# Properties

 Label 637.2.w.b Level $637$ Weight $2$ Character orbit 637.w Analytic conductor $5.086$ Analytic rank $0$ Dimension $174$ CM no Inner twists $2$

# Related objects

## 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: $$174$$ Relative dimension: $$29$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$174 q - 3 q^{2} - 31 q^{4} - 4 q^{5} - 2 q^{6} + 9 q^{7} - 15 q^{8} - 31 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$174 q - 3 q^{2} - 31 q^{4} - 4 q^{5} - 2 q^{6} + 9 q^{7} - 15 q^{8} - 31 q^{9} - 10 q^{10} - 5 q^{11} + 25 q^{12} - 29 q^{13} + 15 q^{14} - 10 q^{15} - 51 q^{16} - 9 q^{17} + 44 q^{18} + 24 q^{19} + 63 q^{20} - 28 q^{21} - 8 q^{22} - 13 q^{23} - 48 q^{24} - 49 q^{25} - 3 q^{26} - 9 q^{27} - 44 q^{28} + 2 q^{29} - 22 q^{30} + 10 q^{31} + 24 q^{32} - 26 q^{33} + 118 q^{34} + 5 q^{35} - 55 q^{36} - 32 q^{37} + 16 q^{38} + 42 q^{40} - 14 q^{41} + 4 q^{42} - 50 q^{43} + 35 q^{44} - q^{45} + 4 q^{46} - 24 q^{47} - 116 q^{48} - 25 q^{49} + 156 q^{50} + 12 q^{51} - 31 q^{52} - 30 q^{53} - 78 q^{54} + 25 q^{55} + 3 q^{56} - 63 q^{57} - 12 q^{58} - 4 q^{59} + 128 q^{60} - 42 q^{61} - 38 q^{62} - 85 q^{63} - 105 q^{64} - 4 q^{65} + 15 q^{66} + 94 q^{67} + 214 q^{68} + 32 q^{69} - 57 q^{70} - 29 q^{71} - 64 q^{72} - 66 q^{73} - 90 q^{74} + 131 q^{75} - 21 q^{76} - 82 q^{77} + 19 q^{78} + 6 q^{79} + 22 q^{80} + 49 q^{81} - 50 q^{82} + 25 q^{83} + 89 q^{84} - 86 q^{85} - 28 q^{86} + 24 q^{87} + 48 q^{88} - 50 q^{89} - 155 q^{90} - 5 q^{91} - 98 q^{92} + 89 q^{93} - 28 q^{94} - 130 q^{95} - 105 q^{96} - 42 q^{97} + 195 q^{98} + 438 q^{99} + O(q^{100})$$

## 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 −1.68715 + 2.11563i −0.820369 0.395069i −1.18434 5.18892i −0.379531 0.182773i 2.21991 1.06905i −1.63169 2.08269i 8.09995 + 3.90073i −1.35354 1.69729i 1.02701 0.494580i
92.2 −1.57084 + 1.96978i 2.34650 + 1.13001i −0.967425 4.23856i 3.79256 + 1.82640i −5.91185 + 2.84700i 0.107940 2.64355i 5.32882 + 2.56623i 2.35865 + 2.95765i −9.55512 + 4.60150i
92.3 −1.44058 + 1.80643i −0.133830 0.0644489i −0.742877 3.25476i −3.89600 1.87621i 0.309215 0.148910i −1.21227 + 2.35168i 2.78627 + 1.34180i −1.85671 2.32824i 9.00174 4.33501i
