# Properties

 Label 3.37.b Level 3 Weight 37 Character orbit b Rep. character $$\chi_{3}(2,\cdot)$$ Character field $$\Q$$ Dimension 11 Newform subspaces 2 Sturm bound 12 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$37$$ Character orbit: $$[\chi]$$ $$=$$ 3.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$3$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$12$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{37}(3, [\chi])$$.

Total New Old
Modular forms 13 13 0
Cusp forms 11 11 0
Eisenstein series 2 2 0

## Trace form

 $$11q - 164736261q^{3} - 366480813328q^{4} + 77238909590832q^{6} + 1529656139588998q^{7} + 219422675471657499q^{9} + O(q^{10})$$ $$11q - 164736261q^{3} - 366480813328q^{4} + 77238909590832q^{6} + 1529656139588998q^{7} + 219422675471657499q^{9} + 976353968400999840q^{10} + 12713663984972326704q^{12} - 26664183895525067642q^{13} + 7346306914204243680q^{15} + 12912742276961748775040q^{16} - 116380061882656121268000q^{18} + 138821577295896839366518q^{19} - 2247956081834234679653850q^{21} + 6375713195756021276887200q^{22} - 18479722715707703137259904q^{24} + 7279508127194401821131675q^{25} + 95231216966669978862527019q^{27} - 391577988767589793638714272q^{28} + 832077359475776822335514400q^{30} - 1733608557857573255379602138q^{31} - 1833168501223628340439879200q^{33} + 11044532239488942328340194176q^{34} - 21948402658393775440817520912q^{36} + 12786450877411433164392216358q^{37} - 9936827303902532409454966170q^{39} + 14850499435375431475622142720q^{40} + 551003488989169331884355647200q^{42} - 515487319179727929010406869802q^{43} - 75992211973807694886478996800q^{45} - 887182867438274648036289139776q^{46} - 4346257047411219928317289501056q^{48} + 9110404204240350847814031518529q^{49} - 2349347159740543712848217109888q^{51} - 13203626071464804773830678836512q^{52} + 35390782846752825865826083751568q^{54} - 30568574550133842572339360395200q^{55} + 96963716675033228271210975522198q^{57} - 112605799975805654976196455444000q^{58} + 19469866547367779716225803605760q^{60} + 66999347224237022458014520403782q^{61} - 20218845986102637586445162314842q^{63} - 304236018528391125615295992251392q^{64} + 1808913396822349027193824888380960q^{66} - 3004194906649294236056896403510282q^{67} + 5226032060124551681525948207907648q^{69} - 11529100185846942980259125797809600q^{70} + 19879821238248816464160255047788800q^{72} - 10390545366461872145216469125969162q^{73} + 21376990929305061495027761443252875q^{75} - 39862084315848733842646101136440224q^{76} + 66756441224105756984056960627624800q^{78} - 71385979381146382550075928433902362q^{79} + 