# Properties

 Label 637.2.w.a Level $637$ Weight $2$ Character orbit 637.w Analytic conductor $5.086$ Analytic rank $0$ Dimension $162$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.w (of order $$7$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$162$$ Relative dimension: $$27$$ over $$\Q(\zeta_{7})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$162 q + 3 q^{2} - 25 q^{4} - 4 q^{5} + 18 q^{6} - 15 q^{7} + 3 q^{8} - 5 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$162 q + 3 q^{2} - 25 q^{4} - 4 q^{5} + 18 q^{6} - 15 q^{7} + 3 q^{8} - 5 q^{9} - 10 q^{10} + 15 q^{11} + 25 q^{12} + 27 q^{13} + 33 q^{14} + 18 q^{15} - 5 q^{16} + 3 q^{17} - 64 q^{18} + 24 q^{19} - 47 q^{20} + 24 q^{22} + 27 q^{23} - 8 q^{24} - 35 q^{25} - 3 q^{26} + 15 q^{27} + 2 q^{28} + 46 q^{29} - 30 q^{30} + 46 q^{31} + 16 q^{32} - 18 q^{33} - 62 q^{34} - 51 q^{35} + 39 q^{36} + 16 q^{37} - 54 q^{38} + 74 q^{40} - 2 q^{41} + 88 q^{42} + 14 q^{43} - 95 q^{44} + 83 q^{45} + 56 q^{46} - 4 q^{47} - 20 q^{48} - 3 q^{49} - 216 q^{50} - 56 q^{51} + 25 q^{52} + 38 q^{53} - 6 q^{54} + 73 q^{55} - 35 q^{56} + 41 q^{57} + 72 q^{58} - 44 q^{59} + 24 q^{60} - 6 q^{61} - 36 q^{62} - q^{63} - 11 q^{64} + 4 q^{65} + 95 q^{66} - 126 q^{67} - 382 q^{68} - 108 q^{69} - 47 q^{70} + 51 q^{71} + 130 q^{72} + 14 q^{73} - 26 q^{74} + 3 q^{75} + 75 q^{76} - 6 q^{77} + 31 q^{78} - 58 q^{79} + 110 q^{80} - 5 q^{81} - 90 q^{82} - 35 q^{83} + 21 q^{84} + 18 q^{85} + 76 q^{86} - 100 q^{87} + 6 q^{88} + 32 q^{89} + 13 q^{90} + q^{91} + 46 q^{92} + 19 q^{93} + 72 q^{94} + 38 q^{95} + 95 q^{96} + 6 q^{97} - 299 q^{98} - 334 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
92.1 −1.64981 + 2.06880i 0.0373896 + 0.0180059i −1.11300 4.87639i 1.69442 + 0.815989i −0.0989361 + 0.0476451i 2.20489 + 1.46235i 7.15641 + 3.44634i −1.86940 2.34415i −4.48358 + 2.15918i
92.2 −1.60516 + 2.01281i −2.57103 1.23814i −1.02981 4.51189i −1.31474 0.633146i 6.61905 3.18757i −0.0925892 + 2.64413i 6.09552 + 2.93545i 3.20673 + 4.02111i 3.38477 1.63002i
92.3 −1.55889 + 1.95478i 1.69142 + 0.814547i −0.946002 4.14471i 0.780772 + 0.376000i −4.22900 + 2.03658i −1.80494 + 1.93447i 5.07139 + 2.44225i 0.326959 + 0.409994i −1.95213 + 0.940098i
92.4 −1.50697 + 1.88968i 2.72555 + 1.31256i −0.854886 3.74550i −3.20148 1.54175i −6.58762 + 3.17243i −1.89391 1.84746i 4.01081 + 1.93150i 3.83535 + 4.80937i 7.73794 3.72639i
92.5 −1.40295 + 1.75924i −2.32362 1.11900i −0.681629 2.98641i 3.41692 + 1.64550i 5.22852 2.51792i −2.48146 0.917792i 2.15548 + 1.03802i 2.27660 + 2.85477i −7.68860 + 3.70264i
