# Properties

 Label 3.36.a.b Level $3$ Weight $36$ Character orbit 3.a Self dual yes Analytic conductor $23.279$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$36$$ Character orbit: $$[\chi]$$ $$=$$ 3.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$23.2785391901$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - x^{2} - 1847580440 x + 20051963761200$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{7}\cdot 3^{5}\cdot 5$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -29110 + \beta_{1} ) q^{2} -129140163 q^{3} + ( 10829584300 - 155888 \beta_{1} + 23 \beta_{2} ) q^{4} + ( 922892078470 - 7872460 \beta_{1} - 1434 \beta_{2} ) q^{5} + ( 3759270144930 - 129140163 \beta_{1} ) q^{6} + ( 162745949512688 - 1530829676 \beta_{1} + 705590 \beta_{2} ) q^{7} + ( -6227715559376984 + 8864941856 \beta_{1} - 2008590 \beta_{2} ) q^{8} + 16677181699666569 q^{9} +O(q^{10})$$ $$q +(-29110 + \beta_{1}) q^{2} -129140163 q^{3} +(10829584300 - 155888 \beta_{1} + 23 \beta_{2}) q^{4} +(922892078470 - 7872460 \beta_{1} - 1434 \beta_{2}) q^{5} +(3759270144930 - 129140163 \beta_{1}) q^{6} +(162745949512688 - 1530829676 \beta_{1} + 705590 \beta_{2}) q^{7} +(-6227715559376984 + 8864941856 \beta_{1} - 2008590 \beta_{2}) q^{8} +16677181699666569 q^{9} +(-375926542309622340 + 1133373620870 \beta_{1} - 279378752 \beta_{2}) q^{10} +(1142118725041408268 - 172276958744 \beta_{1} + 2103404556 \beta_{2}) q^{11} +(-1398534281724240900 + 20131401729744 \beta_{1} - 2970223749 \beta_{2}) q^{12} +(16695906143752846502 + 104639888932984 \beta_{1} - 12217424188 \beta_{2}) q^{13} +(-72626787653481350048 + 744341535666416 \beta_{1} + 13164756672 \beta_{2}) q^{14} +(-$$$$11\!\cdots\!10$$$$+ 1016650767610980 \beta_{1} + 185186993742 \beta_{2}) q^{15} +($$$$20\!\cdots\!48$$$$- 3098470453435968 \beta_{1} - 724085232996 \beta_{2}) q^{16} +($$$$91\!\cdots\!14$$$$- 7154824633530840 \beta_{1} + 1416590862828 \beta_{2}) q^{17} +(-$$$$48\!\cdots\!90$$$$+ 16677181699666569 \beta_{1}) q^{18} +($$$$95\!\cdots\!64$$$$- 59286509654693496 \beta_{1} - 15574423845828 \beta_{2}) q^{19} +($$$$29\!\cdots\!20$$$$- 402556500667987360 \beta_{1} + 56185809620106 \beta_{2}) q^{20} +(-$$$$21\!\cdots\!44$$$$+ 197691593883877188 \beta_{1} - 91120007611170 \beta_{2}) q^{21} +(-$$$$40\!