# Properties

 Label 3.34.a.b Level $3$ Weight $34$ Character orbit 3.a Self dual yes Analytic conductor $20.695$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$34$$ Character orbit: $$[\chi]$$ $$=$$ 3.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$20.6948486643$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - x^{2} - 5357605 x + 842871622$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{11}\cdot 3^{6}\cdot 11$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 45540 - \beta_{1} ) q^{2} -43046721 q^{3} + ( 4863185200 - 67826 \beta_{1} - 10 \beta_{2} ) q^{4} + ( -86829345378 + 1305823 \beta_{1} + 297 \beta_{2} ) q^{5} + ( -1960347674340 + 43046721 \beta_{1} ) q^{6} + ( 3586899630944 + 291628981 \beta_{1} + 147875 \beta_{2} ) q^{7} + ( 602117349936192 - 2933686424 \beta_{1} - 1366200 \beta_{2} ) q^{8} + 1853020188851841 q^{9} +O(q^{10})$$ $$q +(45540 - \beta_{1}) q^{2} -43046721 q^{3} +(4863185200 - 67826 \beta_{1} - 10 \beta_{2}) q^{4} +(-86829345378 + 1305823 \beta_{1} + 297 \beta_{2}) q^{5} +(-1960347674340 + 43046721 \beta_{1}) q^{6} +(3586899630944 + 291628981 \beta_{1} + 147875 \beta_{2}) q^{7} +(602117349936192 - 2933686424 \beta_{1} - 1366200 \beta_{2}) q^{8} +1853020188851841 q^{9} +(-18814238065025352 + 268852224482 \beta_{1} + 33490048 \beta_{2}) q^{10} +(-115160789733595452 + 1374009141218 \beta_{1} - 117158130 \beta_{2}) q^{11} +(-209344176475729200 + 2919686898546 \beta_{1} + 430467210 \beta_{2}) q^{12} +(243349234327264574 + 9029058791434 \beta_{1} - 3407034970 \beta_{2}) q^{13} +(-3155549579462270592 + 79051193603872 \beta_{1} + 13089202560 \beta_{2}) q^{14} +(3737718605099405538 - 56211398356383 \beta_{1} - 12784876137 \beta_{2}) q^{15} +(19032618795066604288 - 788314597484064 \beta_{1} - 37423881120 \beta_{2}) q^{16} +(-80567381117531622078 - 401349169596570 \beta_{1} + 85047521130 \beta_{2}) q^{17} +(84386539400312839140 - 1853020188851841 \beta_{1}) q^{18} +(-$$$$10\!\cdots\!44$$$$- 852472207041702 \beta_{1} - 394062747690 \beta_{2}) q^{19} +(-$$$$31\!\cdots\!92$$$$+ 30832520123076772 \beta_{1} + 2441226033108 \beta_{2}) q^{20} +(-$$$$15\!\cdots\!24$$$$- 12553671380621301 \beta_{1} - 6365533867875 \beta_{2}) q^{21} +(-$$$$20\!\cdots\!96$$$$+ 85458811157363260 \beta_{1} + 5680315016960 \beta_{2}) q^{22} +(-$$$$20\!