92.6 −1.27694 + 1.60123i 1.03070 + 0.496360i −0.488322 2.13948i −0.0782783 0.0376968i −2.11093 + 1.01657i 0.138811 2.64211i 0.358894 + 0.172834i −1.05450 1.32230i 0.160318 0.0772049i
92.7 −0.928616 + 1.16445i −1.16240 0.559785i −0.0485689 0.212794i −2.30199 1.10858i 1.73127 0.833735i −2.33750 1.23939i −2.39089 1.15139i −0.832643 1.04410i 3.42855 1.65110i
92.8 −0.911679 + 1.14321i −1.60612 0.773466i −0.0307265 0.134622i −2.27795 1.09700i 2.34850 1.13098i 2.64144 + 0.150997i −2.45291 1.18126i 0.110899 + 0.139063i 3.33086 1.60406i
92.9 −0.807265 + 1.01228i −0.741630 0.357150i 0.0720117 + 0.315504i 0.711905 + 0.342835i 0.960227 0.462421i −0.971640 + 2.46088i −2.71057 1.30534i −1.44801 1.81575i −0.921740 + 0.443887i
92.10 −0.706465 + 0.885879i 1.43197 + 0.689598i 0.159353 + 0.698171i 3.16254 + 1.52300i −1.62253 + 0.781371i 2.63659 + 0.220013i −2.77282 1.33532i −0.295490 0.370533i −3.58341 + 1.72568i
92.11 −0.633585 + 0.794490i 2.34732 + 1.13041i 0.215257 + 0.943102i 2.05977 + 0.991933i −2.38533 + 1.14871i −2.62531 + 0.328254i −2.71678 1.30833i 2.36163 + 2.96138i −2.09312 + 1.00799i
92.12 −0.403219 + 0.505621i 2.54540 + 1.22580i 0.351975 + 1.54210i −3.71879 1.79087i −1.64615 + 0.792742i 2.60419 0.467092i −2.08698 1.00504i 3.10602 + 3.89482i 2.40499 1.15818i
92.13 −0.146099 + 0.183202i 0.315054 + 0.151722i 0.432824 + 1.89632i −1.68254 0.810269i −0.0738247 + 0.0355521i 0.399609 2.61540i −0.832883 0.401095i −1.79423 2.24989i 0.394260 0.189865i
92.14 0.0552404 0.0692692i −2.67810 1.28970i 0.443295 + 1.94220i −3.26519 1.57243i −0.237276 + 0.114266i −0.177815 + 2.63977i 0.318672 + 0.153464i 3.63840 + 4.56241i −0.289292 + 0.139316i
92.15 0.161166 0.202095i −2.41515 1.16307i 0.430174 + 1.88471i 1.29813 + 0.625147i −0.624290 + 0.300642i −2.63945 + 0.182451i 0.916003 + 0.441124i 2.60973 + 3.27250i 0.335553 0.161594i
92.16 0.251865 0.315829i −0.214027 0.103070i 0.408730 + 1.79076i 0.747465 + 0.359960i −0.0864582 + 0.0416361i 1.94492 + 1.79368i 1.39643 + 0.672485i −1.83529 2.30138i 0.301946 0.145409i
92.17 0.258756 0.324470i −2.07294 0.998275i 0.406716 + 1.78194i 2.95478 + 1.42295i −0.860297 + 0.414297i 2.39153 + 1.13164i 1.43125 + 0.689256i 1.43005 + 1.79323i 1.22627 0.590542i
92.18 0.431982 0.541688i 2.32009 + 1.11729i 0.338224 + 1.48186i 1.38659 + 0.667747i 1.60746 0.774112i −2.10133 1.60760i 2.19727 + 1.05815i 2.26399 + 2.83895i 0.960693 0.462646i
92.19 0.505100 0.633375i 0.463406 + 0.223165i 0.299004 + 1.31002i −2.43329 1.17181i 0.375413 0.180790i −2.60368 + 0.469932i 2.44054 + 1.17530i −1.70553 2.13866i −1.97125 + 0.949304i
92.20 0.789572 0.990091i 0.188017 + 0.0905443i 0.0881841 + 0.386360i 3.30397 + 1.59111i 0.238100 0.114663i −0.667783 2.56009i 2.73409 + 1.31667i −1.84332 2.31145i 4.18406 2.01494i