\cdots\!32$$$$+ 2319179345885041420 \beta_{1} + 140242839499136 \beta_{2}) q^{22} +($$$$60\!\cdots\!36$$$$+ 786887228305973288 \beta_{1} - 307740421145364 \beta_{2}) q^{23} +($$$$80\!\cdots\!92$$$$- 1144820036269362528 \beta_{1} + 259389640000170 \beta_{2}) q^{24} +($$$$28\!\cdots\!75$$$$- 3388961875629683600 \beta_{1} - 145625899280440 \beta_{2}) q^{25} +($$$$41\!\cdots\!72$$$$- 3280112791214552410 \beta_{1} + 1569115277977728 \beta_{2}) q^{26} -$$$$21\!\cdots\!47$$$$q^{27} +($$$$29\!\cdots\!64$$$$-$$$$10\!\cdots\!88$$$$\beta_{1} - 6221483086830576 \beta_{2}) q^{28} +($$$$26\!\cdots\!38$$$$+$$$$10\!\cdots\!48$$$$\beta_{1} - 14297266947051054 \beta_{2}) q^{29} +($$$$48\!\cdots\!20$$$$-$$$$14\!\cdots\!10$$$$\beta_{1} + 36079017572016576 \beta_{2}) q^{30} +(-$$$$53\!\cdots\!16$$$$+$$$$76\!\cdots\!92$$$$\beta_{1} + 58006914115659806 \beta_{2}) q^{31} +($$$$70\!\cdots\!68$$$$-$$$$10\!\cdots\!76$$$$\beta_{1} - 51892028944185912 \beta_{2}) q^{32} +(-$$$$14\!\cdots\!84$$$$+ 22247874533344435272 \beta_{1} - 271634007216782628 \beta_{2}) q^{33} +(-$$$$34\!\cdots\!40$$$$+$$$$25\!\cdots\!94$$$$\beta_{1} - 67442330197447296 \beta_{2}) q^{34} +(-$$$$38\!\cdots\!00$$$$-$$$$80\!\cdots\!00$$$$\beta_{1} + 922574061615363300 \beta_{2}) q^{35} +($$$$18\!\cdots\!00$$$$-$$$$25\!\cdots\!72$$$$\beta_{1} + 383575179092331087 \beta_{2}) q^{36} +(-$$$$17\!\cdots\!62$$$$-$$$$10\!\cdots\!96$$$$\beta_{1} - 1231959827831347488 \beta_{2}) q^{37} +(-$$$$29\!\cdots\!88$$$$+$$$$85\!\cdots\!92$$$$\beta_{1} - 2431341072080226432 \beta_{2}) q^{38} +(-$$$$21\!\cdots\!26$$$$-$$$$13\!\cdots\!92$$$$\beta_{1} + 1577760151078462644 \beta_{2}) q^{39} +(-$$$$57\!\cdots\!20$$$$+$$$$72\!\cdots\!60$$$$\beta_{1} + 4192568044869874604 \beta_{2}) q^{40} +(-$$$$67\!\cdots\!86$$$$+$$$$24\!\cdots\!32$$$$\beta_{1} - 632694542246452476 \beta_{2}) q^{41} +($$$$93\!\cdots\!24$$$$-$$$$96\!\cdots\!08$$$$\beta_{1} - 1700098822477417536 \beta_{2}) q^{42} +($$$$15\!\cdots\!24$$$$+$$$$15\!\cdots\!56$$$$\beta_{1} + 4060988346763694516 \beta_{2}) q^{43} +($$$$64\!\cdots\!96$$$$-$$$$25\!\cdots\!80$$$$\beta_{1} - 9316536680481486060 \beta_{2}) q^{44} +($$$$15\!\cdots\!30$$$$-$$$$13\!\cdots\!40$$$$\beta_{1} - 23915078557321859946 \beta_{2}) q^{45} +($$$$17\!\cdots\!64$$$$+$$$$33\!