\cdots\!08$$$$- 138440615323674422 \beta_{1} - 12222579683610 \beta_{2}) q^{23} +(-$$$$25\!\cdots\!32$$$$+ 126285580995415704 \beta_{1} + 58810430230200 \beta_{2}) q^{24} +(-$$$$37\!\cdots\!73$$$$- 692128799755206332 \beta_{1} - 41247911010148 \beta_{2}) q^{25} +(-$$$$91\!\cdots\!08$$$$- 1796364138790299710 \beta_{1} - 144092975811840 \beta_{2}) q^{26} -$$$$79\!\cdots\!61$$$$q^{27} +(-$$$$10\!\cdots\!32$$$$+ 9151665061279739712 \beta_{1} + 420733959159360 \beta_{2}) q^{28} +(-$$$$41\!\cdots\!98$$$$+ 7356185636766476021 \beta_{1} + 650382979885155 \beta_{2}) q^{29} +($$$$80\!\cdots\!92$$$$- 11573206697506023522 \beta_{1} - 1441636752532608 \beta_{2}) q^{30} +($$$$37\!\cdots\!84$$$$+ 5996767654496208601 \beta_{1} - 1942248070367905 \beta_{2}) q^{31} +($$$$46\!\cdots\!04$$$$- 30669876755876060544 \beta_{1} + 1277884186980480 \beta_{2}) q^{32} +($$$$49\!\cdots\!92$$$$- 59146588153460846178 \beta_{1} + 5043273334991730 \beta_{2}) q^{33} +($$$$89\!\cdots\!80$$$$+$$$$11\!\cdots\!98$$$$\beta_{1} + 1837267472651520 \beta_{2}) q^{34} +($$$$29\!\cdots\!60$$$$-$$$$20\!\cdots\!10$$$$\beta_{1} + 526068048546810 \beta_{2}) q^{35} +($$$$90\!\cdots\!00$$$$-$$$$12\!\cdots\!66$$$$\beta_{1} - 18530201888518410 \beta_{2}) q^{36} +(-$$$$15\!\cdots\!06$$$$+$$$$99\!\cdots\!56$$$$\beta_{1} - 13657279933217280 \beta_{2}) q^{37} +($$$$51\!\cdots\!44$$$$-$$$$12\!\cdots\!48$$$$\beta_{1} - 35633874735002880 \beta_{2}) q^{38} +(-$$$$10\!\cdots\!54$$$$-$$$$38\!\cdots\!14$$$$\beta_{1} + 146661683790833370 \beta_{2}) q^{39} +(-$$$$33\!\cdots\!44$$$$+$$$$28\!\cdots\!04$$$$\beta_{1} + 188589583149459056 \beta_{2}) q^{40} +(-$$$$17\!\cdots\!06$$$$-$$$$44\!\cdots\!94$$$$\beta_{1} - 276313934012595570 \beta_{2}) q^{41} +($$$$13\!\cdots\!32$$$$-$$$$34\!\cdots\!12$$$$\beta_{1} - 563447250712805760 \beta_{2}) q^{42} +(-$$$$80\!\cdots\!92$$$$-$$$$76\!\cdots\!34$$$$\beta_{1} - 374805537341295310 \beta_{2}) q^{43} +(-$$$$93\!\cdots\!56$$$$+$$$$13\!\cdots\!80$$$$\beta_{1} + 2251740376471411800 \beta_{2}) q^{44} +(-$$$$16\!\cdots\!98$$$$+$$$$24\!\cdots\!43$$$$\beta_{1} + 550346996088996777 \beta_{2}) q^{45} +($$$$63\!\cdots\!84$$$$+$$$$11\!\cdots\!36$$$$\beta_{1} - 2225246299991010560 \beta_{2}) q^{46} +($$$$64\!\cdots\!76$$$$-$$$$13\!