See next 80 embeddings (of 162 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 547.27 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.w.a 162
49.e even 7 1 inner 637.2.w.a 162

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.w.a 162 1.a even 1 1 trivial
637.2.w.a 162 49.e even 7 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$22\!\cdots\!59$$$$T_{2}^{140} -$$$$50\!\cdots\!05$$$$T_{2}^{139} +$$$$16\!\cdots\!85$$$$T_{2}^{138} -$$$$37\!\cdots\!95$$$$T_{2}^{137} +$$$$11\!\cdots\!87$$$$T_{2}^{136} -$$$$25\!\cdots\!53$$$$T_{2}^{135} +$$$$79\!\cdots\!32$$$$T_{2}^{134} -$$$$17\!\cdots\!52$$$$T_{2}^{133} +$$$$51\!\cdots\!92$$$$T_{2}^{132} -$$$$10\!\cdots\!63$$$$T_{2}^{131} +$$$$31\!\cdots\!80$$$$T_{2}^{130} -$$$$66\!\cdots\!71$$$$T_{2}^{129} +$$$$18\!\cdots\!94$$$$T_{2}^{128} -$$$$38\!\cdots\!71$$$$T_{2}^{127} +$$$$10\!\cdots\!39$$$$T_{2}^{126} -$$$$21\!\cdots\!21$$$$T_{2}^{125} +$$$$59\!\cdots\!12$$$$T_{2}^{124} -$$$$11\!\cdots\!85$$$$T_{2}^{123} +$$$$31\!\cdots\!25$$$$T_{2}^{122} -$$$$59\!\cdots\!52$$$$T_{2}^{121} +$$$$15\!\cdots\!55$$$$T_{2}^{120} -$$$$29\!\cdots\!31$$$$T_{2}^{119} +$$$$75\!\cdots\!67$$$$T_{2}^{118} -$$$$13\!\cdots\!15$$$$T_{2}^{117} +$$$$35\!\cdots\!37$$$$T_{2}^{116} -$$$$63\!\cdots\!92$$$$T_{2}^{115} +$$$$15\!\cdots\!97$$$$T_{2}^{114} -$$$$27\!\cdots\!09$$$$T_{2}^{113} +$$$$68\!\cdots\!23$$$$T_{2}^{112} -$$$$11\!\cdots\!52$$$$T_{2}^{111} +$$$$28\!\cdots\!22$$$$T_{2}^{110} -$$$$46\!\cdots\!75$$$$T_{2}^{109} +$$$$11\!\cdots\!65$$$$T_{2}^{108} -$$$$17\!\cdots\!58$$$$T_{2}^{107} +$$$$42\!\cdots\!99$$$$T_{2}^{106} -$$$$66\!\cdots\!51$$$$T_{2}^{105} +$$$$15\!\cdots\!84$$$$T_{2}^{104} -$$$$23\!\cdots\!57$$$$T_{2}^{103} +$$$$53\!\cdots\!26$$$$T_{2}^{102} -$$$$78\!\cdots\!06$$$$T_{2}^{101} +$$$$17\!\cdots\!27$$$$T_{2}^{100} -$$$$25\!\cdots\!60$$$$T_{2}^{99} +$$$$57\!\cdots\!28$$$$T_{2}^{98} -$$$$77\!\cdots\!07$$$$T_{2}^{97} +$$$$17\!\cdots\!91$$$$T_{2}^{96} -$$$$22\!\cdots\!73$$$$T_{2}^{95} +$$$$50\!\cdots\!42$$$$T_{2}^{94} -$$$$63\!\cdots\!86$$$$T_{2}^{93} +$$$$13\!\cdots\!91$$$$T_{2}^{92} -$$$$16\!\cdots\!64$$$$T_{2}^{91} +$$$$36\!\cdots\!67$$$$T_{2}^{90} -$$$$42\!\cdots\!24$$$$T_{2}^{89} +$$$$92\!\cdots\!71$$$$T_{2}^{88} -$$$$10\!\cdots\!36$$$$T_{2}^{87} +$$$$22\!\cdots\!00$$$$T_{2}^{86} -$$$$23\!\cdots\!32$$$$T_{2}^{85} +$$$$50\!\cdots\!98$$$$T_{2}^{84} -$$$$50\!\cdots\!35$$$$T_{2}^{83} +$$$$10\!\cdots\!23$$$$T_{2}^{82} -$$$$10\!\cdots\!14$$$$T_{2}^{81} +$$$$22\!\cdots\!40$$$$T_{2}^{80} -$$$$20\!\cdots\!27$$$$T_{2}^{79} +$$$$43\!\cdots\!95$$$$T_{2}^{78} -$$$$38\!\cdots\!17$$$$T_{2}^{77} +$$$$79\!\cdots\!18$$$$T_{2}^{76} -$$$$69\!