# Properties

 Label 3.39.b Level $3$ Weight $39$ Character orbit 3.b Rep. character $\chi_{3}(2,\cdot)$ Character field $\Q$ Dimension $12$ Newform subspaces $1$ Sturm bound $13$ Trace bound $0$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$39$$ Character orbit: $$[\chi]$$ $$=$$ 3.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$3$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$13$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{39}(3, [\chi])$$.

Total New Old
Modular forms 14 14 0
Cusp forms 12 12 0
Eisenstein series 2 2 0

## Trace form

 $$12q - 114742404q^{3} - 1699274528448q^{4} - 483611204680128q^{6} + 8107872236538648q^{7} - 424319151461513940q^{9} + O(q^{10})$$ $$12q - 114742404q^{3} - 1699274528448q^{4} - 483611204680128q^{6} + 8107872236538648q^{7} - 424319151461513940q^{9} + 8521437485093339520q^{10} - 2862564534392665536q^{12} + 1069098773333431501752q^{13} + 6325133233762551847680q^{15} + 67078537203201948051456q^{16} + 1420463738762719667349120q^{18} - 4607058619794992781108360q^{19} + 33320142533758881258920952q^{21} - 136880063896016789094648960q^{22} + 783927349783175843577206784q^{24} - 1296747019384761463507120020q^{25} + 3527882385497241000493515804q^{27} - 772965222736808266417757568q^{28} - 31212306699175351074822848640q^{30} + 62426311865853800230409578584q^{31} - 201583764927512776992433501440q^{33} + 279954989831379847783072264704q^{34} + 524806089486681368742761588544q^{36} - 1044821541252033349072004239752q^{37} + 3798009668755107990641466418968q^{39} - 8640808308189045028871484272640q^{40} + 3739179021877842175021096471680q^{42} + 10033244180304067819165288107192q^{43} - 61618154391458016749088317176320q^{45} + 133876649746070235210936111300864q^{46} - 169673882786170669254051959076864q^{48} - 74274509875804693019854807543452q^{49} + 717994133339248153525793387369472q^{51} - 993185387699150183190245387663232q^{52} + 1237237684228372752705008546718912q^{54} - 147621918295658178296158670231040q^{55} - 1942894316331924260346088046339112q^{57} + 5456952458981587224214732301397120q^{58} - 21422716753388621512702364409876480q^{60} + 19261150877798591818539348576756024q^{61} - 6850204156609697882698173452536488q^{63} - 3354811274780051137580237339295744q^{64} + 29491133770268183961996554653096320q^{66} - 12293160890248249992597347473356552q^{67} + 14368205812565646738112192879237632q^{69} + 131887910392782907972479210848551680q^{70} - 849782187269944662605305038851543040q^{72} + 904634266610182985560078196407011672q^{73} - 1950889856240531910659528579515703460q^{75} + 