\cdots\!38$$$$\beta_{1} - 3162089172287251710 \beta_{2}) q^{47} +(-$$$$81\!\cdots\!48$$$$+$$$$33\!\cdots\!44$$$$\beta_{1} + 1610975369309807520 \beta_{2}) q^{48} +($$$$60\!\cdots\!41$$$$-$$$$66\!\cdots\!28$$$$\beta_{1} + 9463742798274712640 \beta_{2}) q^{49} +($$$$61\!\cdots\!68$$$$+$$$$11\!\cdots\!37$$$$\beta_{1} - 9758896787584184832 \beta_{2}) q^{50} +($$$$34\!\cdots\!38$$$$+$$$$17\!\cdots\!70$$$$\beta_{1} - 3661016913824714730 \beta_{2}) q^{51} +($$$$14\!\cdots\!12$$$$-$$$$10\!\cdots\!00$$$$\beta_{1} + 1389833979053964180 \beta_{2}) q^{52} +($$$$26\!\cdots\!22$$$$-$$$$43\!\cdots\!87$$$$\beta_{1} + 25951231157220031815 \beta_{2}) q^{53} +(-$$$$36\!\cdots\!40$$$$+$$$$79\!\cdots\!61$$$$\beta_{1}) q^{54} +($$$$10\!\cdots\!52$$$$-$$$$31\!\cdots\!32$$$$\beta_{1} - 65820023746468742648 \beta_{2}) q^{55} +(-$$$$12\!\cdots\!00$$$$+$$$$81\!\cdots\!20$$$$\beta_{1} + 8025228747367453440 \beta_{2}) q^{56} +($$$$43\!\cdots\!24$$$$+$$$$36\!\cdots\!42$$$$\beta_{1} + 16963109156304824490 \beta_{2}) q^{57} +(-$$$$10\!\cdots\!92$$$$+$$$$91\!\cdots\!94$$$$\beta_{1} +$$$$11\!\cdots\!80$$$$\beta_{2}) q^{58} +(-$$$$82\!\cdots\!56$$$$-$$$$75\!\cdots\!28$$$$\beta_{1} - 20648918645650648440 \beta_{2}) q^{59} +($$$$13\!\cdots\!32$$$$-$$$$13\!\cdots\!12$$$$\beta_{1} -$$$$10\!\cdots\!68$$$$\beta_{2}) q^{60} +($$$$12\!\cdots\!06$$$$+$$$$11\!\cdots\!92$$$$\beta_{1} - 75857882953725781660 \beta_{2}) q^{61} +($$$$10\!\cdots\!08$$$$-$$$$45\!\cdots\!88$$$$\beta_{1} - 73647337207927570560 \beta_{2}) q^{62} +($$$$66\!\cdots\!04$$$$+$$$$54\!\cdots\!21$$$$\beta_{1} +$$$$27\!\cdots\!75$$$$\beta_{2}) q^{63} +($$$$39\!\cdots\!72$$$$+$$$$20\!\cdots\!40$$$$\beta_{1} +$$$$10\!\cdots\!20$$$$\beta_{2}) q^{64} +(-$$$$47\!\cdots\!04$$$$+$$$$19\!\cdots\!14$$$$\beta_{1} -$$$$22\!\cdots\!54$$$$\beta_{2}) q^{65} +($$$$89\!\cdots\!16$$$$-$$$$36\!\cdots\!60$$$$\beta_{1} -$$$$24\!\cdots\!60$$$$\beta_{2}) q^{66} +($$$$33\!\cdots\!32$$$$+$$$$13\!\cdots\!40$$$$\beta_{1} +$$$$28\!\cdots\!00$$$$\beta_{2}) q^{67} +(-$$$$58\!\cdots\!00$$$$+$$$$60\!\cdots\!48$$$$\beta_{1} +$$$$54\!\cdots\!00$$$$\beta_{2}) q^{68} +($$$$89\!\cdots\!68$$$$+$$$$59\!\cdots\!62$$$$\beta_{1} +$$$$52\!\cdots\!10$$$$\beta_{2}) q^{69} +($$$$37\!\cdots\!40$$$$-$$$$34\!\cdots\!40$$$$\beta_{1} -$$$$20\!\cdots\!60$$$$\beta_{2}) q^{70} +(-$$$$43\!