92.4 −1.42465 + 1.78645i −2.04852 0.986513i −0.716746 3.14027i 0.342127 + 0.164760i 4.68077 2.25414i 2.57961 0.587895i 2.51370 + 1.21053i 1.35274 + 1.69628i −0.781746 + 0.376469i
92.5 −1.18201 + 1.48220i −0.0940579 0.0452959i −0.354712 1.55409i 1.30132 + 0.626680i 0.178315 0.0858720i −2.43875 + 1.02591i −0.693369 0.333909i −1.86367 2.33697i −2.46703 + 1.18806i
92.6 −1.14796 + 1.43949i 1.04469 + 0.503095i −0.309290 1.35509i −0.563090 0.271170i −1.92345 + 0.926286i 1.79557 1.94318i −1.01200 0.487355i −1.03220 1.29434i 1.03675 0.499271i
92.7 −1.12298 + 1.40817i −2.74049 1.31975i −0.276823 1.21284i −2.53396 1.22029i 4.93595 2.37703i −1.05285 2.42724i −1.22675 0.590772i 3.89808 + 4.88803i 4.56397 2.19789i
92.8 −1.10143 + 1.38115i 3.05365 + 1.47056i −0.249384 1.09262i 0.775588 + 0.373504i −5.39445 + 2.59783i 0.705731 + 2.54989i −1.39947 0.673949i 5.29177 + 6.63567i −1.37012 + 0.659815i
92.9 −0.807700 + 1.01282i −2.74330 1.32111i 0.0716088 + 0.313739i 2.02464 + 0.975017i 3.55382 1.71143i 0.327305 + 2.62543i −2.70992 1.30503i 3.90993 + 4.90289i −2.62283 + 1.26309i
92.10 −0.487175 + 0.610898i 1.39678 + 0.672652i 0.309185 + 1.35463i −2.44805 1.17892i −1.09140 + 0.525588i −2.52492 0.790430i −2.38614 1.14911i −0.371947 0.466407i 1.91283 0.921169i
92.11 −0.440322 + 0.552147i −1.61690 0.778657i 0.334060 + 1.46361i 2.37681 + 1.14461i 1.14189 0.549905i 2.31524 1.28050i −2.22779 1.07285i 0.137583 + 0.172524i −1.67856 + 0.808350i
92.12 −0.347630 + 0.435914i −0.809113 0.389648i 0.375867 + 1.64678i −1.77186 0.853282i 0.451125 0.217250i 2.01500 + 1.71458i −1.85320 0.892452i −1.36763 1.71495i 0.987907 0.475751i
92.13 −0.165915 + 0.208051i 2.56994 + 1.23762i 0.429285 + 1.88082i 0.602735 + 0.290262i −0.683878 + 0.329338i 1.20359 2.35614i −0.942039 0.453662i 3.20241 + 4.01570i −0.160392 + 0.0772407i
92.14 −0.118827 + 0.149004i 1.11947 + 0.539107i 0.436959 + 1.91444i 0.156007 + 0.0751292i −0.213352 + 0.102745i 0.0412747 + 2.64543i −0.680602 0.327760i −0.907899 1.13847i −0.0297324 + 0.0143184i
92.15 0.153433 0.192399i −1.67003 0.804242i 0.431566 + 1.89082i 0.217813 + 0.104893i −0.410972 + 0.197914i −2.59813 0.499745i 0.873441 + 0.420627i 0.271713 + 0.340717i 0.0536009 0.0258129i