124901631937240804827635146103482731q^{81} - 295731520386077266321378186874491200q^{82} + 565465498335095720662452392509252320q^{84} - 416995278796335191384264774379644160q^{85} + 502161755301600073278417594406864800q^{87} - 1043368725545479380062123692743456000q^{88} + 1148239664725140620121403824942828960q^{90} - 676075249402808126607982688855079700q^{91} + 1136822552118204808049516782296691398q^{93} - 1775468098694354855547362712124166784q^{94} + 2402404280227868861578148783194622976q^{96} - 1638469121614694224091416117067972522q^{97} + 2995711137935834844910948248309577920q^{99} + O(q^{100})$$

## Decomposition of $$S_{37}^{\mathrm{new}}(3, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
3.37.b.a $$1$$ $$24.627$$ $$\Q$$ $$\Q(\sqrt{-3})$$ $$0$$ $$387420489$$ $$0$$ $$27\!\cdots\!98$$ $$q+3^{18}q^{3}+2^{36}q^{4}+2757049053441698q^{7}+\cdots$$
3.37.b.b $$10$$ $$24.627$$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$0$$ $$-552156750$$ $$0$$ $$-1\!\cdots\!00$$ $$q+\beta _{1}q^{2}+(-55215675-69\beta _{1}-\beta _{2}+\cdots)q^{3}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$( 1 - 262144 T )( 1 + 262144 T )$$)($$1 - 125997238648 T^{2} +$$$$13\!\cdots\!92$$$$T^{4} -$$$$10\!\cdots\!08$$$$T^{6} +$$$$91\!\cdots\!52$$$$T^{8} -$$$$60\!\cdots\!28$$$$T^{10} +$$$$43\!\cdots\!92$$$$T^{12} -$$$$22\!\cdots\!28$$$$T^{14} +$$$$14\!\cdots\!12$$$$T^{16} -$$$$62\!\cdots\!88$$$$T^{18} +$$$$23\!\cdots\!76$$$$T^{20}$$)
$3$ ($$1 - 387420489 T$$)($$1 + 552156750 T + 117774518197952061 T^{2} -$$$$34\!\cdots\!00$$$$T^{3} -$$$$33\!\cdots\!62$$$$T^{4} -$$$$20\!\cdots\!00$$$$T^{5} -$$$$49\!\cdots\!02$$$$T^{6} -$$$$77\!\cdots\!00$$$$T^{7} +$$$$39\!\cdots\!21$$$$T^{8} +$$$$28\!\cdots\!50$$$$T^{9} +$$$$76\!\cdots\!01$$$$T^{10}$$)
$5$ ($$( 1 - 3814697265625 T )( 1 + 3814697265625 T )$$)($$1 -$$$$69\!\cdots\!50$$$$T^{2} +$$$$24\!\cdots\!25$$$$T^{4} -$$$$57\!\cdots\!00$$$$T^{6} +$$$$10\!\cdots\!50$$$$T^{8} -$$$$16\!\cdots\!00$$$$T^{10} +$$$$22\!\cdots\!50$$$$T^{12} -$$$$25\!\cdots\!00$$$$T^{14} +$$$$23\!\cdots\!25$$$$T^{16} -$$$$13\!\cdots\!50$$$$T^{18} +$$$$42\!\cdots\!25$$$$T^{20}$$)
$7$ ($$1 - 2757049053441698 T +$$$$26\!\cdots\!01$$$$T^{2}$$)($$( 1 + 613696456926350 T +$$$$57\!\cdots\!21$$$$T^{2} -$$$$18\!\cdots\!00$$$$T^{3} +$$$$11\!\cdots\!58$$$$T^{4} -$$$$17\!\cdots\!00$$$$T^{5} +$$$$30\!\cdots\!58$$$$T^{6} -$$$$13\!\cdots\!00$$$$T^{7} +$$$$10\!\cdots\!21$$$$T^{8} +$$$$30\!\cdots\!50$$$$T^{9} +$$$$13\!\cdots\!01$$$$T^{10} )^{2}$$)