3737782451304773339414088480976030848q^{76} - 4783964522796872138966708226773738880q^{78} + 3380994931932561121624661644777634520q^{79} - 3842105375893366153709000791027089588q^{81} + 9707181808332667829526886407699144960q^{82} - 23112447300276128146951334880402529152q^{84} + 16191233322474057038450787601892136960q^{85} - 46435849480387750961684301913616613120q^{87} + 111257699492994117641050357545187921920q^{88} - 164148346324517317604081085509260974720q^{90} + 128973762284915213743998856562037216624q^{91} - 173596942201215499347713600620947459528q^{93} + 323629532832623192615016136649966897664q^{94} - 450074405332355244291177655815409434624q^{96} + 248679388944018942060778750016570824728q^{97} - 346605715634587898579110273074604669440q^{99} + O(q^{100})$$

## Decomposition of $$S_{39}^{\mathrm{new}}(3, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
3.39.b.a $$12$$ $$27.439$$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$-114742404$$ $$0$$ $$81\!\cdots\!48$$ $$q+\beta _{1}q^{2}+(-9561867+97\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 799630177440 T^{2} +$$$$41\!\cdots\!64$$$$T^{4} -$$$$16\!\cdots\!80$$$$T^{6} +$$$$53\!\cdots\!20$$$$T^{8} -$$$$15\!\cdots\!80$$$$T^{10} +$$$$43\!\cdots\!60$$$$T^{12} -$$$$11\!\cdots\!80$$$$T^{14} +$$$$30\!\cdots\!20$$$$T^{16} -$$$$72\!\cdots\!80$$$$T^{18} +$$$$13\!\cdots\!24$$$$T^{20} -$$$$19\!\cdots\!40$$$$T^{22} +$$$$18\!\cdots\!36$$$$T^{24}$$
$3$ $$1 + 114742404 T + 218742485368606578 T^{2} -$$$$11\!\cdots\!44$$$$T^{3} +$$$$84\!\cdots\!79$$$$T^{4} -$$$$20\!\cdots\!68$$$$T^{5} +$$$$15\!\cdots\!24$$$$T^{6} -$$$$27\!\cdots\!52$$$$T^{7} +$$$$15\!\cdots\!59$$$$T^{8} -$$$$28\!\cdots\!36$$$$T^{9} +$$$$72\!\cdots\!98$$$$T^{10} +$$$$51\!\cdots\!96$$$$T^{11} +$$$$60\!\cdots\!61$$$$T^{12}$$
$5$ $$1 -$$$$15\!\cdots\!40$$$$T^{2} +$$$$13\!\cdots\!50$$$$T^{4} -$$$$88\!\cdots\!00$$$$T^{6} +$$$$47\!\cdots\!75$$$$T^{8} -$$$$21\!\cdots\!00$$$$T^{10} +$$$$85\!\cdots\!00$$$$T^{12} -$$$$28\!\cdots\!00$$$$T^{14} +$$$$83\!\cdots\!75$$$$T^{16} -$$$$20\!\cdots\!00$$$$T^{18} +$$$$41\!\cdots\!50$$$$T^{20} -$$$$62\!\cdots\!00$$$$T^{22} +$$$$53\!\cdots\!25$$$$T^{24}$$
$7$ $$( 1 - 4053936118269324 T +$$$$41\!\cdots\!98$$$$T^{2} -$$$$33\!\cdots\!76$$$$T^{3} +$$$$90\!\cdots\!99$$$$T^{4} +$$$$61\!\cdots\!48$$$$T^{5} +$$$$13\!\cdots\!04$$$$T^{6} +$$$$79\!\cdots\!52$$$$T^{7} +$$$$15\!\cdots\!99$$$$T^{8} -$$$$73\!\cdots\!24$$$$T^{9} +$$$$11\!\cdots\!98$$$$T^{10} -$$$$15\!\cdots\!76$$$$T^{11} +$$$$48\!\cdots\!01$$$$T^{12} )^{2}$$