\cdots\!92$$$$\beta_{1} - 2999661541846479488 \beta_{2}) q^{46} +($$$$97\!\cdots\!72$$$$-$$$$69\!\cdots\!12$$$$\beta_{1} + 87782805605485124484 \beta_{2}) q^{47} +(-$$$$26\!\cdots\!24$$$$+$$$$40\!\cdots\!84$$$$\beta_{1} + 93508485014996418348 \beta_{2}) q^{48} +($$$$27\!\cdots\!69$$$$-$$$$75\!\cdots\!44$$$$\beta_{1} -$$$$20\!\cdots\!52$$$$\beta_{2}) q^{49} +(-$$$$23\!\cdots\!50$$$$+$$$$32\!\cdots\!75$$$$\beta_{1} - 87929943542351128320 \beta_{2}) q^{50} +(-$$$$11\!\cdots\!82$$$$+$$$$92\!\cdots\!20$$$$\beta_{1} -$$$$18\!\cdots\!64$$$$\beta_{2}) q^{51} +(-$$$$84\!\cdots\!16$$$$+$$$$18\!\cdots\!00$$$$\beta_{1} +$$$$45\!\cdots\!78$$$$\beta_{2}) q^{52} +(-$$$$36\!\cdots\!94$$$$-$$$$33\!\cdots\!12$$$$\beta_{1} +$$$$24\!\cdots\!26$$$$\beta_{2}) q^{53} +($$$$62\!\cdots\!70$$$$-$$$$21\!\cdots\!47$$$$\beta_{1}) q^{54} +(-$$$$20\!\cdots\!80$$$$-$$$$25\!\cdots\!60$$$$\beta_{1} +$$$$39\!\cdots\!56$$$$\beta_{2}) q^{55} +(-$$$$31\!\cdots\!00$$$$+$$$$14\!\cdots\!20$$$$\beta_{1} -$$$$33\!\cdots\!28$$$$\beta_{2}) q^{56} +(-$$$$12\!\cdots\!32$$$$+$$$$76\!\cdots\!48$$$$\beta_{1} +$$$$20\!\cdots\!64$$$$\beta_{2}) q^{57} +($$$$37\!\cdots\!24$$$$+$$$$54\!\cdots\!14$$$$\beta_{1} +$$$$13\!\cdots\!72$$$$\beta_{2}) q^{58} +($$$$37\!\cdots\!16$$$$+$$$$14\!\cdots\!36$$$$\beta_{1} +$$$$55\!\cdots\!52$$$$\beta_{2}) q^{59} +(-$$$$38\!\cdots\!60$$$$+$$$$51\!\cdots\!80$$$$\beta_{1} -$$$$72\!\cdots\!78$$$$\beta_{2}) q^{60} +($$$$42\!\cdots\!06$$$$-$$$$33\!\cdots\!76$$$$\beta_{1} -$$$$16\!\cdots\!08$$$$\beta_{2}) q^{61} +($$$$33\!\cdots\!56$$$$-$$$$65\!\cdots\!72$$$$\beta_{1} +$$$$21\!\cdots\!64$$$$\beta_{2}) q^{62} +($$$$27\!\cdots\!72$$$$-$$$$25\!\cdots\!44$$$$\beta_{1} +$$$$11\!\cdots\!10$$$$\beta_{2}) q^{63} +(-$$$$13\!\cdots\!92$$$$+$$$$16\!\cdots\!80$$$$\beta_{1} +$$$$18\!\cdots\!84$$$$\beta_{2}) q^{64} +(-$$$$25\!\cdots\!20$$$$+$$$$56\!\cdots\!60$$$$\beta_{1} -$$$$69\!\cdots\!56$$$$\beta_{2}) q^{65} +($$$$52\!\cdots\!16$$$$-$$$$29\!\cdots\!60$$$$\beta_{1} -$$$$18\!\cdots\!68$$$$\beta_{2}) q^{66} +($$$$58\!\cdots\!04$$$$-$$$$11\!\cdots\!60$$$$\beta_{1} +$$$$67\!\cdots\!00$$$$\beta_{2}) q^{67} +($$$$93\!\cdots\!00$$$$-$$$$46\!\cdots\!72$$$$\beta_{1} +$$$$63\!\cdots\!90$$$$\beta_{2}) q^{68} +(-$$$$77\!\cdots\!68$$$$-$$$$10\!\cdots\!44$$$$\beta_{1} +$$$$39\!\cdots\!32$$$$\beta_{2}) q^{69} +(-$$$$34\!