$11$ ($$( 1 - 5559917313492231481 T )( 1 + 5559917313492231481 T )$$)($$1 -$$$$18\!\cdots\!90$$$$T^{2} +$$$$18\!\cdots\!85$$$$T^{4} -$$$$11\!\cdots\!60$$$$T^{6} +$$$$52\!\cdots\!90$$$$T^{8} -$$$$18\!\cdots\!52$$$$T^{10} +$$$$50\!\cdots\!90$$$$T^{12} -$$$$10\!\cdots\!60$$$$T^{14} +$$$$15\!\cdots\!85$$$$T^{16} -$$$$15\!\cdots\!90$$$$T^{18} +$$$$79\!\cdots\!01$$$$T^{20}$$)
$13$ ($$1 +$$$$17\!\cdots\!42$$$$T +$$$$12\!\cdots\!41$$$$T^{2}$$)($$( 1 - 73221010691734315450 T +$$$$36\!\cdots\!81$$$$T^{2} -$$$$37\!\cdots\!00$$$$T^{3} +$$$$66\!\cdots\!58$$$$T^{4} -$$$$70\!\cdots\!00$$$$T^{5} +$$$$83\!\cdots\!78$$$$T^{6} -$$$$59\!\cdots\!00$$$$T^{7} +$$$$74\!\cdots\!01$$$$T^{8} -$$$$18\!\cdots\!50$$$$T^{9} +$$$$32\!\cdots\!01$$$$T^{10} )^{2}$$)
$17$ ($$( 1 -$$$$14\!\cdots\!09$$$$T )( 1 +$$$$14\!\cdots\!09$$$$T )$$)($$1 -$$$$12\!\cdots\!18$$$$T^{2} +$$$$80\!\cdots\!37$$$$T^{4} -$$$$33\!\cdots\!68$$$$T^{6} +$$$$10\!\cdots\!22$$$$T^{8} -$$$$23\!\cdots\!48$$$$T^{10} +$$$$40\!\cdots\!42$$$$T^{12} -$$$$51\!\cdots\!28$$$$T^{14} +$$$$48\!\cdots\!97$$$$T^{16} -$$$$29\!\cdots\!38$$$$T^{18} +$$$$91\!\cdots\!01$$$$T^{20}$$)
$19$ ($$1 +$$$$11\!\cdots\!18$$$$T +$$$$10\!\cdots\!81$$$$T^{2}$$)($$( 1 -$$$$12\!\cdots\!18$$$$T +$$$$33\!\cdots\!97$$$$T^{2} -$$$$45\!\cdots\!88$$$$T^{3} +$$$$64\!\cdots\!82$$$$T^{4} -$$$$65\!\cdots\!48$$$$T^{5} +$$$$70\!\cdots\!42$$$$T^{6} -$$$$53\!\cdots\!68$$$$T^{7} +$$$$43\!\cdots\!77$$$$T^{8} -$$$$17\!\cdots\!78$$$$T^{9} +$$$$14\!\cdots\!01$$$$T^{10} )^{2}$$)
$23$ ($$( 1 -$$$$32\!\cdots\!69$$$$T )( 1 +$$$$32\!\cdots\!69$$$$T )$$)($$1 -$$$$44\!\cdots\!18$$$$T^{2} +$$$$12\!\cdots\!17$$$$T^{4} -$$$$23\!\cdots\!28$$$$T^{6} +$$$$35\!\cdots\!02$$$$T^{8} -$$$$41\!\cdots\!48$$$$T^{10} +$$$$38\!\cdots\!42$$$$T^{12} -$$$$28\!\cdots\!48$$$$T^{14} +$$$$16\!\cdots\!37$$$$T^{16} -$$$$66\!\cdots\!58$$$$T^{18} +$$$$16\!\cdots\!01$$$$T^{20}$$)
$29$ ($$( 1 -$$$$21\!\cdots\!61$$$$T )( 1 +$$$$21\!\cdots\!61$$$$T )$$)($$1 -$$$$15\!\cdots\!30$$$$T^{2} +$$$$13\!\cdots\!05$$$$T^{4} -$$$$91\!\cdots\!80$$$$T^{6} +$$$$51\!\cdots\!30$$$$T^{8} -$$$$24\!\cdots\!52$$$$T^{10} +$$$$10\!\cdots\!30$$$$T^{12} -$$$$35\!\cdots\!80$$$$T^{14} +$$$$10\!\cdots\!05$$$$T^{16} -$$$$22\!\cdots\!30$$$$T^{18} +$$$$29\!\cdots\!01$$$$T^{20}$$)
$31$ ($$1 -$$$$11\!\cdots\!82$$$$T +$$$$48\!\cdots\!81$$$$T^{2}$$)($$( 1 +$$$$14\!\cdots\!10$$$$T +$$$$20\!\cdots\!85$$$$T^{2} +$$$$19\!\cdots\!40$$$$T^{3} +$$$$16\!\cdots\!90$$$$T^{4} +$$$$12\!\cdots\!48$$$$T^{5} +$$$$82\!\cdots\!90$$$$T^{6} +$$$$46\!\cdots\!40$$$$T^{7} +$$$$23\!\cdots\!85$$$$T^{8} +$$$$80\!\cdots\!10$$$$T^{9} +$$$$27\!\cdots\!01$$$$T^{10} )^{2}$$)
