# Properties

 Label 471.2.m.a Level $471$ Weight $2$ Character orbit 471.m Analytic conductor $3.761$ Analytic rank $0$ Dimension $156$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$471 = 3 \cdot 157$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 471.m (of order $$13$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.76095393520$$ Analytic rank: $$0$$ Dimension: $$156$$ Relative dimension: $$13$$ over $$\Q(\zeta_{13})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{13}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$156 q - q^{2} + 13 q^{3} - 9 q^{4} - 4 q^{5} + q^{6} + 2 q^{7} - 9 q^{8} - 13 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$156 q - q^{2} + 13 q^{3} - 9 q^{4} - 4 q^{5} + q^{6} + 2 q^{7} - 9 q^{8} - 13 q^{9} - 2 q^{11} - 160 q^{12} - 24 q^{13} - 8 q^{14} - 22 q^{15} - q^{16} - 8 q^{17} - q^{18} + 9 q^{19} - 16 q^{20} - 2 q^{21} + 4 q^{22} - 15 q^{23} + 9 q^{24} - 67 q^{25} - 61 q^{26} + 13 q^{27} + 88 q^{28} + 11 q^{29} + 24 q^{31} - 19 q^{32} + 2 q^{33} + 10 q^{34} + 6 q^{35} - 9 q^{36} + 38 q^{37} + q^{38} - 2 q^{39} + 89 q^{40} + 58 q^{41} + 8 q^{42} - 12 q^{43} + 12 q^{44} - 17 q^{45} - 10 q^{46} + 6 q^{47} + q^{48} + 15 q^{49} + 66 q^{50} - 44 q^{51} - 4 q^{52} - 2 q^{53} + q^{54} + 40 q^{55} + 2 q^{56} - 9 q^{57} - 164 q^{58} + 52 q^{59} + 16 q^{60} + 28 q^{61} + 74 q^{62} + 15 q^{63} + 9 q^{64} - 12 q^{65} + 22 q^{66} - 12 q^{67} + 64 q^{68} + 2 q^{69} - 4 q^{70} - 91 q^{71} - 9 q^{72} + 3 q^{73} - 106 q^{74} + 15 q^{75} - 143 q^{76} - 7 q^{77} + 61 q^{78} + 87 q^{79} + 152 q^{80} - 13 q^{81} - 24 q^{82} + 4 q^{83} - 62 q^{84} + 99 q^{85} + 158 q^{86} + 2 q^{87} - 136 q^{88} - 71 q^{89} - 78 q^{90} - 62 q^{91} - 126 q^{92} - 24 q^{93} - 58 q^{94} - 39 q^{95} + 84 q^{96} + 2 q^{97} + 121 q^{98} - 2 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1 −1.95805 + 1.73468i −0.120537 0.992709i 0.583768 4.80776i 1.42316 3.75255i 1.95805 + 1.73468i −0.446437 3.67674i 4.22485 + 6.12075i −0.970942 + 0.239316i 3.72287 + 9.81641i
16.2 −1.72550 + 1.52866i −0.120537 0.992709i 0.399475 3.28997i 0.291242 0.767943i 1.72550 + 1.52866i 0.434722 + 3.58026i 1.72090 + 2.49315i −0.970942 + 0.239316i 0.671384 + 1.77029i
16.3 −1.11889 + 0.991250i −0.120537 0.992709i 0.0282647 0.232781i 0.0758684 0.200049i 1.11889 + 0.991250i 0.0960865 + 0.791344i −1.49919 2.17195i −0.970942 + 0.239316i 0.113410 + 0.299037i
16.4 −1.09915 + 0.973761i −0.120537 0.992709i 0.0188445 0.155199i −1.22156 + 3.22099i 1.09915 + 0.973761i −0.492440 4.05561i −1.53793 2.22808i −0.970942 + 0.239316i −1.79380 4.72985i