\cdots\!00$$$$+$$$$11\!\cdots\!00$$$$\beta_{1} -$$$$12\!\cdots\!00$$$$\beta_{2}) q^{70} +(-$$$$91\!\cdots\!08$$$$+$$$$11\!\cdots\!80$$$$\beta_{1} +$$$$14\!\cdots\!32$$$$\beta_{2}) q^{71} +(-$$$$10\!\cdots\!96$$$$+$$$$14\!\cdots\!64$$$$\beta_{1} -$$$$33\!\cdots\!10$$$$\beta_{2}) q^{72} +(-$$$$19\!\cdots\!74$$$$+$$$$39\!\cdots\!08$$$$\beta_{1} -$$$$18\!\cdots\!80$$$$\beta_{2}) q^{73} +($$$$44\!\cdots\!72$$$$-$$$$23\!\cdots\!34$$$$\beta_{1} -$$$$86\!\cdots\!12$$$$\beta_{2}) q^{74} +(-$$$$37\!\cdots\!25$$$$+$$$$43\!\cdots\!00$$$$\beta_{1} +$$$$18\!\cdots\!20$$$$\beta_{2}) q^{75} +($$$$13\!\cdots\!04$$$$-$$$$32\!\cdots\!76$$$$\beta_{1} +$$$$56\!\cdots\!64$$$$\beta_{2}) q^{76} +($$$$17\!\cdots\!36$$$$-$$$$17\!\cdots\!52$$$$\beta_{1} +$$$$12\!\cdots\!72$$$$\beta_{2}) q^{77} +(-$$$$53\!\cdots\!36$$$$+$$$$42\!\cdots\!30$$$$\beta_{1} -$$$$20\!\cdots\!64$$$$\beta_{2}) q^{78} +(-$$$$88\!\cdots\!00$$$$+$$$$96\!\cdots\!20$$$$\beta_{1} -$$$$38\!\cdots\!78$$$$\beta_{2}) q^{79} +($$$$23\!\cdots\!80$$$$+$$$$11\!\cdots\!60$$$$\beta_{1} +$$$$23\!\cdots\!04$$$$\beta_{2}) q^{80} +$$$$27\!\cdots\!61$$$$q^{81} +($$$$30\!\cdots\!96$$$$-$$$$73\!\cdots\!02$$$$\beta_{1} +$$$$13\!\cdots\!28$$$$\beta_{2}) q^{82} +($$$$21\!\cdots\!60$$$$+$$$$25\!\cdots\!32$$$$\beta_{1} -$$$$15\!\cdots\!60$$$$\beta_{2}) q^{83} +(-$$$$38\!\cdots\!32$$$$+$$$$13\!\cdots\!44$$$$\beta_{1} +$$$$80\!\cdots\!88$$$$\beta_{2}) q^{84} +($$$$11\!\cdots\!80$$$$-$$$$24\!\cdots\!40$$$$\beta_{1} +$$$$23\!\cdots\!64$$$$\beta_{2}) q^{85} +($$$$62\!\cdots\!08$$$$-$$$$13\!\cdots\!24$$$$\beta_{1} +$$$$37\!\cdots\!16$$$$\beta_{2}) q^{86} +(-$$$$33\!\cdots\!94$$$$-$$$$13\!\cdots\!24$$$$\beta_{1} +$$$$18\!\cdots\!02$$$$\beta_{2}) q^{87} +(-$$$$11\!\cdots\!24$$$$+$$$$11\!\cdots\!76$$$$\beta_{1} -$$$$11\!\cdots\!68$$$$\beta_{2}) q^{88} +($$$$38\!\cdots\!74$$$$-$$$$57\!\cdots\!36$$$$\beta_{1} -$$$$75\!\cdots\!84$$$$\beta_{2}) q^{89} +(-$$$$62\!\cdots\!60$$$$+$$$$18\!\cdots\!30$$$$\beta_{1} -$$$$46\!\cdots\!88$$$$\beta_{2}) q^{90} +(-$$$$13\!\cdots\!16$$$$+$$$$37\!\cdots\!12$$$$\beta_{1} +$$$$19\!\cdots\!68$$$$\beta_{2}) q^{91} +(-$$$$64\!\cdots\!52$$$$-$$$$53\!\cdots\!56$$$$\beta_{1} +$$$$18\!\cdots\!64$$$$\beta_{2}) q^{92} +($$$$68\!\cdots\!08$$$$-$$$$98\!\cdots\!96$$$$\beta_{1} -$$$$74\!\cdots\!78$$$$\beta_{2}) q^{93} +(-$$$$33\!