\cdots\!28$$$$+$$$$17\!\cdots\!30$$$$\beta_{1} - 32207871855713784210 \beta_{2}) q^{71} +($$$$11\!\cdots\!72$$$$-$$$$54\!\cdots\!84$$$$\beta_{1} -$$$$25\!\cdots\!00$$$$\beta_{2}) q^{72} +(-$$$$70\!\cdots\!98$$$$-$$$$29\!\cdots\!92$$$$\beta_{1} +$$$$18\!\cdots\!00$$$$\beta_{2}) q^{73} +(-$$$$12\!\cdots\!52$$$$+$$$$31\!\cdots\!82$$$$\beta_{1} +$$$$90\!\cdots\!40$$$$\beta_{2}) q^{74} +($$$$16\!\cdots\!33$$$$+$$$$29\!\cdots\!72$$$$\beta_{1} +$$$$17\!\cdots\!08$$$$\beta_{2}) q^{75} +($$$$24\!\cdots\!64$$$$-$$$$18\!\cdots\!28$$$$\beta_{1} -$$$$27\!\cdots\!20$$$$\beta_{2}) q^{76} +(-$$$$60\!\cdots\!92$$$$-$$$$85\!\cdots\!48$$$$\beta_{1} -$$$$34\!\cdots\!80$$$$\beta_{2}) q^{77} +($$$$39\!\cdots\!68$$$$+$$$$77\!\cdots\!10$$$$\beta_{1} +$$$$62\!\cdots\!40$$$$\beta_{2}) q^{78} +($$$$74\!\cdots\!00$$$$+$$$$62\!\cdots\!05$$$$\beta_{1} +$$$$23\!\cdots\!35$$$$\beta_{2}) q^{79} +(-$$$$19\!\cdots\!32$$$$+$$$$22\!\cdots\!12$$$$\beta_{1} +$$$$20\!\cdots\!68$$$$\beta_{2}) q^{80} +$$$$34\!\cdots\!81$$$$q^{81} +($$$$42\!\cdots\!68$$$$-$$$$71\!\cdots\!38$$$$\beta_{1} -$$$$63\!\cdots\!20$$$$\beta_{2}) q^{82} +(-$$$$12\!\cdots\!20$$$$-$$$$24\!\cdots\!58$$$$\beta_{1} +$$$$74\!\cdots\!50$$$$\beta_{2}) q^{83} +($$$$46\!\cdots\!72$$$$-$$$$39\!\cdots\!52$$$$\beta_{1} -$$$$18\!\cdots\!60$$$$\beta_{2}) q^{84} +($$$$15\!\cdots\!24$$$$-$$$$10\!\cdots\!34$$$$\beta_{1} -$$$$12\!\cdots\!26$$$$\beta_{2}) q^{85} +($$$$83\!\cdots\!28$$$$-$$$$28\!\cdots\!12$$$$\beta_{1} -$$$$10\!\cdots\!80$$$$\beta_{2}) q^{86} +($$$$18\!\cdots\!58$$$$-$$$$31\!\cdots\!41$$$$\beta_{1} -$$$$27\!\cdots\!55$$$$\beta_{2}) q^{87} +(-$$$$21\!\cdots\!68$$$$+$$$$16\!\cdots\!16$$$$\beta_{1} +$$$$24\!\cdots\!80$$$$\beta_{2}) q^{88} +(-$$$$10\!\cdots\!74$$$$+$$$$87\!\cdots\!48$$$$\beta_{1} -$$$$18\!\cdots\!20$$$$\beta_{2}) q^{89} +(-$$$$34\!\cdots\!32$$$$+$$$$49\!\cdots\!62$$$$\beta_{1} +$$$$62\!\cdots\!68$$$$\beta_{2}) q^{90} +(-$$$$26\!\cdots\!36$$$$+$$$$59\!\cdots\!66$$$$\beta_{1} -$$$$26\!\cdots\!90$$$$\beta_{2}) q^{91} +($$$$77\!\cdots\!64$$$$-$$$$33\!\cdots\!44$$$$\beta_{1} +$$$$65\!\cdots\!40$$$$\beta_{2}) q^{92} +(-$$$$15\!\cdots\!64$$$$-$$$$25\!\cdots\!21$$$$\beta_{1} +$$$$83\!\cdots\!05$$$$\beta_{2}) q^{93} +($$$$44\!\cdots\!16$$$$-$$$$22\!\cdots\!24$$$$\beta_{1} -$$$$23\!\cdots\!20$$$$\beta_{2}) q^{94} +(-$$$$72\!