$37$ ($$1 +$$$$33\!\cdots\!42$$$$T +$$$$28\!\cdots\!41$$$$T^{2}$$)($$( 1 -$$$$23\!\cdots\!50$$$$T +$$$$98\!\cdots\!81$$$$T^{2} -$$$$21\!\cdots\!00$$$$T^{3} +$$$$45\!\cdots\!58$$$$T^{4} -$$$$82\!\cdots\!00$$$$T^{5} +$$$$13\!\cdots\!78$$$$T^{6} -$$$$17\!\cdots\!00$$$$T^{7} +$$$$22\!\cdots\!01$$$$T^{8} -$$$$15\!\cdots\!50$$$$T^{9} +$$$$18\!\cdots\!01$$$$T^{10} )^{2}$$)
$41$ ($$( 1 -$$$$10\!\cdots\!21$$$$T )( 1 +$$$$10\!\cdots\!21$$$$T )$$)($$1 -$$$$23\!\cdots\!90$$$$T^{2} +$$$$10\!\cdots\!85$$$$T^{4} +$$$$58\!\cdots\!40$$$$T^{6} +$$$$19\!\cdots\!90$$$$T^{8} -$$$$48\!\cdots\!52$$$$T^{10} +$$$$25\!\cdots\!90$$$$T^{12} +$$$$10\!\cdots\!40$$$$T^{14} +$$$$24\!\cdots\!85$$$$T^{16} -$$$$71\!\cdots\!90$$$$T^{18} +$$$$40\!\cdots\!01$$$$T^{20}$$)
$43$ ($$1 -$$$$28\!\cdots\!98$$$$T +$$$$63\!\cdots\!01$$$$T^{2}$$)($$( 1 +$$$$27\!\cdots\!50$$$$T +$$$$27\!\cdots\!81$$$$T^{2} +$$$$57\!\cdots\!00$$$$T^{3} +$$$$31\!\cdots\!38$$$$T^{4} +$$$$50\!\cdots\!00$$$$T^{5} +$$$$20\!\cdots\!38$$$$T^{6} +$$$$23\!\cdots\!00$$$$T^{7} +$$$$71\!\cdots\!81$$$$T^{8} +$$$$45\!\cdots\!50$$$$T^{9} +$$$$10\!\cdots\!01$$$$T^{10} )^{2}$$)
$47$ ($$( 1 -$$$$12\!\cdots\!89$$$$T )( 1 +$$$$12\!\cdots\!89$$$$T )$$)($$1 -$$$$90\!\cdots\!18$$$$T^{2} +$$$$34\!\cdots\!97$$$$T^{4} -$$$$76\!\cdots\!08$$$$T^{6} +$$$$10\!\cdots\!42$$$$T^{8} -$$$$14\!\cdots\!28$$$$T^{10} +$$$$26\!\cdots\!22$$$$T^{12} -$$$$46\!\cdots\!48$$$$T^{14} +$$$$52\!\cdots\!37$$$$T^{16} -$$$$33\!\cdots\!98$$$$T^{18} +$$$$90\!\cdots\!01$$$$T^{20}$$)
$53$ ($$( 1 -$$$$10\!\cdots\!89$$$$T )( 1 +$$$$10\!\cdots\!89$$$$T )$$)($$1 -$$$$62\!\cdots\!18$$$$T^{2} +$$$$19\!\cdots\!97$$$$T^{4} -$$$$38\!\cdots\!08$$$$T^{6} +$$$$57\!\cdots\!42$$$$T^{8} -$$$$72\!\cdots\!28$$$$T^{10} +$$$$81\!\cdots\!22$$$$T^{12} -$$$$76\!\cdots\!48$$$$T^{14} +$$$$53\!\cdots\!37$$$$T^{16} -$$$$24\!\cdots\!98$$$$T^{18} +$$$$54\!\cdots\!01$$$$T^{20}$$)
$59$ ($$( 1 -$$$$75\!\cdots\!21$$$$T )( 1 +$$$$75\!\cdots\!21$$$$T )$$)($$1 -$$$$26\!\cdots\!90$$$$T^{2} +$$$$39\!\cdots\!85$$$$T^{4} -$$$$40\!\cdots\!60$$$$T^{6} +$$$$31\!\cdots\!90$$$$T^{8} -$$$$19\!\cdots\!52$$$$T^{10} +$$$$10\!\cdots\!90$$$$T^{12} -$$$$40\!\cdots\!60$$$$T^{14} +$$$$12\!\cdots\!85$$$$T^{16} -$$$$27\!\cdots\!90$$$$T^{18} +$$$$32\!\cdots\!01$$$$T^{20}$$)
$61$ ($$1 +$$$$18\!\cdots\!38$$$$T +$$$$18\!\cdots\!61$$$$T^{2}$$)($$( 1 -$$$$12\!\cdots\!10$$$$T +$$$$40\!\cdots\!45$$$$T^{2} -$$$$51\!\cdots\!20$$$$T^{3} +$$$$12\!\cdots\!10$$$$T^{4} -$$$$12\!\cdots\!52$$$$T^{5} +$$$$23\!\cdots\!10$$$$T^{6} -$$$$17\!\cdots\!20$$$$T^{7} +$$$$26\!\cdots\!45$$$$T^{8} -$$$$15\!\cdots\!10$$$$T^{9} +$$$$22\!\cdots\!01$$$$T^{10} )^{2}$$)