\cdots\!76$$$$+$$$$23\!\cdots\!88$$$$\beta_{1} -$$$$10\!\cdots\!04$$$$\beta_{2}) q^{94} +($$$$53\!\cdots\!20$$$$-$$$$97\!\cdots\!60$$$$\beta_{1} -$$$$93\!\cdots\!04$$$$\beta_{2}) q^{95} +(-$$$$91\!\cdots\!84$$$$+$$$$13\!\cdots\!88$$$$\beta_{1} +$$$$67\!\cdots\!56$$$$\beta_{2}) q^{96} +($$$$70\!\cdots\!14$$$$+$$$$44\!\cdots\!00$$$$\beta_{1} -$$$$23\!\cdots\!92$$$$\beta_{2}) q^{97} +(-$$$$41\!\cdots\!62$$$$+$$$$26\!\cdots\!61$$$$\beta_{1} -$$$$31\!\cdots\!28$$$$\beta_{2}) q^{98} +($$$$19\!\cdots\!92$$$$-$$$$28\!\cdots\!36$$$$\beta_{1} +$$$$35\!\cdots\!64$$$$\beta_{2}) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 87330q^{2} - 387420489q^{3} + 32488752900q^{4} + 2768676235410q^{5} + 11277810434790q^{6} + 488237848538064q^{7} - 18683146678130952q^{8} + 50031545098999707q^{9} + O(q^{10})$$ $$3q - 87330q^{2} - 387420489q^{3} + 32488752900q^{4} + 2768676235410q^{5} + 11277810434790q^{6} + 488237848538064q^{7} - 18683146678130952q^{8} + 50031545098999707q^{9} - 1127779626928867020q^{10} + 3426356175124224804q^{11} - 4195602845172722700q^{12} + 50087718431258539506q^{13} -$$$$21\!\cdots\!44$$$$q^{14} -$$$$35\!\cdots\!30$$$$q^{15} +$$$$60\!\cdots\!44$$$$q^{16} +$$$$27\!\cdots\!42$$$$q^{17} -$$$$14\!\cdots\!70$$$$q^{18} +$$$$28\!\cdots\!92$$$$q^{19} +$$$$88\!\cdots\!60$$$$q^{20} -$$$$63\!\cdots\!32$$$$q^{21} -$$$$12\!\cdots\!96$$$$q^{22} +$$$$18\!\cdots\!08$$$$q^{23} +$$$$24\!\cdots\!76$$$$q^{24} +$$$$85\!\cdots\!25$$$$q^{25} +$$$$12\!\cdots\!16$$$$q^{26} -$$$$64\!\cdots\!41$$$$q^{27} +$$$$88\!\cdots\!92$$$$q^{28} +$$$$78\!\cdots\!14$$$$q^{29} +$$$$14\!\cdots\!60$$$$q^{30} -$$$$15\!\cdots\!48$$$$q^{31} +$$$$21\!\cdots\!04$$$$q^{32} -$$$$44\!\cdots\!52$$$$q^{33} -$$$$10\!\cdots\!20$$$$q^{34} -$$$$11\!\cdots\!00$$$$q^{35} +$$$$54\!\cdots\!00$$$$q^{36} -$$$$51\!\cdots\!86$$$$q^{37} -$$$$87\!\cdots\!64$$$$q^{38} -$$$$64\!\cdots\!78$$$$q^{39} -$$$$17\!\cdots\!60$$$$q^{40} -$$$$20\!\cdots\!58$$$$q^{41} +$$$$28\!\cdots\!72$$$$q^{42} +$$$$46\!\cdots\!72$$$$q^{43} +$$$$19\!\cdots\!88$$$$q^{44} +$$$$46\!\cdots\!90$$$$q^{45} +$$$$52\!\cdots\!92$$$$q^{46} +$$$$29\!\cdots\!16$$$$q^{47} -$$$$78\!\cdots\!72$$$$q^{48} +$$$$83\!\cdots\!07$$$$q^{49} -$$$$70\!\cdots\!50$$$$q^{50} -$$$$35\!\cdots\!46$$$$q^{51} -$$$$25\!\cdots\!48$$$$q^{52} -$$$$10\!