16.5 −0.780454 + 0.691422i −0.120537 0.992709i −0.110029 + 0.906172i 0.132793 0.350146i 0.780454 + 0.691422i −0.307448 2.53206i −1.72529 2.49951i −0.970942 + 0.239316i 0.138460 + 0.365088i
16.6 −0.704389 + 0.624034i −0.120537 0.992709i −0.134328 + 1.10629i 1.47904 3.89991i 0.704389 + 0.624034i −0.0203227 0.167372i −1.66490 2.41203i −0.970942 + 0.239316i 1.39186 + 3.67002i
16.7 0.145924 0.129278i −0.120537 0.992709i −0.236492 + 1.94769i −1.06871 + 2.81795i −0.145924 0.129278i −0.0208582 0.171783i 0.438774 + 0.635674i −0.970942 + 0.239316i 0.208347 + 0.549366i
16.8 0.258202 0.228747i −0.120537 0.992709i −0.226730 + 1.86729i −0.237935 + 0.627382i −0.258202 0.228747i −0.0579046 0.476887i 0.760508 + 1.10179i −0.970942 + 0.239316i 0.0820766 + 0.216418i
16.9 0.404158 0.358053i −0.120537 0.992709i −0.205931 + 1.69600i 1.34301 3.54121i −0.404158 0.358053i 0.496054 + 4.08537i 1.13748 + 1.64793i −0.970942 + 0.239316i −0.725155 1.91208i
16.10 0.886634 0.785489i −0.120537 0.992709i −0.0719469 + 0.592536i 0.576938 1.52126i −0.886634 0.785489i −0.437614 3.60407i 1.74742 + 2.53157i −0.970942 + 0.239316i −0.683401 1.80198i
16.11 1.23748 1.09631i −0.120537 0.992709i 0.0883853 0.727918i −1.03403 + 2.72650i −1.23748 1.09631i 0.407516 + 3.35619i 1.18966 + 1.72353i −0.970942 + 0.239316i 1.70951 + 4.50761i
16.12 1.68653 1.49413i −0.120537 0.992709i 0.370869 3.05438i 0.725667 1.91343i −1.68653 1.49413i 0.0983934 + 0.810342i −1.37827 1.99677i −0.970942 + 0.239316i −1.63506 4.31129i
16.13 2.01899 1.78867i −0.120537 0.992709i 0.635909 5.23718i −0.491944 + 1.29715i −2.01899 1.78867i −0.109123 0.898708i −5.01915 7.27150i −0.970942 + 0.239316i 1.32694 + 3.49885i
46.1 −0.289472 2.38401i 0.970942 + 0.239316i −3.65785 + 0.901579i 0.828579 0.734057i 0.289472 2.38401i 3.03355 + 0.747704i 1.50504 + 3.96846i 0.885456 + 0.464723i −1.98985 1.76286i
46.2 −0.269717 2.22132i 0.970942 + 0.239316i −2.91962 + 0.719622i −2.42824 + 2.15123i 0.269717 2.22132i 1.25723 + 0.309880i 0.799029 + 2.10687i 0.885456 + 0.464723i 5.43350 + 4.81367i
46.3 −0.189990 1.56471i 0.970942 + 0.239316i −0.470342 + 0.115929i 2.57156 2.27820i 0.189990 1.56471i −0.499501 0.123116i −0.847103 2.23363i 0.885456 + 0.464723i −4.05330 3.59091i
46.4 −0.176901 1.45691i 0.970942 + 0.239316i −0.149421 + 0.0368289i −1.94465 + 1.72281i 0.176901 1.45691i −4.37963 1.07948i −0.960757 2.53331i 0.885456 + 0.464723i 2.85400 + 2.52842i
46.5 −0.106448 0.876679i 0.970942 + 0.239316i 1.18465 0.291990i −1.79521 + 1.59042i 0.106448 0.876679i 0.820353 + 0.202199i −1.00840 2.65893i 0.885456 + 0.464723i 1.58538 + 1.40453i