92.16 0.252803 0.317004i 1.07472 + 0.517556i 0.408459 + 1.78958i 3.73621 + 1.79926i 0.435758 0.209850i −1.42709 + 2.22787i 1.40118 + 0.674774i −0.983320 1.23304i 1.51490 0.729536i
92.17 0.311250 0.390295i −1.89343 0.911826i 0.389588 + 1.70690i −2.41457 1.16280i −0.945210 + 0.455189i 0.718518 2.54632i 1.68699 + 0.812413i 0.883165 + 1.10745i −1.20537 + 0.580476i
92.18 0.529450 0.663910i 2.82603 + 1.36094i 0.284583 + 1.24684i −2.32207 1.11825i 2.39979 1.15568i −1.55674 + 2.13929i 2.50862 + 1.20809i 4.26380 + 5.34664i −1.97184 + 0.949586i
92.19 0.708170 0.888017i 0.627836 + 0.302350i 0.157972 + 0.692122i −3.97243 1.91302i 0.713107 0.343414i 2.60413 + 0.467470i 2.77316 + 1.33548i −1.56771 1.96584i −4.51195 + 2.17284i
92.20 0.743539 0.932368i 1.30142 + 0.626732i 0.128582 + 0.563353i 0.955511 + 0.460150i 1.55200 0.747405i 2.35959 1.19679i 2.76975 + 1.33384i −0.569562 0.714208i 1.13949 0.548748i
See next 80 embeddings (of 174 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 547.29 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.b 174
49.e even 7 1 inner 637.2.w.b 174

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.w.b 174 1.a even 1 1 trivial
637.2.w.b 174 49.e even 7 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$18\!\cdots\!75$$$$T_{2}^{153} +$$$$69\!\cdots\!68$$$$T_{2}^{152} +$$$$15\!\cdots\!79$$$$T_{2}^{151} +$$$$57\!\cdots\!41$$$$T_{2}^{150} +$$$$12\!\cdots\!13$$$$T_{2}^{149} +$$$$45\!\cdots\!62$$$$T_{2}^{148} +$$$$93\!\cdots\!50$$$$T_{2}^{147} +$$$$33\!\cdots\!26$$$$T_{2}^{146} +$$$$68\!\cdots\!83$$$$T_{2}^{145} +$$$$24\!\cdots\!23$$$$T_{2}^{144} +$$$$47\!\cdots\!16$$$$T_{2}^{143} +$$$$16\!\cdots\!89$$$$T_{2}^{142} +$$$$31\!\cdots\!59$$$$T_{2}^{141} +$$$$11\!\cdots\!37$$$$T_{2}^{140} +$$$$20\!\cdots\!36$$$$T_{2}^{139} +$$$$70\!\cdots\!69$$$$T_{2}^{138} +$$$$12\!\cdots\!40$$$$T_{2}^{137} +$$$$43\!\cdots\!06$$$$T_{2}^{136} +$$$$74\!\cdots\!76$$$$T_{2}^{135} +$$$$25\!\cdots\!92$$$$T_{2}^{134} +$$$$42\!\cdots\!35$$$$T_{2}^{133} +$$$$14\!\cdots\!43$$$$T_{2}^{132} +$$$$23\!\cdots\!12$$$$T_{2}^{131} +$$$$78\!\cdots\!40$$$$T_{2}^{130} +$$$$12\!\cdots\!70$$$$T_{2}^{129} +$$$$41\!\cdots\!59$$$$T_{2}^{128} +$$$$61\!\cdots\!91$$$$T_{2}^{127} +$$$$20\!