\cdots\!72$$$$+$$$$45\!\cdots\!52$$$$\beta_{1} -$$$$26\!\cdots\!72$$$$\beta_{2}) q^{95} +(-$$$$20\!\cdots\!84$$$$+$$$$13\!\cdots\!24$$$$\beta_{1} -$$$$55\!\cdots\!80$$$$\beta_{2}) q^{96} +(-$$$$29\!\cdots\!98$$$$-$$$$32\!\cdots\!20$$$$\beta_{1} +$$$$75\!\cdots\!80$$$$\beta_{2}) q^{97} +($$$$10\!\cdots\!96$$$$-$$$$26\!\cdots\!29$$$$\beta_{1} -$$$$14\!\cdots\!20$$$$\beta_{2}) q^{98} +(-$$$$21\!\cdots\!32$$$$+$$$$25\!\cdots\!38$$$$\beta_{1} -$$$$21\!\cdots\!30$$$$\beta_{2}) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 136620q^{2} - 129140163q^{3} + 14589555600q^{4} - 260488036134q^{5} - 5881043023020q^{6} + 10760698892832q^{7} + 1806352049808576q^{8} + 5559060566555523q^{9} + O(q^{10})$$ $$3q + 136620q^{2} - 129140163q^{3} + 14589555600q^{4} - 260488036134q^{5} - 5881043023020q^{6} + 10760698892832q^{7} + 1806352049808576q^{8} + 5559060566555523q^{9} - 56442714195076056q^{10} - 345482369200786356q^{11} - 628032529427187600q^{12} + 730047702981793722q^{13} - 9466648738386811776q^{14} + 11213155815298216614q^{15} + 57097856385199812864q^{16} -$$$$24\!\cdots\!34$$$$q^{17} +$$$$25\!\cdots\!20$$$$q^{18} -$$$$30\!\cdots\!32$$$$q^{19} -$$$$95\!\cdots\!76$$$$q^{20} -$$$$46\!\cdots\!72$$$$q^{21} -$$$$62\!\cdots\!88$$$$q^{22} -$$$$62\!\cdots\!24$$$$q^{23} -$$$$77\!\cdots\!96$$$$q^{24} -$$$$11\!\cdots\!19$$$$q^{25} -$$$$27\!\cdots\!24$$$$q^{26} -$$$$23\!\cdots\!83$$$$q^{27} -$$$$32\!\cdots\!96$$$$q^{28} -$$$$12\!\cdots\!94$$$$q^{29} +$$$$24\!\cdots\!76$$$$q^{30} +$$$$11\!\cdots\!52$$$$q^{31} +$$$$13\!\cdots\!12$$$$q^{32} +$$$$14\!\cdots\!76$$$$q^{33} +$$$$26\!\cdots\!40$$$$q^{34} +$$$$89\!\cdots\!80$$$$q^{35} +$$$$27\!\cdots\!00$$$$q^{36} -$$$$47\!\cdots\!18$$$$q^{37} +$$$$15\!\cdots\!32$$$$q^{38} -$$$$31\!\cdots\!62$$$$q^{39} -$$$$10\!\cdots\!32$$$$q^{40} -$$$$51\!\cdots\!18$$$$q^{41} +$$$$40\!\cdots\!96$$$$q^{42} -$$$$24\!\cdots\!76$$$$q^{43} -$$$$28\!\cdots\!68$$$$q^{44} -$$$$48\!\cdots\!94$$$$q^{45} +$$$$18\!\cdots\!52$$$$q^{46} +$$$$19\!\cdots\!28$$$$q^{47} -$$$$24\!\cdots\!44$$$$q^{48} +$$$$18\!\cdots\!23$$$$q^{49} +$$$$18\!\cdots\!04$$$$q^{50} +$$$$10\!\cdots\!14$$$$q^{51} +$$$$42\!\cdots\!36$$$$q^{52} +$$$$80\!\cdots\!66$$$$q^{53} -$$$$10\!\cdots\!20$$$$q^{54} +$$$$30\!\cdots\!56$$$$q^{55} -$$$$37\!\cdots\!00$$$$q^{56} +$$$$13\!