$11$ $$1 -$$$$13\!\cdots\!32$$$$T^{2} +$$$$71\!\cdots\!26$$$$T^{4} -$$$$13\!\cdots\!20$$$$T^{6} -$$$$30\!\cdots\!05$$$$T^{8} +$$$$30\!\cdots\!08$$$$T^{10} -$$$$12\!\cdots\!36$$$$T^{12} +$$$$41\!\cdots\!88$$$$T^{14} -$$$$60\!\cdots\!05$$$$T^{16} -$$$$38\!\cdots\!20$$$$T^{18} +$$$$27\!\cdots\!66$$$$T^{20} -$$$$72\!\cdots\!32$$$$T^{22} +$$$$75\!\cdots\!61$$$$T^{24}$$
$13$ $$( 1 -$$$$53\!\cdots\!76$$$$T +$$$$59\!\cdots\!58$$$$T^{2} -$$$$45\!\cdots\!04$$$$T^{3} +$$$$21\!\cdots\!59$$$$T^{4} -$$$$16\!\cdots\!68$$$$T^{5} +$$$$55\!\cdots\!44$$$$T^{6} -$$$$35\!\cdots\!72$$$$T^{7} +$$$$99\!\cdots\!19$$$$T^{8} -$$$$44\!\cdots\!56$$$$T^{9} +$$$$12\!\cdots\!98$$$$T^{10} -$$$$23\!\cdots\!24$$$$T^{11} +$$$$95\!\cdots\!21$$$$T^{12} )^{2}$$
$17$ $$1 -$$$$25\!\cdots\!60$$$$T^{2} +$$$$44\!\cdots\!34$$$$T^{4} -$$$$53\!\cdots\!20$$$$T^{6} +$$$$51\!\cdots\!35$$$$T^{8} -$$$$39\!\cdots\!20$$$$T^{10} +$$$$24\!\cdots\!00$$$$T^{12} -$$$$12\!\cdots\!20$$$$T^{14} +$$$$55\!\cdots\!35$$$$T^{16} -$$$$18\!\cdots\!20$$$$T^{18} +$$$$51\!\cdots\!14$$$$T^{20} -$$$$95\!\cdots\!60$$$$T^{22} +$$$$12\!\cdots\!81$$$$T^{24}$$
$19$ $$( 1 +$$$$23\!\cdots\!80$$$$T +$$$$21\!\cdots\!94$$$$T^{2} +$$$$38\!\cdots\!60$$$$T^{3} +$$$$19\!\cdots\!55$$$$T^{4} +$$$$27\!\cdots\!60$$$$T^{5} +$$$$97\!\cdots\!20$$$$T^{6} +$$$$10\!\cdots\!60$$$$T^{7} +$$$$29\!\cdots\!55$$$$T^{8} +$$$$22\!\cdots\!60$$$$T^{9} +$$$$49\!\cdots\!34$$$$T^{10} +$$$$21\!\cdots\!80$$$$T^{11} +$$$$35\!\cdots\!41$$$$T^{12} )^{2}$$
$23$ $$1 -$$$$48\!\cdots\!40$$$$T^{2} +$$$$11\!\cdots\!14$$$$T^{4} -$$$$18\!\cdots\!80$$$$T^{6} +$$$$19\!\cdots\!95$$$$T^{8} -$$$$16\!\cdots\!80$$$$T^{10} +$$$$10\!\cdots\!60$$$$T^{12} -$$$$50\!\cdots\!80$$$$T^{14} +$$$$19\!\cdots\!95$$$$T^{16} -$$$$53\!\cdots\!80$$$$T^{18} +$$$$10\!\cdots\!74$$$$T^{20} -$$$$13\!\cdots\!40$$$$T^{22} +$$$$88\!\cdots\!61$$$$T^{24}$$
$29$ $$1 -$$$$22\!\cdots\!72$$$$T^{2} +$$$$28\!\cdots\!86$$$$T^{4} -$$$$24\!\cdots\!20$$$$T^{6} +$$$$15\!\cdots\!95$$$$T^{8} -$$$$79\!\cdots\!92$$$$T^{10} +$$$$32\!\cdots\!44$$$$T^{12} -$$$$11\!\cdots\!32$$$$T^{14} +$$$$29\!\cdots\!95$$$$T^{16} -$$$$64\!\cdots\!20$$$$T^{18} +$$$$10\!\cdots\!66$$$$T^{20} -$$$$11\!\cdots\!72$$$$T^{22} +$$$$71\!\cdots\!21$$$$T^{24}$$
$31$ $$( 1 -$$$$31\!\cdots\!92$$$$T +$$$$14\!\cdots\!06$$$$T^{2} -$$$$17\!\cdots\!20$$$$T^{3} +$$$$45\!\cdots\!95$$$$T^{4} +$$$$33\!\cdots\!08$$$$T^{5} +$$$$56\!\cdots\!84$$$$T^{6} +$$$$15\!\cdots\!28$$$$T^{7} +$$$$99\!\cdots\!95$$$$T^{8} -$$$$17\!\cdots\!20$$$$T^{9} +$$$$69\!\cdots\!66$$$$T^{10} -$$$$71\!\cdots\!92$$$$T^{11} +$$$$10\!\cdots\!41$$$$T^{12} )^{2}$$