\cdots\!82$$$$q^{53} +$$$$18\!\cdots\!10$$$$q^{54} -$$$$62\!\cdots\!40$$$$q^{55} -$$$$93\!\cdots\!00$$$$q^{56} -$$$$37\!\cdots\!96$$$$q^{57} +$$$$11\!\cdots\!72$$$$q^{58} +$$$$11\!\cdots\!48$$$$q^{59} -$$$$11\!\cdots\!80$$$$q^{60} +$$$$12\!\cdots\!18$$$$q^{61} +$$$$10\!\cdots\!68$$$$q^{62} +$$$$81\!\cdots\!16$$$$q^{63} -$$$$41\!\cdots\!76$$$$q^{64} -$$$$77\!\cdots\!60$$$$q^{65} +$$$$15\!\cdots\!48$$$$q^{66} +$$$$17\!\cdots\!12$$$$q^{67} +$$$$28\!\cdots\!00$$$$q^{68} -$$$$23\!\cdots\!04$$$$q^{69} -$$$$10\!\cdots\!00$$$$q^{70} -$$$$27\!\cdots\!24$$$$q^{71} -$$$$31\!\cdots\!88$$$$q^{72} -$$$$58\!\cdots\!22$$$$q^{73} +$$$$13\!\cdots\!16$$$$q^{74} -$$$$11\!\cdots\!75$$$$q^{75} +$$$$40\!\cdots\!12$$$$q^{76} +$$$$53\!\cdots\!08$$$$q^{77} -$$$$16\!\cdots\!08$$$$q^{78} -$$$$26\!\cdots\!00$$$$q^{79} +$$$$71\!\cdots\!40$$$$q^{80} +$$$$83\!\cdots\!83$$$$q^{81} +$$$$91\!\cdots\!88$$$$q^{82} +$$$$65\!\cdots\!80$$$$q^{83} -$$$$11\!\cdots\!96$$$$q^{84} +$$$$35\!\cdots\!40$$$$q^{85} +$$$$18\!\cdots\!24$$$$q^{86} -$$$$10\!\cdots\!82$$$$q^{87} -$$$$34\!\cdots\!72$$$$q^{88} +$$$$11\!\cdots\!22$$$$q^{89} -$$$$18\!\cdots\!80$$$$q^{90} -$$$$40\!\cdots\!48$$$$q^{91} -$$$$19\!\cdots\!56$$$$q^{92} +$$$$20\!\cdots\!24$$$$q^{93} -$$$$10\!\cdots\!28$$$$q^{94} +$$$$15\!\cdots\!60$$$$q^{95} -$$$$27\!\cdots\!52$$$$q^{96} +$$$$21\!\cdots\!42$$$$q^{97} -$$$$12\!\cdots\!86$$$$q^{98} +$$$$57\!\cdots\!76$$$$q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 1847580440 x + 20051963761200$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$6 \nu - 2$$ $$\beta_{2}$$ $$=$$ $$($$$$36 \nu^{2} + 585984 \nu - 44342125900$$$$)/23$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 2$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$23 \beta_{2} - 97664 \beta_{1} + 44341930572$$$$)/36$$

## 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
 −47629.1 11725.6 35904.5
−314887. −1.29140e8 6.47940e10 2.58564e12 4.06645e13 8.89060e14 −9.58334e15 1.66772e16 −8.14184e17
1.2 41241.5 −1.29140e8 −3.26589e10 2.39670e12 −5.32594e12 −9.42639e14 −2.76395e15 1.66772e16 9.88435e16
1.3 186315. −1.29140e8 3.53648e8 −2.21366e12 −2.40608e13 5.41817e14 −6.33585e15 1.66772e16 −4.12439e17
 $$n$$: 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 3.36.a.b 3