\cdots\!72$$$$q^{57} -$$$$30\!\cdots\!76$$$$q^{58} -$$$$24\!\cdots\!68$$$$q^{59} +$$$$40\!\cdots\!96$$$$q^{60} +$$$$37\!\cdots\!18$$$$q^{61} +$$$$30\!\cdots\!24$$$$q^{62} +$$$$19\!\cdots\!12$$$$q^{63} +$$$$11\!\cdots\!16$$$$q^{64} -$$$$14\!\cdots\!12$$$$q^{65} +$$$$26\!\cdots\!48$$$$q^{66} +$$$$10\!\cdots\!96$$$$q^{67} -$$$$17\!\cdots\!00$$$$q^{68} +$$$$26\!\cdots\!04$$$$q^{69} +$$$$11\!\cdots\!20$$$$q^{70} -$$$$13\!\cdots\!84$$$$q^{71} +$$$$33\!\cdots\!16$$$$q^{72} -$$$$21\!\cdots\!94$$$$q^{73} -$$$$36\!\cdots\!56$$$$q^{74} +$$$$48\!\cdots\!99$$$$q^{75} +$$$$74\!\cdots\!92$$$$q^{76} -$$$$18\!\cdots\!76$$$$q^{77} +$$$$11\!\cdots\!04$$$$q^{78} +$$$$22\!\cdots\!00$$$$q^{79} -$$$$59\!\cdots\!96$$$$q^{80} +$$$$10\!\cdots\!43$$$$q^{81} +$$$$12\!\cdots\!04$$$$q^{82} -$$$$36\!\cdots\!60$$$$q^{83} +$$$$13\!\cdots\!16$$$$q^{84} +$$$$47\!\cdots\!72$$$$q^{85} +$$$$24\!\cdots\!84$$$$q^{86} +$$$$54\!\cdots\!74$$$$q^{87} -$$$$64\!\cdots\!04$$$$q^{88} -$$$$30\!\cdots\!22$$$$q^{89} -$$$$10\!\cdots\!96$$$$q^{90} -$$$$79\!\cdots\!08$$$$q^{91} +$$$$23\!\cdots\!92$$$$q^{92} -$$$$47\!\cdots\!92$$$$q^{93} +$$$$13\!\cdots\!48$$$$q^{94} -$$$$21\!\cdots\!16$$$$q^{95} -$$$$60\!\cdots\!52$$$$q^{96} -$$$$87\!\cdots\!94$$$$q^{97} +$$$$30\!\cdots\!88$$$$q^{98} -$$$$64\!\cdots\!96$$$$q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5357605 x + 842871622$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-4 \nu^{2} + 1788 \nu + 14286352$$$$)/105$$ $$\beta_{2}$$ $$=$$ $$($$$$14876 \nu^{2} + 41250588 \nu - 53146909808$$$$)/105$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 3719 \beta_{1} + 152064$$$$)/456192$$ $$\nu^{2}$$ $$=$$ $$($$$$149 \beta_{2} - 3437549 \beta_{1} + 543132615168$$$$)/152064$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 158.055 2232.09 −2389.14
−92260.3 −4.30467e7 −7.79766e7 −3.77234e10 3.97150e12 −2.13696e13 7.99704e14 1.85302e15 3.48037e15
1.2 61269.1 −4.30467e7 −4.83603e9 2.12383e11 −2.63744e12 1.58203e14 −8.22597e14 1.85302e15 1.30125e16
1.3 167611. −4.30467e7 1.95036e10 −4.35148e11 −7.21511e12 −1.26073e14 1.82925e15 1.85302e15 −7.29356e16
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.34.a.b 3
3.b odd 2 1 9.34.a.c 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.34.a.b 3 1.a even 1 1 trivial