$37$ $$( 1 +$$$$52\!\cdots\!76$$$$T +$$$$22\!\cdots\!58$$$$T^{2} +$$$$91\!\cdots\!04$$$$T^{3} +$$$$20\!\cdots\!59$$$$T^{4} +$$$$67\!\cdots\!68$$$$T^{5} +$$$$10\!\cdots\!44$$$$T^{6} +$$$$26\!\cdots\!72$$$$T^{7} +$$$$31\!\cdots\!19$$$$T^{8} +$$$$54\!\cdots\!56$$$$T^{9} +$$$$51\!\cdots\!98$$$$T^{10} +$$$$47\!\cdots\!24$$$$T^{11} +$$$$35\!\cdots\!21$$$$T^{12} )^{2}$$
$41$ $$1 -$$$$69\!\cdots\!12$$$$T^{2} +$$$$25\!\cdots\!06$$$$T^{4} -$$$$62\!\cdots\!20$$$$T^{6} +$$$$13\!\cdots\!95$$$$T^{8} -$$$$30\!\cdots\!92$$$$T^{10} +$$$$66\!\cdots\!24$$$$T^{12} -$$$$11\!\cdots\!72$$$$T^{14} +$$$$19\!\cdots\!95$$$$T^{16} -$$$$32\!\cdots\!20$$$$T^{18} +$$$$49\!\cdots\!66$$$$T^{20} -$$$$50\!\cdots\!12$$$$T^{22} +$$$$26\!\cdots\!41$$$$T^{24}$$
$43$ $$( 1 -$$$$50\!\cdots\!96$$$$T +$$$$42\!\cdots\!98$$$$T^{2} -$$$$15\!\cdots\!04$$$$T^{3} +$$$$97\!\cdots\!99$$$$T^{4} -$$$$27\!\cdots\!08$$$$T^{5} +$$$$13\!\cdots\!04$$$$T^{6} -$$$$32\!\cdots\!92$$$$T^{7} +$$$$13\!\cdots\!99$$$$T^{8} -$$$$25\!\cdots\!96$$$$T^{9} +$$$$83\!\cdots\!98$$$$T^{10} -$$$$11\!\cdots\!04$$$$T^{11} +$$$$26\!\cdots\!01$$$$T^{12} )^{2}$$
$47$ $$1 -$$$$20\!\cdots\!60$$$$T^{2} +$$$$22\!\cdots\!94$$$$T^{4} -$$$$17\!\cdots\!20$$$$T^{6} +$$$$10\!\cdots\!35$$$$T^{8} -$$$$46\!\cdots\!20$$$$T^{10} +$$$$17\!\cdots\!00$$$$T^{12} -$$$$56\!\cdots\!20$$$$T^{14} +$$$$14\!\cdots\!35$$$$T^{16} -$$$$29\!\cdots\!20$$$$T^{18} +$$$$46\!\cdots\!14$$$$T^{20} -$$$$51\!\cdots\!60$$$$T^{22} +$$$$29\!\cdots\!21$$$$T^{24}$$
$53$ $$1 -$$$$21\!\cdots\!60$$$$T^{2} +$$$$21\!\cdots\!94$$$$T^{4} -$$$$14\!\cdots\!20$$$$T^{6} +$$$$66\!\cdots\!35$$$$T^{8} -$$$$26\!\cdots\!20$$$$T^{10} +$$$$90\!\cdots\!00$$$$T^{12} -$$$$28\!\cdots\!20$$$$T^{14} +$$$$82\!\cdots\!35$$$$T^{16} -$$$$19\!\cdots\!20$$$$T^{18} +$$$$32\!\cdots\!14$$$$T^{20} -$$$$35\!\cdots\!60$$$$T^{22} +$$$$18\!\cdots\!21$$$$T^{24}$$
$59$ $$1 -$$$$11\!\cdots\!12$$$$T^{2} +$$$$61\!\cdots\!06$$$$T^{4} -$$$$24\!\cdots\!20$$$$T^{6} +$$$$72\!\cdots\!95$$$$T^{8} -$$$$18\!\cdots\!92$$$$T^{10} +$$$$38\!\cdots\!24$$$$T^{12} -$$$$69\!\cdots\!72$$$$T^{14} +$$$$10\!\cdots\!95$$$$T^{16} -$$$$13\!\cdots\!20$$$$T^{18} +$$$$13\!\cdots\!66$$$$T^{20} -$$$$92\!\cdots\!12$$$$T^{22} +$$$$32\!\cdots\!41$$$$T^{24}$$
$61$ $$( 1 -$$$$96\!\cdots\!12$$$$T +$$$$25\!\cdots\!46$$$$T^{2} -$$$$16\!\cdots\!20$$$$T^{3} +$$$$28\!\cdots\!95$$$$T^{4} -$$$$14\!\cdots\!92$$$$T^{5} +$$$$21\!\cdots\!24$$$$T^{6} -$$$$98\!\cdots\!52$$$$T^{7} +$$$$13\!\cdots\!95$$$$T^{8} -$$$$57\!\cdots\!20$$$$T^{9} +$$$$59\!\cdots\!66$$$$T^{10} -$$$$15\!\cdots\!12$$$$T^{11} +$$$$11\!\cdots\!81$$$$T^{12} )^{2}$$
