 83077015820107388 T^{9} + 3639755044566377066 T^{10} - 99868007632803564140 T^{11} +$$$$26\!\cdots\!21$$$$T^{12} )^{2}$$
$67$ $$1 - 324 T + 52488 T^{2} - 5922460 T^{3} + 578352418 T^{4} - 55559962188 T^{5} + 5182632258728 T^{6} - 447217160038868 T^{7} + 35870583162738015 T^{8} - 2754817067496129224 T^{9} +$$$$20\!\cdots\!04$$$$T^{10} -$$$$14\!\cdots\!28$$$$T^{11} +$$$$99\!\cdots\!72$$$$T^{12} -$$$$65\!\cdots\!92$$$$T^{13} +$$$$41\!\cdots\!84$$$$T^{14} -$$$$24\!\cdots\!56$$$$T^{15} +$$$$14\!\cdots\!15$$$$T^{16} -$$$$81\!\cdots\!32$$$$T^{17} +$$$$42\!\cdots\!08$$$$T^{18} -$$$$20\!\cdots\!52$$$$T^{19} +$$$$95\!\cdots\!58$$$$T^{20} -$$$$43\!\cdots\!40$$$$T^{21} +$$$$17\!\cdots\!88$$$$T^{22} -$$$$48\!\cdots\!36$$$$T^{23} +$$$$66\!\cdots\!21$$$$T^{24}$$
$71$ $$( 1 - 72 T + 18698 T^{2} - 925624 T^{3} + 153299139 T^{4} - 5976553984 T^{5} + 867379084884 T^{6} - 30127808633344 T^{7} + 3895588817842659 T^{8} - 118572697204091704 T^{9} + 12074299527233239178 T^{10} -$$$$23\!\cdots\!72$$$$T^{11} +$$$$16\!\cdots\!41$$$$T^{12} )^{2}$$
$73$ $$1 + 248 T + 30752 T^{2} + 3080952 T^{3} + 254433510 T^{4} + 15775678040 T^{5} + 834161467552 T^{6} + 36580397057880 T^{7} + 633277440059215 T^{8} - 33802054838880848 T^{9} - 2957631245611274944 T^{10} -$$$$15\!\cdots\!12$$$$T^{11} -$$$$73\!\cdots\!92$$$$T^{12} -$$$$81\!\cdots\!48$$$$T^{13} -$$$$83\!\cdots\!04$$$$T^{14} -$$$$51\!\cdots\!72$$$$T^{15} +$$$$51\!\cdots\!15$$$$T^{16} +$$$$15\!\cdots\!20$$$$T^{17} +$$$$19\!\cdots\!92$$$$T^{18} +$$$$19\!\cdots\!60$$$$T^{19} +$$$$16\!\cdots\!10$$$$T^{20} +$$$$10\!\cdots\!88$$$$T^{21} +$$$$56\!\cdots\!52$$$$T^{22} +$$$$24\!\cdots\!92$$$$T^{23} +$$$$52\!\cdots\!41$$$$T^{24}$$
$79$ $$1 - 30162 T^{2} + 585976951 T^{4} - 7906141610906 T^{6} + 83787919669756051 T^{8} -$$$$70\!\cdots\!12$$$$T^{10} +$$$$48\!\cdots\!94$$$$T^{12} -$$$$27\!\cdots\!72$$$$T^{14} +$$$$12\!\cdots\!11$$$$T^{16} -$$$$46\!\cdots\!46$$$$T^{18} +$$$$13\!\cdots\!71$$$$T^{20} -$$$$27\!\cdots\!62$$$$T^{22} +$$$$34\!\cdots\!81$$$$T^{24}$$
$83$ $$1 + 224 T + 25088 T^{2} + 4058120 T^{3} + 584315970 T^{4} + 50039102632 T^{5} + 4783608901408 T^{6} + 554051313045392 T^{7} + 38707506803414307 T^{8} + 2439696738179139952 T^{9} +$$$$28\!\cdots\!24$$$$T^{10} +$$$$19\!\cdots\!36$$$$T^{11} +$$$$97\!\cdots\!44$$$$T^{12} +$$$$13\!\cdots\!04$$$$T^{13} +$$$$13\!\cdots\!04$$$$T^{14} +$$$$79\!\cdots\!88$$$$T^{15} +$$$$87\!\cdots\!87$$$$T^{16} +$$$$85\!\cdots\!08$$$$T^{17} +$$$$51\!\cdots\!88$$$$T^{18} +$$$$36\!\cdots\!28$$$$T^{19} +$$$$29\!\cdots\!70$$$$T^{20} +$$$$14\!\cdots\!80$$$$T^{21} +$$$$60\!\cdots\!88$$$$T^{22} +$$$$37\!\cdots\!36$$$$T^{23} +$$$$11\!\cdots\!21$$$$T^{24}$$
$89$ $$1 - 55228 T^{2} + 1551237378 T^{4} - 29097366482252 T^{6} + 406334885845730607 T^{8} -$$$$44\!\cdots\!64$$$$T^{10} +$$$$39\!\cdots\!36$$$$T^{12} -$$$$27\!\cdots\!24$$$$T^{14} +$$$$15\!\cdots\!67$$$$T^{16} -$$$$71\!\cdots\!92$$$$T^{18} +$$$$24\!\cdots\!58$$$$T^{20} -$$$$53\!\cdots\!28$$$$T^{22} +$$$$61\!\cdots\!41$$$$T^{24}$$
$97$ $$1 - 564 T + 159048 T^{2} - 28992320 T^{3} + 3611144475 T^{4} - 287383360192 T^{5} + 8016218179688 T^{6} + 1344437255154268 T^{7} - 202827254806258013 T^{8} + 6263140923061522168 T^{9} +$$$$20\!\cdots\!44$$$$T^{10} -$$$$45\!\cdots\!96$$$$T^{11} +$$$$54\!\cdots\!74$$$$T^{12} -$$$$43\!\cdots\!64$$$$T^{13} +$$$$18\!\cdots\!64$$$$T^{14} +$$$$52\!\cdots\!72$$$$T^{15} -$$$$15\!\cdots\!93$$$$T^{16} +$$$$99\!\cdots\!32$$$$T^{17} +$$$$55\!\cdots\!08$$$$T^{18} -$$$$18\!\cdots\!48$$$$T^{19} +$$$$22\!\cdots\!75$$$$T^{20} -$$$$16\!\cdots\!80$$$$T^{21} +$$$$86\!\cdots\!48$$$$T^{22} -$$$$28\!\cdots\!76$$$$T^{23} +$$$$48\!\cdots\!81$$$$T^{24}$$