9.34.a.c 3 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 136620 T_{2}^{2} - 10847167488 T_{2} +$$947456640811008

'>$$94\!\cdots\!08$$ acting on $$S_{34}^{\mathrm{new}}(\Gamma_0(3))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 136620 T + 14922636288 T^{2} - 1399657087107072 T^{3} +$$$$12\!\cdots\!96$$$$T^{4} -$$$$10\!\cdots\!80$$$$T^{5} +$$$$63\!\cdots\!88$$$$T^{6}$$
$3$ $$( 1 + 43046721 T )^{3}$$
$5$ $$1 + 260488036134 T +$$$$26\!\cdots\!75$$$$T^{2} +$$$$57\!\cdots\!00$$$$T^{3} +$$$$30\!\cdots\!75$$$$T^{4} +$$$$35\!\cdots\!50$$$$T^{5} +$$$$15\!\cdots\!25$$$$T^{6}$$
$7$ $$1 - 10760698892832 T +$$$$25\!\cdots\!61$$$$T^{2} -$$$$59\!\cdots\!48$$$$T^{3} +$$$$19\!\cdots\!27$$$$T^{4} -$$$$64\!\cdots\!68$$$$T^{5} +$$$$46\!\cdots\!43$$$$T^{6}$$
$11$ $$1 + 345482369200786356 T +$$$$65\!\cdots\!53$$$$T^{2} +$$$$91\!\cdots\!80$$$$T^{3} +$$$$15\!\cdots\!43$$$$T^{4} +$$$$18\!\cdots\!16$$$$T^{5} +$$$$12\!\cdots\!91$$$$T^{6}$$
$13$ $$1 - 730047702981793722 T +$$$$58\!\cdots\!79$$$$T^{2} +$$$$55\!\cdots\!04$$$$T^{3} +$$$$33\!\cdots\!87$$$$T^{4} -$$$$24\!\cdots\!98$$$$T^{5} +$$$$19\!\cdots\!77$$$$T^{6}$$
$17$ $$1 +$$$$24\!\cdots\!34$$$$T +$$$$13\!\cdots\!63$$$$T^{2} +$$$$19\!\cdots\!68$$$$T^{3} +$$$$52\!\cdots\!31$$$$T^{4} +$$$$39\!\cdots\!46$$$$T^{5} +$$$$65\!\cdots\!53$$$$T^{6}$$
$19$ $$1 +$$$$30\!\cdots\!32$$$$T +$$$$46\!\cdots\!13$$$$T^{2} +$$$$94\!\cdots\!56$$$$T^{3} +$$$$73\!\cdots\!67$$$$T^{4} +$$$$75\!\cdots\!92$$$$T^{5} +$$$$39\!\cdots\!79$$$$T^{6}$$
$23$ $$1 +$$$$62\!\cdots\!24$$$$T +$$$$34\!\cdots\!69$$$$T^{2} +$$$$10\!\cdots\!84$$$$T^{3} +$$$$29\!\cdots\!27$$$$T^{4} +$$$$46\!\cdots\!36$$$$T^{5} +$$$$64\!\cdots\!87$$$$T^{6}$$
$29$ $$1 +$$$$12\!\cdots\!94$$$$T +$$$$46\!\cdots\!51$$$$T^{2} +$$$$46\!\cdots\!32$$$$T^{3} +$$$$85\!\cdots\!39$$$$T^{4} +$$$$41\!\cdots\!74$$$$T^{5} +$$$$59\!\cdots\!69$$$$T^{6}$$
$31$ $$1 -$$$$11\!\cdots\!52$$$$T +$$$$86\!\cdots\!73$$$$T^{2} -$$$$40\!\cdots\!64$$$$T^{3} +$$$$14\!\cdots\!43$$$$T^{4} -$$$$29\!\cdots\!12$$$$T^{5} +$$$$44\!\cdots\!71$$$$T^{6}$$
$37$ $$1 +$$$$47\!\cdots\!18$$$$T +$$$$52\!\cdots\!51$$$$T^{2} -$$$$22\!\cdots\!08$$$$T^{3} +$$$$29\!\cdots\!47$$$$T^{4} +$$$$15\!\cdots\!62$$$$T^{5} +$$$$17\!\cdots\!73$$$$T^{6}$$