\cdots\!04$$$$T_{2}^{126} +$$$$29\!\cdots\!21$$$$T_{2}^{125} +$$$$10\!\cdots\!24$$$$T_{2}^{124} +$$$$13\!\cdots\!48$$$$T_{2}^{123} +$$$$47\!\cdots\!50$$$$T_{2}^{122} +$$$$61\!\cdots\!72$$$$T_{2}^{121} +$$$$21\!\cdots\!60$$$$T_{2}^{120} +$$$$26\!\cdots\!82$$$$T_{2}^{119} +$$$$92\!\cdots\!98$$$$T_{2}^{118} +$$$$10\!\cdots\!82$$$$T_{2}^{117} +$$$$38\!\cdots\!45$$$$T_{2}^{116} +$$$$43\!\cdots\!66$$$$T_{2}^{115} +$$$$15\!\cdots\!48$$$$T_{2}^{114} +$$$$16\!\cdots\!24$$$$T_{2}^{113} +$$$$60\!\cdots\!44$$$$T_{2}^{112} +$$$$60\!\cdots\!21$$$$T_{2}^{111} +$$$$22\!\cdots\!95$$$$T_{2}^{110} +$$$$21\!\cdots\!76$$$$T_{2}^{109} +$$$$80\!\cdots\!88$$$$T_{2}^{108} +$$$$73\!\cdots\!08$$$$T_{2}^{107} +$$$$27\!\cdots\!71$$$$T_{2}^{106} +$$$$24\!\cdots\!48$$$$T_{2}^{105} +$$$$92\!\cdots\!53$$$$T_{2}^{104} +$$$$77\!\cdots\!89$$$$T_{2}^{103} +$$$$29\!\cdots\!30$$$$T_{2}^{102} +$$$$24\!\cdots\!66$$$$T_{2}^{101} +$$$$91\!\cdots\!79$$$$T_{2}^{100} +$$$$72\!\cdots\!56$$$$T_{2}^{99} +$$$$27\!\cdots\!40$$$$T_{2}^{98} +$$$$21\!\cdots\!22$$$$T_{2}^{97} +$$$$77\!\cdots\!86$$$$T_{2}^{96} +$$$$60\!\cdots\!38$$$$T_{2}^{95} +$$$$21\!\cdots\!74$$$$T_{2}^{94} +$$$$16\!\cdots\!45$$$$T_{2}^{93} +$$$$56\!\cdots\!86$$$$T_{2}^{92} +$$$$44\!\cdots\!58$$$$T_{2}^{91} +$$$$14\!\cdots\!82$$$$T_{2}^{90} +$$$$11\!\cdots\!77$$$$T_{2}^{89} +$$$$34\!\cdots\!85$$$$T_{2}^{88} +$$$$28\!\cdots\!53$$$$T_{2}^{87} +$$$$81\!\cdots\!23$$$$T_{2}^{86} +$$$$68\!\cdots\!58$$$$T_{2}^{85} +$$$$18\!\cdots\!80$$$$T_{2}^{84} +$$$$15\!\cdots\!29$$$$T_{2}^{83} +$$$$39\!\cdots\!60$$$$T_{2}^{82} +$$$$35\!\cdots\!87$$$$T_{2}^{81} +$$$$81\!\cdots\!73$$$$T_{2}^{80} +$$$$74\!\cdots\!10$$$$T_{2}^{79} +$$$$16\!\cdots\!73$$$$T_{2}^{78} +$$$$15\!\cdots\!85$$$$T_{2}^{77} +$$$$30\!\cdots\!68$$$$T_{2}^{76} +$$$$29\!\cdots\!04$$$$T_{2}^{75} +$$$$55\!\cdots\!44$$$$T_{2}^{74} +$$$$53\!\cdots\!95$$$$T_{2}^{73} +$$$$94\!\cdots\!26$$$$T_{2}^{72} +$$$$91\!\cdots\!49$$$$T_{2}^{71} +$$$$15\!\cdots\!05$$$$T_{2}^{70} +$$$$14\!\cdots\!55$$$$T_{2}^{69} +$$$$23\!\cdots\!28$$$$T_{2}^{68} +$$$$22\!\cdots\!67$$$$T_{2}^{67} +$$$$32\!\cdots\!20$$$$T_{2}^{66} +$$$$30\!\cdots\!77$$$$T_{2}^{65} +$$$$42\!\cdots\!01$$$$T_{2}^{64} +$$$$39\!\cdots\!33$$$$T_{2}^{63} +$$$$51\!