3.b odd 2 1 9.36.a.c 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 87330 T_{2}^{2} - 63970719552 T_{2} +$$2419568332406784

'>$$24\!\cdots\!84$$ acting on $$S_{36}^{\mathrm{new}}(\Gamma_0(3))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 87330 T + 39108495552 T^{2} + 8420840235761664 T^{3} +$$$$13\!\cdots\!36$$$$T^{4} +$$$$10\!\cdots\!20$$$$T^{5} +$$$$40\!\cdots\!32$$$$T^{6}$$
$3$ $$( 1 + 129140163 T )^{3}$$
$5$ $$1 - 2768676235410 T +$$$$38\!\cdots\!75$$$$T^{2} -$$$$23\!\cdots\!00$$$$T^{3} +$$$$11\!\cdots\!75$$$$T^{4} -$$$$23\!\cdots\!50$$$$T^{5} +$$$$24\!\cdots\!25$$$$T^{6}$$
$7$ $$1 - 488237848538064 T +$$$$26\!\cdots\!09$$$$T^{2} +$$$$84\!\cdots\!96$$$$T^{3} +$$$$10\!\cdots\!87$$$$T^{4} -$$$$70\!\cdots\!36$$$$T^{5} +$$$$54\!\cdots\!07$$$$T^{6}$$
$11$ $$1 - 3426356175124224804 T +$$$$53\!\cdots\!33$$$$T^{2} -$$$$70\!\cdots\!60$$$$T^{3} +$$$$14\!\cdots\!83$$$$T^{4} -$$$$27\!\cdots\!04$$$$T^{5} +$$$$22\!\cdots\!51$$$$T^{6}$$
$13$ $$1 - 50087718431258539506 T +$$$$27\!\cdots\!11$$$$T^{2} -$$$$77\!\cdots\!12$$$$T^{3} +$$$$27\!\cdots\!27$$$$T^{4} -$$$$47\!\cdots\!94$$$$T^{5} +$$$$92\!\cdots\!93$$$$T^{6}$$
$17$ $$1 -$$$$27\!\cdots\!42$$$$T +$$$$30\!\cdots\!67$$$$T^{2} -$$$$59\!\cdots\!56$$$$T^{3} +$$$$35\!\cdots\!31$$$$T^{4} -$$$$36\!\cdots\!58$$$$T^{5} +$$$$15\!\cdots\!57$$$$T^{6}$$
$19$ $$1 -$$$$28\!\cdots\!92$$$$T +$$$$13\!\cdots\!93$$$$T^{2} -$$$$22\!\cdots\!96$$$$T^{3} +$$$$78\!\cdots\!07$$$$T^{4} -$$$$93\!\cdots\!92$$$$T^{5} +$$$$18\!\cdots\!99$$$$T^{6}$$
$23$ $$1 -$$$$18\!\cdots\!08$$$$T +$$$$22\!\cdots\!61$$$$T^{2} -$$$$17\!\cdots\!12$$$$T^{3} +$$$$10\!\cdots\!27$$$$T^{4} -$$$$37\!\cdots\!92$$$$T^{5} +$$$$95\!\cdots\!43$$$$T^{6}$$
$29$ $$1 -$$$$78\!\cdots\!14$$$$T +$$$$56\!\cdots\!11$$$$T^{2} -$$$$22\!\cdots\!52$$$$T^{3} +$$$$85\!\cdots\!39$$$$T^{4} -$$$$18\!\cdots\!14$$$$T^{5} +$$$$35\!\cdots\!49$$$$T^{6}$$
$31$ $$1 +$$$$15\!\cdots\!48$$$$T +$$$$33\!\cdots\!53$$$$T^{2} -$$$$11\!\cdots\!04$$$$T^{3} +$$$$52\!\cdots\!03$$$$T^{4} +$$$$39\!\cdots\!48$$$$T^{5} +$$$$39\!\cdots\!51$$$$T^{6}$$
$37$ $$1 +$$$$51\!\cdots\!86$$$$T +$$$$29\!\cdots\!19$$$$T^{2} +$$$$78\!\cdots\!96$$$$T^{3} +$$$$22\!\cdots\!67$$$$T^{4} +$$$$30\!\cdots\!14$$$$T^{5} +$$$$45\!\cdots\!57$$$$T^{6}$$
