# Properties

 Label 531.2.i.c Level $531$ Weight $2$ Character orbit 531.i Analytic conductor $4.240$ Analytic rank $0$ Dimension $140$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$531 = 3^{2} \cdot 59$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 531.i (of order $$29$$, degree $$28$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.24005634733$$ Analytic rank: $$0$$ Dimension: $$140$$ Relative dimension: $$5$$ over $$\Q(\zeta_{29})$$ Twist minimal: no (minimal twist has level 177) Sato-Tate group: $\mathrm{SU}(2)[C_{29}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$140q + q^{2} - 9q^{4} + 2q^{5} - 2q^{7} + 9q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$140q + q^{2} - 9q^{4} + 2q^{5} - 2q^{7} + 9q^{8} + 88q^{10} + 14q^{11} - 12q^{13} + q^{14} - 41q^{16} + 16q^{17} - 10q^{19} + 32q^{20} - 26q^{22} + 22q^{23} + 27q^{25} + 56q^{26} - 50q^{28} + 24q^{29} - 24q^{31} - 106q^{32} - 54q^{34} + 70q^{35} - 28q^{37} + 80q^{38} - 50q^{40} + 40q^{41} + 4q^{43} + 104q^{44} - 28q^{46} - 31q^{47} - q^{49} - 39q^{50} + 62q^{52} - 4q^{53} + 5q^{55} - 96q^{56} + 128q^{58} + q^{59} - 16q^{61} - 223q^{62} + 97q^{64} - 121q^{65} - 12q^{67} - 10q^{68} - 2q^{70} + 22q^{71} + 179q^{73} + 38q^{74} + 112q^{76} + 62q^{77} - 84q^{79} - 204q^{80} - 152q^{82} + 88q^{83} - 118q^{85} + 118q^{86} + 18q^{88} + 86q^{89} + 78q^{91} + 174q^{92} - 164q^{94} - 218q^{95} - 84q^{97} - 129q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −1.31462 + 2.47964i 0 −3.29802 4.86421i −1.27307 + 0.280224i 0 1.14406 0.869689i 10.8169 1.17641i 0 0.978753 3.52515i
19.2 −0.655840 + 1.23705i 0 0.0222185 + 0.0327698i −2.32817 + 0.512469i 0 −1.36308 + 1.03619i −2.83899 + 0.308758i 0 0.892959 3.21615i
19.3 −0.0292574 + 0.0551852i 0 1.12018 + 1.65215i 2.41343 0.531236i 0 −2.91820 + 2.21836i −0.248138 + 0.0269866i 0 −0.0412942 + 0.148728i
19.4 0.454038 0.856407i 0 0.595092 + 0.877695i −3.60206 + 0.792874i 0 1.56865 1.19245i 2.94914 0.320738i 0 −0.956451 + 3.44483i
19.5 1.07727 2.03195i 0 −1.84595 2.72257i 3.23491 0.712057i 0 1.31099 0.996592i −2.94796 + 0.320610i 0 2.03802 7.34027i
28.1 −1.31462 2.47964i 0 −3.29802 + 4.86421i −1.27307 0.280224i 0 1.14406 + 0.869689i 10.8169 + 1.17641i 0 0.978753 + 3.52515i
28.2 −0.655840 1.23705i 0 0.0222185 0.0327698i −2.32817 0.512469i 0 −1.36308 1.03619i −2.83899 0.308758i 0 0.892959 + 3.21615i
28.3 −0.0292574 0.0551852i 0 1.12018 1.65215i 2.41343 + 0.531236i 0 −2.91820 2.21836i −0.248138 0.0269866i 0 −0.0412942 0.148728i
28.4 0.454038 + 0.856407i 0 0.595092 0.877695i −3.60206 0.792874i 0 1.56865 + 1.19245i 2.94914 + 0.320738i 0 −0.956451 3.44483i
28.5 1.07727 + 2.03195i 0 −1.84595 + 2.72257i 3.23491 + 0.712057i 0 1.31099 + 0.996592i −2.94796 0.320610i 0 2.03802 + 7.34027i