$67$ ($$1 -$$$$26\!\cdots\!18$$$$T +$$$$54\!\cdots\!81$$$$T^{2}$$)($$( 1 +$$$$16\!\cdots\!50$$$$T +$$$$32\!\cdots\!41$$$$T^{2} +$$$$34\!\cdots\!00$$$$T^{3} +$$$$38\!\cdots\!58$$$$T^{4} +$$$$27\!\cdots\!00$$$$T^{5} +$$$$20\!\cdots\!98$$$$T^{6} +$$$$10\!\cdots\!00$$$$T^{7} +$$$$53\!\cdots\!81$$$$T^{8} +$$$$14\!\cdots\!50$$$$T^{9} +$$$$49\!\cdots\!01$$$$T^{10} )^{2}$$)
$71$ ($$( 1 -$$$$21\!\cdots\!61$$$$T )( 1 +$$$$21\!\cdots\!61$$$$T )$$)($$1 -$$$$31\!\cdots\!30$$$$T^{2} +$$$$49\!\cdots\!05$$$$T^{4} -$$$$48\!\cdots\!80$$$$T^{6} +$$$$34\!\cdots\!30$$$$T^{8} -$$$$17\!\cdots\!52$$$$T^{10} +$$$$66\!\cdots\!30$$$$T^{12} -$$$$18\!\cdots\!80$$$$T^{14} +$$$$36\!\cdots\!05$$$$T^{16} -$$$$46\!\cdots\!30$$$$T^{18} +$$$$28\!\cdots\!01$$$$T^{20}$$)
$73$ ($$1 +$$$$60\!\cdots\!62$$$$T +$$$$12\!\cdots\!61$$$$T^{2}$$)($$( 1 +$$$$21\!\cdots\!50$$$$T +$$$$41\!\cdots\!41$$$$T^{2} +$$$$83\!\cdots\!00$$$$T^{3} +$$$$84\!\cdots\!98$$$$T^{4} +$$$$13\!\cdots\!00$$$$T^{5} +$$$$10\!\cdots\!78$$$$T^{6} +$$$$12\!\cdots\!00$$$$T^{7} +$$$$72\!\cdots\!21$$$$T^{8} +$$$$45\!\cdots\!50$$$$T^{9} +$$$$25\!\cdots\!01$$$$T^{10} )^{2}$$)
$79$ ($$1 +$$$$89\!\cdots\!78$$$$T +$$$$20\!\cdots\!21$$$$T^{2}$$)($$( 1 +$$$$31\!\cdots\!42$$$$T +$$$$86\!\cdots\!57$$$$T^{2} +$$$$16\!\cdots\!12$$$$T^{3} +$$$$29\!\cdots\!62$$$$T^{4} +$$$$42\!\cdots\!52$$$$T^{5} +$$$$60\!\cdots\!02$$$$T^{6} +$$$$70\!\cdots\!92$$$$T^{7} +$$$$75\!\cdots\!77$$$$T^{8} +$$$$56\!\cdots\!02$$$$T^{9} +$$$$37\!\cdots\!01$$$$T^{10} )^{2}$$)
$83$ ($$( 1 -$$$$34\!\cdots\!09$$$$T )( 1 +$$$$34\!\cdots\!09$$$$T )$$)($$1 -$$$$70\!\cdots\!98$$$$T^{2} +$$$$26\!\cdots\!37$$$$T^{4} -$$$$64\!\cdots\!48$$$$T^{6} +$$$$11\!\cdots\!22$$$$T^{8} -$$$$16\!\cdots\!28$$$$T^{10} +$$$$17\!\cdots\!42$$$$T^{12} -$$$$14\!\cdots\!08$$$$T^{14} +$$$$86\!\cdots\!97$$$$T^{16} -$$$$34\!\cdots\!18$$$$T^{18} +$$$$73\!\cdots\!01$$$$T^{20}$$)
$89$ ($$( 1 -$$$$12\!\cdots\!81$$$$T )( 1 +$$$$12\!\cdots\!81$$$$T )$$)($$1 -$$$$39\!\cdots\!90$$$$T^{2} +$$$$94\!\cdots\!85$$$$T^{4} -$$$$22\!\cdots\!60$$$$T^{6} +$$$$40\!\cdots\!90$$$$T^{8} -$$$$61\!\cdots\!52$$$$T^{10} +$$$$92\!\cdots\!90$$$$T^{12} -$$$$11\!\cdots\!60$$$$T^{14} +$$$$11\!\cdots\!85$$$$T^{16} -$$$$10\!\cdots\!90$$$$T^{18} +$$$$60\!\cdots\!01$$$$T^{20}$$)
$97$ ($$1 +$$$$11\!\cdots\!22$$$$T +$$$$33\!\cdots\!21$$$$T^{2}$$)($$( 1 +$$$$25\!\cdots\!50$$$$T +$$$$11\!\cdots\!81$$$$T^{2} +$$$$43\!\cdots\!00$$$$T^{3} +$$$$60\!\cdots\!98$$$$T^{4} +$$$$23\!\cdots\!00$$$$T^{5} +$$$$20\!\cdots\!58$$$$T^{6} +$$$$48\!\cdots\!00$$$$T^{7} +$$$$42\!\cdots\!41$$$$T^{8} +$$$$31\!\cdots\!50$$$$T^{9} +$$$$41\!\cdots\!01$$$$T^{10} )^{2}$$)