\cdots\!70$$$$T_{2}^{62} +$$$$44\!\cdots\!03$$$$T_{2}^{61} +$$$$55\!\cdots\!08$$$$T_{2}^{60} +$$$$44\!\cdots\!41$$$$T_{2}^{59} +$$$$53\!\cdots\!17$$$$T_{2}^{58} +$$$$40\!\cdots\!73$$$$T_{2}^{57} +$$$$46\!\cdots\!06$$$$T_{2}^{56} +$$$$30\!\cdots\!20$$$$T_{2}^{55} +$$$$35\!\cdots\!12$$$$T_{2}^{54} +$$$$20\!\cdots\!26$$$$T_{2}^{53} +$$$$24\!\cdots\!46$$$$T_{2}^{52} +$$$$11\!\cdots\!18$$$$T_{2}^{51} +$$$$15\!\cdots\!96$$$$T_{2}^{50} +$$$$56\!\cdots\!32$$$$T_{2}^{49} +$$$$97\!\cdots\!52$$$$T_{2}^{48} +$$$$22\!\cdots\!22$$$$T_{2}^{47} +$$$$58\!\cdots\!37$$$$T_{2}^{46} +$$$$46\!\cdots\!04$$$$T_{2}^{45} +$$$$33\!\cdots\!03$$$$T_{2}^{44} -$$$$27\!\cdots\!95$$$$T_{2}^{43} +$$$$19\!\cdots\!22$$$$T_{2}^{42} -$$$$46\!\cdots\!02$$$$T_{2}^{41} +$$$$10\!\cdots\!63$$$$T_{2}^{40} -$$$$40\!\cdots\!96$$$$T_{2}^{39} +$$$$56\!\cdots\!32$$$$T_{2}^{38} -$$$$27\!\cdots\!99$$$$T_{2}^{37} +$$$$29\!\cdots\!40$$$$T_{2}^{36} -$$$$16\!\cdots\!36$$$$T_{2}^{35} +$$$$14\!\cdots\!54$$$$T_{2}^{34} -$$$$88\!\cdots\!84$$$$T_{2}^{33} +$$$$68\!\cdots\!90$$$$T_{2}^{32} -$$$$41\!\cdots\!18$$$$T_{2}^{31} +$$$$28\!\cdots\!35$$$$T_{2}^{30} -$$$$16\!\cdots\!80$$$$T_{2}^{29} +$$$$10\!\cdots\!78$$$$T_{2}^{28} -$$$$56\!\cdots\!62$$$$T_{2}^{27} +$$$$30\!\cdots\!94$$$$T_{2}^{26} -$$$$15\!\cdots\!73$$$$T_{2}^{25} +$$$$73\!\cdots\!65$$$$T_{2}^{24} -$$$$32\!\cdots\!54$$$$T_{2}^{23} +$$$$13\!\cdots\!90$$$$T_{2}^{22} -$$$$55\!\cdots\!70$$$$T_{2}^{21} +$$$$20\!\cdots\!91$$$$T_{2}^{20} -$$$$71\!\cdots\!96$$$$T_{2}^{19} +$$$$25\!\cdots\!79$$$$T_{2}^{18} -$$$$98\!\cdots\!35$$$$T_{2}^{17} +$$$$38\!\cdots\!62$$$$T_{2}^{16} -$$$$14\!\cdots\!40$$$$T_{2}^{15} +$$$$48\!\cdots\!91$$$$T_{2}^{14} -$$$$14\!\cdots\!75$$$$T_{2}^{13} +$$$$43\!\cdots\!76$$$$T_{2}^{12} -$$$$12\!\cdots\!55$$$$T_{2}^{11} +$$$$30\!\cdots\!34$$$$T_{2}^{10} -$$$$70\!\cdots\!44$$$$T_{2}^{9} +$$$$14\!\cdots\!30$$$$T_{2}^{8} -$$$$28\!\cdots\!03$$$$T_{2}^{7} +$$$$51\!\cdots\!72$$$$T_{2}^{6} -$$$$82\!\cdots\!07$$$$T_{2}^{5} +$$$$12\!\cdots\!83$$$$T_{2}^{4} -$$$$16\!\cdots\!47$$$$T_{2}^{3} +$$$$17\!\cdots\!13$$$$T_{2}^{2} -$$$$14\!\cdots\!04$$$$T_{2} +$$$$84\!\cdots\!81$$">$$T_{2}^{174} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$.