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 $2$

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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: \(174\)
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.

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

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:

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 \(S_{2}^{\mathrm{new}}(637, [\chi])\).