46.1 −1.65338 1.94651i 0 −0.731672 + 4.46300i −0.865278 + 1.63209i 0 −2.96873 + 0.322869i 5.52030 3.32145i 0 4.60751 1.01419i
46.2 −1.15330 1.35777i 0 −0.189873 + 1.15817i 1.46043 2.75467i 0 1.72018 0.187081i −1.26142 + 0.758972i 0 −5.42452 + 1.19403i
46.3 −0.398193 0.468789i 0 0.262358 1.60032i −0.780962 + 1.47305i 0 −2.98728 + 0.324887i −1.90875 + 1.14846i 0 1.00152 0.220452i
46.4 0.814427 + 0.958818i 0 0.0675241 0.411879i −0.588484 + 1.11000i 0 2.16620 0.235588i 2.60580 1.56786i 0 −1.54356 + 0.339764i
46.5 1.74306 + 2.05209i 0 −0.849247 + 5.18018i 1.70562 3.21713i 0 3.51311 0.382074i −7.49637 + 4.51042i 0 9.57483 2.10758i
64.1 −2.28036 0.768343i 0 3.01751 + 2.29385i −0.739682 + 1.85646i 0 −0.807669 + 0.373667i −2.41776 3.56592i 0 3.11314 3.66507i
64.2 −0.827908 0.278955i 0 −0.984570 0.748451i −0.302390 + 0.758941i 0 3.54386 1.63956i 1.58690 + 2.34050i 0 0.462061 0.543981i
64.3 −0.288668 0.0972636i 0 −1.51832 1.15419i 1.30154 3.26662i 0 −3.34578 + 1.54792i 0.667919 + 0.985107i 0 −0.693437 + 0.816377i
64.4 1.73089 + 0.583204i 0 1.06366 + 0.808574i 0.835093 2.09593i 0 2.04844 0.947709i −0.680501 1.00366i 0 2.66781 3.14078i
64.5 2.61370 + 0.880659i 0 4.46370 + 3.39321i −0.422705 + 1.06091i 0 −3.24336 + 1.50054i 5.58292 + 8.23419i 0 −2.03912 + 2.40064i
See next 80 embeddings (of 140 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 523.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.c even 29 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 531.2.i.c 140
3.b odd 2 1 177.2.e.a 140
59.c even 29 1 inner 531.2.i.c 140
177.h odd 58 1 177.2.e.a 140

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.e.a 140 3.b odd 2 1
177.2.e.a 140 177.h odd 58 1
531.2.i.c 140 1.a even 1 1 trivial
531.2.i.c 140 59.c even 29 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$17\!\cdots\!76$$$$T_{2}^{109} +$$$$15\!\cdots\!73$$$$T_{2}^{108} +$$$$54\!\cdots\!81$$$$T_{2}^{107} +$$$$26\!\cdots\!18$$$$T_{2}^{106} -$$$$19\!\cdots\!24$$$$T_{2}^{105} -$$$$44\!\cdots\!52$$$$T_{2}^{104} -$$$$95\!\cdots\!64$$$$T_{2}^{103} -$$$$52\!\cdots\!48$$$$T_{2}^{102} -$$$$85\!\cdots\!70$$$$T_{2}^{101} +$$$$21\!\cdots\!05$$$$T_{2}^{100} -$$$$40\!\cdots\!45$$$$T_{2}^{99} +$$$$13\!\cdots\!06$$$$T_{2}^{98} +$$$$26\!\cdots\!97$$$$T_{2}^{97} +$$$$21\!\cdots\!47$$$$T_{2}^{96} -$$$$57\!\cdots\!05$$$$T_{2}^{95} +$$$$84\!\cdots\!26$$$$T_{2}^{94} -$$$$28\!\cdots\!67$$$$T_{2}^{93} +$$$$35\!\cdots\!42$$$$T_{2}^{92} +$$$$33\!\cdots\!80$$$$T_{2}^{91} +$$$$43\!\cdots\!99$$$$T_{2}^{90} +$$$$18\!\cdots\!41$$$$T_{2}^{89} +$$$$11\!\cdots\!44$$$$T_{2}^{88} -$$$$66\!\cdots\!53$$$$T_{2}^{87} -$$$$48\!\cdots\!26$$$$T_{2}^{86} +$$$$46\!\cdots\!87$$$$T_{2}^{85} +$$$$48\!\cdots\!56$$$$T_{2}^{84} -$$$$83\!\cdots\!66$$$$T_{2}^{83} +$$$$62\!\cdots\!14$$$$T_{2}^{82} +$$$$44\!\cdots\!26$$$$T_{2}^{81} +$$$$91\!\cdots\!50$$$$T_{2}^{80} -$$$$49\!