46.6 −0.0293555 0.241764i 0.970942 + 0.239316i 1.88430 0.464437i −1.41784 + 1.25610i 0.0293555 0.241764i 1.92172 + 0.473662i −0.340320 0.897349i 0.885456 + 0.464723i 0.345301 + 0.305910i
46.7 −0.0248032 0.204273i 0.970942 + 0.239316i 1.90077 0.468498i 1.45998 1.29343i 0.0248032 0.204273i 3.09422 + 0.762657i −0.288783 0.761457i 0.885456 + 0.464723i −0.300425 0.266153i
See next 80 embeddings (of 156 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 415.13 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
157.g even 13 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.2.m.a 156
157.g even 13 1 inner 471.2.m.a 156

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.m.a 156 1.a even 1 1 trivial
471.2.m.a 156 157.g even 13 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$13\!\cdots\!32$$$$T_{2}^{130} + 841177685370 T_{2}^{129} +$$$$80\!\cdots\!88$$$$T_{2}^{128} +$$$$49\!\cdots\!07$$$$T_{2}^{127} +$$$$49\!\cdots\!15$$$$T_{2}^{126} +$$$$26\!\cdots\!57$$$$T_{2}^{125} +$$$$29\!\cdots\!06$$$$T_{2}^{124} +$$$$12\!\cdots\!26$$$$T_{2}^{123} +$$$$17\!\cdots\!93$$$$T_{2}^{122} +$$$$52\!\cdots\!15$$$$T_{2}^{121} +$$$$96\!\cdots\!65$$$$T_{2}^{120} +$$$$16\!\cdots\!72$$$$T_{2}^{119} +$$$$53\!\cdots\!68$$$$T_{2}^{118} +$$$$26\!\cdots\!68$$$$T_{2}^{117} +$$$$28\!\cdots\!65$$$$T_{2}^{116} -$$$$10\!\cdots\!96$$$$T_{2}^{115} +$$$$15\!\cdots\!25$$$$T_{2}^{114} -$$$$10\!\cdots\!95$$$$T_{2}^{113} +$$$$75\!\cdots\!97$$$$T_{2}^{112} -$$$$44\!\cdots\!78$$$$T_{2}^{111} +$$$$36\!\cdots\!23$$$$T_{2}^{110} +$$$$61\!\cdots\!63$$$$T_{2}^{109} +$$$$16\!\cdots\!77$$$$T_{2}^{108} +$$$$20\!\cdots\!00$$$$T_{2}^{107} +$$$$66\!\cdots\!62$$$$T_{2}^{106} +$$$$15\!\cdots\!12$$$$T_{2}^{105} +$$$$26\!\cdots\!25$$$$T_{2}^{104} +$$$$74\!\cdots\!03$$$$T_{2}^{103} +$$$$10\!\cdots\!56$$$$T_{2}^{102} +$$$$24\!\cdots\!70$$$$T_{2}^{101} +$$$$38\!\cdots\!86$$$$T_{2}^{100} +$$$$27\!\cdots\!99$$$$T_{2}^{99} +$$$$13\!\cdots\!02$$$$T_{2}^{98} -$$$$23\!\cdots\!62$$$$T_{2}^{97} +$$$$41\!\cdots\!42$$$$T_{2}^{96} -$$$$19\!\cdots\!79$$$$T_{2}^{95} +$$$$12\!\cdots\!59$$$$T_{2}^{94} -$$$$84\!\cdots\!76$$$$T_{2}^{93} +$$$$43\!\cdots\!45$$$$T_{2}^{92} -$$$$28\!\cdots\!34$$$$T_{2}^{91} +$$$$14\!\cdots\!46$$$$T_{2}^{90} -$$$$82\!\cdots\!30$$$$T_{2}^{89} +$$$$44\!\cdots\!24$$$$T_{2}^{88} -$$$$26\!\cdots\!88$$$$T_{2}^{87} +$$$$12\!\cdots\!73$$$$T_{2}^{86} -$$$$77\!\cdots\!84$$$$T_{2}^{85} +$$$$35\!\cdots\!84$$$$T_{2}^{84} -$$$$24\!\cdots\!79$$$$T_{2}^{83} +$$$$90\!\cdots\!11$$$$T_{2}^{82} -$$$$76\!\cdots\!43$$$$T_{2}^{81} +$$$$23\!\cdots\!46$$$$T_{2}^{80} -$$$$20\!\cdots\!98$$$$T_{2}^{79} +$$$$63\!\cdots\!57$$$$T_{2}^{78} -$$$$58\!\cdots\!61$$$$T_{2}^{77} +$$$$16\!\cdots\!16$$$$T_{2}^{76} -$$$$15\!