$67$ $$( 1 +$$$$61\!\cdots\!76$$$$T +$$$$73\!\cdots\!18$$$$T^{2} -$$$$18\!\cdots\!16$$$$T^{3} +$$$$30\!\cdots\!19$$$$T^{4} -$$$$14\!\cdots\!12$$$$T^{5} +$$$$89\!\cdots\!84$$$$T^{6} -$$$$36\!\cdots\!08$$$$T^{7} +$$$$18\!\cdots\!39$$$$T^{8} -$$$$26\!\cdots\!64$$$$T^{9} +$$$$26\!\cdots\!98$$$$T^{10} +$$$$55\!\cdots\!24$$$$T^{11} +$$$$22\!\cdots\!41$$$$T^{12} )^{2}$$
$71$ $$1 -$$$$13\!\cdots\!72$$$$T^{2} +$$$$88\!\cdots\!86$$$$T^{4} -$$$$35\!\cdots\!20$$$$T^{6} +$$$$10\!\cdots\!95$$$$T^{8} -$$$$25\!\cdots\!92$$$$T^{10} +$$$$58\!\cdots\!44$$$$T^{12} -$$$$12\!\cdots\!32$$$$T^{14} +$$$$25\!\cdots\!95$$$$T^{16} -$$$$43\!\cdots\!20$$$$T^{18} +$$$$53\!\cdots\!66$$$$T^{20} -$$$$41\!\cdots\!72$$$$T^{22} +$$$$14\!\cdots\!21$$$$T^{24}$$
$73$ $$( 1 -$$$$45\!\cdots\!36$$$$T +$$$$29\!\cdots\!38$$$$T^{2} -$$$$11\!\cdots\!84$$$$T^{3} +$$$$43\!\cdots\!39$$$$T^{4} -$$$$12\!\cdots\!08$$$$T^{5} +$$$$35\!\cdots\!64$$$$T^{6} -$$$$80\!\cdots\!52$$$$T^{7} +$$$$17\!\cdots\!79$$$$T^{8} -$$$$28\!\cdots\!56$$$$T^{9} +$$$$50\!\cdots\!98$$$$T^{10} -$$$$48\!\cdots\!64$$$$T^{11} +$$$$68\!\cdots\!81$$$$T^{12} )^{2}$$
$79$ $$( 1 -$$$$16\!\cdots\!60$$$$T +$$$$60\!\cdots\!54$$$$T^{2} -$$$$81\!\cdots\!20$$$$T^{3} +$$$$16\!\cdots\!35$$$$T^{4} -$$$$17\!\cdots\!20$$$$T^{5} +$$$$27\!\cdots\!00$$$$T^{6} -$$$$22\!\cdots\!20$$$$T^{7} +$$$$27\!\cdots\!35$$$$T^{8} -$$$$17\!\cdots\!20$$$$T^{9} +$$$$16\!\cdots\!14$$$$T^{10} -$$$$59\!\cdots\!60$$$$T^{11} +$$$$45\!\cdots\!61$$$$T^{12} )^{2}$$
$83$ $$1 -$$$$52\!\cdots\!60$$$$T^{2} +$$$$14\!\cdots\!34$$$$T^{4} -$$$$26\!\cdots\!20$$$$T^{6} +$$$$37\!\cdots\!35$$$$T^{8} -$$$$42\!\cdots\!20$$$$T^{10} +$$$$38\!\cdots\!00$$$$T^{12} -$$$$29\!\cdots\!20$$$$T^{14} +$$$$18\!\cdots\!35$$$$T^{16} -$$$$94\!\cdots\!20$$$$T^{18} +$$$$36\!\cdots\!14$$$$T^{20} -$$$$94\!\cdots\!60$$$$T^{22} +$$$$12\!\cdots\!81$$$$T^{24}$$
$89$ $$1 -$$$$10\!\cdots\!32$$$$T^{2} +$$$$50\!\cdots\!26$$$$T^{4} -$$$$16\!\cdots\!20$$$$T^{6} +$$$$36\!\cdots\!95$$$$T^{8} -$$$$63\!\cdots\!92$$$$T^{10} +$$$$85\!\cdots\!64$$$$T^{12} -$$$$90\!\cdots\!12$$$$T^{14} +$$$$74\!\cdots\!95$$$$T^{16} -$$$$46\!\cdots\!20$$$$T^{18} +$$$$20\!\cdots\!66$$$$T^{20} -$$$$60\!\cdots\!32$$$$T^{22} +$$$$83\!\cdots\!61$$$$T^{24}$$
$97$ $$( 1 -$$$$12\!\cdots\!64$$$$T +$$$$14\!\cdots\!78$$$$T^{2} -$$$$10\!\cdots\!96$$$$T^{3} +$$$$79\!\cdots\!79$$$$T^{4} -$$$$46\!\cdots\!12$$$$T^{5} +$$$$30\!\cdots\!24$$$$T^{6} -$$$$14\!\cdots\!68$$$$T^{7} +$$$$78\!\cdots\!59$$$$T^{8} -$$$$31\!\cdots\!24$$$$T^{9} +$$$$14\!\cdots\!98$$$$T^{10} -$$$$38\!\cdots\!36$$$$T^{11} +$$$$96\!\cdots\!61$$$$T^{12} )^{2}$$