$41$ $$1 +$$$$20\!\cdots\!58$$$$T +$$$$97\!\cdots\!83$$$$T^{2} +$$$$11\!\cdots\!16$$$$T^{3} +$$$$27\!\cdots\!83$$$$T^{4} +$$$$15\!\cdots\!58$$$$T^{5} +$$$$21\!\cdots\!01$$$$T^{6}$$
$43$ $$1 -$$$$46\!\cdots\!72$$$$T +$$$$36\!\cdots\!57$$$$T^{2} -$$$$11\!\cdots\!00$$$$T^{3} +$$$$53\!\cdots\!99$$$$T^{4} -$$$$10\!\cdots\!28$$$$T^{5} +$$$$32\!\cdots\!43$$$$T^{6}$$
$47$ $$1 -$$$$29\!\cdots\!16$$$$T +$$$$84\!\cdots\!53$$$$T^{2} -$$$$18\!\cdots\!60$$$$T^{3} +$$$$28\!\cdots\!79$$$$T^{4} -$$$$32\!\cdots\!84$$$$T^{5} +$$$$37\!\cdots\!07$$$$T^{6}$$
$53$ $$1 +$$$$10\!\cdots\!82$$$$T +$$$$58\!\cdots\!31$$$$T^{2} +$$$$17\!\cdots\!48$$$$T^{3} +$$$$13\!\cdots\!67$$$$T^{4} +$$$$54\!\cdots\!18$$$$T^{5} +$$$$11\!\cdots\!93$$$$T^{6}$$
$59$ $$1 -$$$$11\!\cdots\!48$$$$T +$$$$26\!\cdots\!33$$$$T^{2} -$$$$20\!\cdots\!44$$$$T^{3} +$$$$25\!\cdots\!67$$$$T^{4} -$$$$10\!\cdots\!48$$$$T^{5} +$$$$86\!\cdots\!99$$$$T^{6}$$
$61$ $$1 -$$$$12\!\cdots\!18$$$$T +$$$$44\!\cdots\!99$$$$T^{2} -$$$$42\!\cdots\!64$$$$T^{3} +$$$$13\!\cdots\!99$$$$T^{4} -$$$$12\!\cdots\!18$$$$T^{5} +$$$$28\!\cdots\!01$$$$T^{6}$$
$67$ $$1 -$$$$17\!\cdots\!12$$$$T +$$$$26\!\cdots\!77$$$$T^{2} -$$$$23\!\cdots\!96$$$$T^{3} +$$$$21\!\cdots\!11$$$$T^{4} -$$$$11\!\cdots\!88$$$$T^{5} +$$$$54\!\cdots\!07$$$$T^{6}$$
$71$ $$1 +$$$$27\!\cdots\!24$$$$T +$$$$62\!\cdots\!45$$$$T^{2} -$$$$11\!\cdots\!40$$$$T^{3} +$$$$39\!\cdots\!95$$$$T^{4} +$$$$10\!\cdots\!24$$$$T^{5} +$$$$24\!\cdots\!51$$$$T^{6}$$
$73$ $$1 +$$$$58\!\cdots\!22$$$$T +$$$$54\!\cdots\!71$$$$T^{2} +$$$$18\!\cdots\!08$$$$T^{3} +$$$$89\!\cdots\!47$$$$T^{4} +$$$$15\!\cdots\!78$$$$T^{5} +$$$$44\!\cdots\!93$$$$T^{6}$$
$79$ $$1 +$$$$26\!\cdots\!00$$$$T +$$$$99\!\cdots\!97$$$$T^{2} +$$$$14\!\cdots\!00$$$$T^{3} +$$$$25\!\cdots\!03$$$$T^{4} +$$$$18\!\cdots\!00$$$$T^{5} +$$$$17\!\cdots\!99$$$$T^{6}$$
$83$ $$1 -$$$$65\!\cdots\!80$$$$T +$$$$11\!\cdots\!73$$$$T^{2} +$$$$21\!\cdots\!32$$$$T^{3} +$$$$17\!\cdots\!11$$$$T^{4} -$$$$14\!\cdots\!20$$$$T^{5} +$$$$31\!\cdots\!43$$$$T^{6}$$
$89$ $$1 -$$$$11\!\cdots\!22$$$$T +$$$$24\!\cdots\!03$$$$T^{2} -$$$$97\!\cdots\!16$$$$T^{3} +$$$$40\!\cdots\!47$$$$T^{4} -$$$$32\!\cdots\!22$$$$T^{5} +$$$$48\!\cdots\!49$$$$T^{6}$$
$97$ $$1 -$$$$21\!\cdots\!42$$$$T +$$$$24\!\cdots\!67$$$$T^{2} -$$$$17\!\cdots\!56$$$$T^{3} +$$$$83\!\cdots\!31$$$$T^{4} -$$$$25\!\cdots\!58$$$$T^{5} +$$$$40\!\cdots\!57$$$$T^{6}$$