\cdots\!08$$$$T_{2}^{79} +$$$$13\!\cdots\!30$$$$T_{2}^{78} +$$$$89\!\cdots\!48$$$$T_{2}^{77} +$$$$16\!\cdots\!22$$$$T_{2}^{76} -$$$$16\!\cdots\!85$$$$T_{2}^{75} +$$$$71\!\cdots\!71$$$$T_{2}^{74} -$$$$86\!\cdots\!21$$$$T_{2}^{73} +$$$$13\!\cdots\!80$$$$T_{2}^{72} -$$$$23\!\cdots\!50$$$$T_{2}^{71} +$$$$11\!\cdots\!94$$$$T_{2}^{70} -$$$$19\!\cdots\!59$$$$T_{2}^{69} +$$$$43\!\cdots\!96$$$$T_{2}^{68} -$$$$99\!\cdots\!74$$$$T_{2}^{67} +$$$$16\!\cdots\!48$$$$T_{2}^{66} -$$$$30\!\cdots\!27$$$$T_{2}^{65} +$$$$55\!\cdots\!80$$$$T_{2}^{64} -$$$$90\!\cdots\!16$$$$T_{2}^{63} +$$$$14\!\cdots\!36$$$$T_{2}^{62} -$$$$22\!\cdots\!43$$$$T_{2}^{61} +$$$$31\!\cdots\!29$$$$T_{2}^{60} -$$$$39\!\cdots\!90$$$$T_{2}^{59} +$$$$49\!\cdots\!38$$$$T_{2}^{58} -$$$$53\!\cdots\!38$$$$T_{2}^{57} +$$$$51\!\cdots\!10$$$$T_{2}^{56} -$$$$47\!\cdots\!65$$$$T_{2}^{55} +$$$$48\!\cdots\!67$$$$T_{2}^{54} -$$$$38\!\cdots\!07$$$$T_{2}^{53} +$$$$46\!\cdots\!46$$$$T_{2}^{52} -$$$$53\!\cdots\!75$$$$T_{2}^{51} +$$$$57\!\cdots\!11$$$$T_{2}^{50} -$$$$56\!\cdots\!93$$$$T_{2}^{49} +$$$$59\!\cdots\!87$$$$T_{2}^{48} -$$$$42\!\cdots\!58$$$$T_{2}^{47} +$$$$25\!\cdots\!24$$$$T_{2}^{46} -$$$$13\!\cdots\!04$$$$T_{2}^{45} +$$$$10\!\cdots\!55$$$$T_{2}^{44} +$$$$16\!\cdots\!60$$$$T_{2}^{43} +$$$$38\!\cdots\!51$$$$T_{2}^{42} +$$$$40\!\cdots\!88$$$$T_{2}^{41} +$$$$98\!\cdots\!00$$$$T_{2}^{40} +$$$$97\!\cdots\!56$$$$T_{2}^{39} +$$$$16\!\cdots\!63$$$$T_{2}^{38} +$$$$19\!\cdots\!51$$$$T_{2}^{37} +$$$$25\!\cdots\!35$$$$T_{2}^{36} +$$$$31\!\cdots\!78$$$$T_{2}^{35} +$$$$34\!\cdots\!38$$$$T_{2}^{34} +$$$$35\!\cdots\!25$$$$T_{2}^{33} +$$$$33\!\cdots\!11$$$$T_{2}^{32} +$$$$29\!\cdots\!36$$$$T_{2}^{31} +$$$$24\!\cdots\!65$$$$T_{2}^{30} +$$$$19\!\cdots\!67$$$$T_{2}^{29} +$$$$15\!\cdots\!93$$$$T_{2}^{28} +$$$$12\!\cdots\!74$$$$T_{2}^{27} +$$$$93\!\cdots\!61$$$$T_{2}^{26} +$$$$71\!\cdots\!64$$$$T_{2}^{25} +$$$$53\!\cdots\!00$$$$T_{2}^{24} +$$$$39\!\cdots\!03$$$$T_{2}^{23} +$$$$28\!\cdots\!93$$$$T_{2}^{22} +$$$$19\!\cdots\!77$$$$T_{2}^{21} +$$$$12\!\cdots\!47$$$$T_{2}^{20} +$$$$72\!\cdots\!14$$$$T_{2}^{19} +$$$$40\!\cdots\!84$$$$T_{2}^{18} +$$$$20\!\cdots\!56$$$$T_{2}^{17} +$$$$93\!\cdots\!05$$$$T_{2}^{16} +$$$$38\!\cdots\!68$$$$T_{2}^{15} +$$$$14\!\cdots\!80$$$$T_{2}^{14} +$$$$46\!\cdots\!77$$$$T_{2}^{13} +$$$$13\!\cdots\!12$$$$T_{2}^{12} +$$$$35\!\cdots\!97$$$$T_{2}^{11} +$$$$79\!\cdots\!45$$$$T_{2}^{10} +$$$$15\!\cdots\!98$$$$T_{2}^{9} +$$$$24\!\cdots\!23$$$$T_{2}^{8} +$$$$33\!\cdots\!67$$$$T_{2}^{7} +$$$$37\!\cdots\!26$$$$T_{2}^{6} +$$$$33\!\cdots\!63$$$$T_{2}^{5} +$$$$24\!\cdots\!88$$$$T_{2}^{4} +$$$$13\!\cdots\!02$$$$T_{2}^{3} +$$$$57\!\cdots\!02$$$$T_{2}^{2} +$$$$16\!\cdots\!73$$$$T_{2} + 23694752761$$">$$T_{2}^{140} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(531, [\chi])$$.