\cdots\!81$$$$T_{2}^{75} +$$$$40\!\cdots\!46$$$$T_{2}^{74} -$$$$40\!\cdots\!30$$$$T_{2}^{73} +$$$$97\!\cdots\!84$$$$T_{2}^{72} -$$$$10\!\cdots\!83$$$$T_{2}^{71} +$$$$22\!\cdots\!82$$$$T_{2}^{70} -$$$$22\!\cdots\!36$$$$T_{2}^{69} +$$$$50\!\cdots\!83$$$$T_{2}^{68} -$$$$48\!\cdots\!03$$$$T_{2}^{67} +$$$$10\!\cdots\!93$$$$T_{2}^{66} -$$$$10\!\cdots\!06$$$$T_{2}^{65} +$$$$19\!\cdots\!33$$$$T_{2}^{64} -$$$$19\!\cdots\!54$$$$T_{2}^{63} +$$$$34\!\cdots\!84$$$$T_{2}^{62} -$$$$33\!\cdots\!49$$$$T_{2}^{61} +$$$$57\!\cdots\!80$$$$T_{2}^{60} -$$$$54\!\cdots\!56$$$$T_{2}^{59} +$$$$86\!\cdots\!82$$$$T_{2}^{58} -$$$$82\!\cdots\!77$$$$T_{2}^{57} +$$$$11\!\cdots\!91$$$$T_{2}^{56} -$$$$11\!\cdots\!25$$$$T_{2}^{55} +$$$$14\!\cdots\!97$$$$T_{2}^{54} -$$$$13\!\cdots\!13$$$$T_{2}^{53} +$$$$16\!\cdots\!35$$$$T_{2}^{52} -$$$$14\!\cdots\!95$$$$T_{2}^{51} +$$$$14\!\cdots\!15$$$$T_{2}^{50} -$$$$11\!\cdots\!59$$$$T_{2}^{49} +$$$$12\!\cdots\!31$$$$T_{2}^{48} -$$$$69\!\cdots\!46$$$$T_{2}^{47} +$$$$77\!\cdots\!49$$$$T_{2}^{46} -$$$$15\!\cdots\!43$$$$T_{2}^{45} +$$$$47\!\cdots\!91$$$$T_{2}^{44} +$$$$11\!\cdots\!08$$$$T_{2}^{43} +$$$$14\!\cdots\!07$$$$T_{2}^{42} +$$$$23\!\cdots\!80$$$$T_{2}^{41} +$$$$50\!\cdots\!26$$$$T_{2}^{40} +$$$$52\!\cdots\!82$$$$T_{2}^{39} +$$$$90\!\cdots\!97$$$$T_{2}^{38} -$$$$20\!\cdots\!64$$$$T_{2}^{37} +$$$$42\!\cdots\!33$$$$T_{2}^{36} -$$$$15\!\cdots\!75$$$$T_{2}^{35} +$$$$49\!\cdots\!92$$$$T_{2}^{34} -$$$$53\!\cdots\!38$$$$T_{2}^{33} +$$$$42\!\cdots\!09$$$$T_{2}^{32} -$$$$41\!\cdots\!46$$$$T_{2}^{31} -$$$$22\!\cdots\!34$$$$T_{2}^{30} +$$$$20\!\cdots\!19$$$$T_{2}^{29} -$$$$60\!\cdots\!46$$$$T_{2}^{28} -$$$$31\!\cdots\!81$$$$T_{2}^{27} +$$$$44\!\cdots\!46$$$$T_{2}^{26} -$$$$26\!\cdots\!94$$$$T_{2}^{25} +$$$$12\!\cdots\!75$$$$T_{2}^{24} -$$$$39\!\cdots\!26$$$$T_{2}^{23} +$$$$31\!\cdots\!12$$$$T_{2}^{22} +$$$$52\!\cdots\!46$$$$T_{2}^{21} -$$$$28\!\cdots\!00$$$$T_{2}^{20} +$$$$77\!\cdots\!08$$$$T_{2}^{19} -$$$$17\!\cdots\!15$$$$T_{2}^{18} -$$$$72\!\cdots\!00$$$$T_{2}^{17} +$$$$40\!\cdots\!00$$$$T_{2}^{16} -$$$$11\!\cdots\!78$$$$T_{2}^{15} +$$$$26\!\cdots\!10$$$$T_{2}^{14} -$$$$38\!\cdots\!47$$$$T_{2}^{13} +$$$$51\!\cdots\!74$$$$T_{2}^{12} -$$$$65\!\cdots\!88$$$$T_{2}^{11} +$$$$29\!\cdots\!25$$$$T_{2}^{10} -$$$$85\!\cdots\!61$$$$T_{2}^{9} +$$$$23\!\cdots\!63$$$$T_{2}^{8} -$$$$40\!\cdots\!61$$$$T_{2}^{7} +$$$$66\!\cdots\!72$$$$T_{2}^{6} -$$$$59\!\cdots\!50$$$$T_{2}^{5} +$$$$72\!\cdots\!66$$$$T_{2}^{4} -$$$$51\!\cdots\!67$$$$T_{2}^{3} +$$$$11\!\cdots\!74$$$$T_{2}^{2} +$$$$26\!\cdots\!47$$$$T_{2} + 262936149529$$">$$T_{2}^{156} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(471, [\chi])$$.