\cdots\!78$$$$T_{2}^{75} +$$$$14\!\cdots\!77$$$$T_{2}^{74} -$$$$11\!\cdots\!69$$$$T_{2}^{73} +$$$$23\!\cdots\!47$$$$T_{2}^{72} -$$$$19\!\cdots\!33$$$$T_{2}^{71} +$$$$37\!\cdots\!54$$$$T_{2}^{70} -$$$$30\!\cdots\!13$$$$T_{2}^{69} +$$$$57\!\cdots\!86$$$$T_{2}^{68} -$$$$46\!\cdots\!49$$$$T_{2}^{67} +$$$$82\!\cdots\!34$$$$T_{2}^{66} -$$$$66\!\cdots\!74$$$$T_{2}^{65} +$$$$11\!\cdots\!96$$$$T_{2}^{64} -$$$$92\!\cdots\!12$$$$T_{2}^{63} +$$$$14\!\cdots\!30$$$$T_{2}^{62} -$$$$12\!\cdots\!15$$$$T_{2}^{61} +$$$$18\!\cdots\!51$$$$T_{2}^{60} -$$$$15\!\cdots\!49$$$$T_{2}^{59} +$$$$21\!\cdots\!90$$$$T_{2}^{58} -$$$$18\!\cdots\!35$$$$T_{2}^{57} +$$$$23\!\cdots\!23$$$$T_{2}^{56} -$$$$20\!\cdots\!99$$$$T_{2}^{55} +$$$$25\!\cdots\!98$$$$T_{2}^{54} -$$$$21\!\cdots\!71$$$$T_{2}^{53} +$$$$25\!\cdots\!11$$$$T_{2}^{52} -$$$$21\!\cdots\!54$$$$T_{2}^{51} +$$$$23\!\cdots\!17$$$$T_{2}^{50} -$$$$20\!\cdots\!10$$$$T_{2}^{49} +$$$$21\!\cdots\!44$$$$T_{2}^{48} -$$$$17\!\cdots\!81$$$$T_{2}^{47} +$$$$17\!\cdots\!78$$$$T_{2}^{46} -$$$$14\!\cdots\!87$$$$T_{2}^{45} +$$$$13\!\cdots\!99$$$$T_{2}^{44} -$$$$10\!\cdots\!08$$$$T_{2}^{43} +$$$$91\!\cdots\!07$$$$T_{2}^{42} -$$$$70\!\cdots\!21$$$$T_{2}^{41} +$$$$55\!\cdots\!50$$$$T_{2}^{40} -$$$$39\!\cdots\!57$$$$T_{2}^{39} +$$$$28\!\cdots\!89$$$$T_{2}^{38} -$$$$18\!\cdots\!96$$$$T_{2}^{37} +$$$$12\!\cdots\!13$$$$T_{2}^{36} -$$$$74\!\cdots\!98$$$$T_{2}^{35} +$$$$45\!\cdots\!33$$$$T_{2}^{34} -$$$$26\!\cdots\!61$$$$T_{2}^{33} +$$$$15\!\cdots\!62$$$$T_{2}^{32} -$$$$87\!\cdots\!50$$$$T_{2}^{31} +$$$$49\!\cdots\!34$$$$T_{2}^{30} -$$$$27\!\cdots\!41$$$$T_{2}^{29} +$$$$15\!\cdots\!18$$$$T_{2}^{28} -$$$$80\!\cdots\!79$$$$T_{2}^{27} +$$$$41\!\cdots\!21$$$$T_{2}^{26} -$$$$19\!\cdots\!44$$$$T_{2}^{25} +$$$$89\!\cdots\!07$$$$T_{2}^{24} -$$$$39\!\cdots\!41$$$$T_{2}^{23} +$$$$17\!\cdots\!59$$$$T_{2}^{22} -$$$$81\!\cdots\!40$$$$T_{2}^{21} +$$$$37\!\cdots\!49$$$$T_{2}^{20} -$$$$17\!\cdots\!59$$$$T_{2}^{19} +$$$$72\!\cdots\!23$$$$T_{2}^{18} -$$$$28\!\cdots\!56$$$$T_{2}^{17} +$$$$10\!\cdots\!33$$$$T_{2}^{16} -$$$$32\!\cdots\!25$$$$T_{2}^{15} +$$$$95\!\cdots\!16$$$$T_{2}^{14} -$$$$25\!\cdots\!42$$$$T_{2}^{13} +$$$$64\!\cdots\!82$$$$T_{2}^{12} -$$$$14\!\cdots\!15$$$$T_{2}^{11} +$$$$31\!\cdots\!86$$$$T_{2}^{10} -$$$$60\!\cdots\!55$$$$T_{2}^{9} +$$$$10\!\cdots\!34$$$$T_{2}^{8} -$$$$17\!\cdots\!28$$$$T_{2}^{7} +$$$$24\!\cdots\!77$$$$T_{2}^{6} -$$$$30\!\cdots\!64$$$$T_{2}^{5} +$$$$31\!\cdots\!78$$$$T_{2}^{4} -$$$$28\!\cdots\!43$$$$T_{2}^{3} +$$$$20\!\cdots\!04$$$$T_{2}^{2} -$$$$10\!\cdots\!70$$$$T_{2} + 346328719009$$">$$T_{2}^{162} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$.