$41$ $$1 +$$$$51\!\cdots\!18$$$$T +$$$$18\!\cdots\!83$$$$T^{2} +$$$$26\!\cdots\!56$$$$T^{3} +$$$$30\!\cdots\!43$$$$T^{4} +$$$$14\!\cdots\!38$$$$T^{5} +$$$$46\!\cdots\!61$$$$T^{6}$$
$43$ $$1 +$$$$24\!\cdots\!76$$$$T +$$$$13\!\cdots\!33$$$$T^{2} +$$$$42\!\cdots\!40$$$$T^{3} +$$$$10\!\cdots\!19$$$$T^{4} +$$$$15\!\cdots\!24$$$$T^{5} +$$$$51\!\cdots\!07$$$$T^{6}$$
$47$ $$1 -$$$$19\!\cdots\!28$$$$T +$$$$37\!\cdots\!17$$$$T^{2} -$$$$43\!\cdots\!20$$$$T^{3} +$$$$57\!\cdots\!59$$$$T^{4} -$$$$43\!\cdots\!12$$$$T^{5} +$$$$34\!\cdots\!83$$$$T^{6}$$
$53$ $$1 -$$$$80\!\cdots\!66$$$$T +$$$$17\!\cdots\!19$$$$T^{2} -$$$$16\!\cdots\!36$$$$T^{3} +$$$$14\!\cdots\!87$$$$T^{4} -$$$$50\!\cdots\!14$$$$T^{5} +$$$$50\!\cdots\!17$$$$T^{6}$$
$59$ $$1 +$$$$24\!\cdots\!68$$$$T +$$$$92\!\cdots\!73$$$$T^{2} +$$$$13\!\cdots\!64$$$$T^{3} +$$$$25\!\cdots\!67$$$$T^{4} +$$$$18\!\cdots\!88$$$$T^{5} +$$$$20\!\cdots\!39$$$$T^{6}$$
$61$ $$1 -$$$$37\!\cdots\!18$$$$T +$$$$22\!\cdots\!99$$$$T^{2} -$$$$75\!\cdots\!24$$$$T^{3} +$$$$18\!\cdots\!19$$$$T^{4} -$$$$25\!\cdots\!98$$$$T^{5} +$$$$55\!\cdots\!41$$$$T^{6}$$
$67$ $$1 -$$$$10\!\cdots\!96$$$$T +$$$$24\!\cdots\!33$$$$T^{2} -$$$$23\!\cdots\!72$$$$T^{3} +$$$$45\!\cdots\!71$$$$T^{4} -$$$$33\!\cdots\!24$$$$T^{5} +$$$$60\!\cdots\!03$$$$T^{6}$$
$71$ $$1 +$$$$13\!\cdots\!84$$$$T +$$$$94\!\cdots\!85$$$$T^{2} +$$$$40\!\cdots\!00$$$$T^{3} +$$$$11\!\cdots\!35$$$$T^{4} +$$$$20\!\cdots\!64$$$$T^{5} +$$$$18\!\cdots\!31$$$$T^{6}$$
$73$ $$1 +$$$$21\!\cdots\!94$$$$T +$$$$76\!\cdots\!79$$$$T^{2} +$$$$14\!\cdots\!04$$$$T^{3} +$$$$23\!\cdots\!07$$$$T^{4} +$$$$20\!\cdots\!66$$$$T^{5} +$$$$29\!\cdots\!37$$$$T^{6}$$
$79$ $$1 -$$$$22\!\cdots\!00$$$$T +$$$$70\!\cdots\!17$$$$T^{2} -$$$$23\!\cdots\!00$$$$T^{3} +$$$$29\!\cdots\!63$$$$T^{4} -$$$$38\!\cdots\!00$$$$T^{5} +$$$$73\!\cdots\!19$$$$T^{6}$$
$83$ $$1 +$$$$36\!\cdots\!60$$$$T +$$$$97\!\cdots\!57$$$$T^{2} -$$$$13\!\cdots\!24$$$$T^{3} +$$$$20\!\cdots\!91$$$$T^{4} +$$$$16\!\cdots\!40$$$$T^{5} +$$$$97\!\cdots\!47$$$$T^{6}$$
$89$ $$1 +$$$$30\!\cdots\!22$$$$T +$$$$93\!\cdots\!23$$$$T^{2} +$$$$13\!\cdots\!56$$$$T^{3} +$$$$20\!\cdots\!87$$$$T^{4} +$$$$13\!\cdots\!42$$$$T^{5} +$$$$97\!\cdots\!09$$$$T^{6}$$
$97$ $$1 +$$$$87\!\cdots\!94$$$$T +$$$$67\!\cdots\!43$$$$T^{2} +$$$$41\!\cdots\!68$$$$T^{3} +$$$$24\!\cdots\!11$$$$T^{4} +$$$$11\!\cdots\!26$$$$T^{5} +$$$$49\!\cdots\!33$$$$T^{6}$$