Properties

Label 507.2.m.b
Level $507$
Weight $2$
Character orbit 507.m
Analytic conductor $4.048$
Analytic rank $0$
Dimension $204$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.m (of order \(13\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(204\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 204q - q^{2} - 17q^{3} - 21q^{4} - 6q^{5} - q^{6} - 8q^{7} - 9q^{8} - 17q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 204q - q^{2} - 17q^{3} - 21q^{4} - 6q^{5} - q^{6} - 8q^{7} - 9q^{8} - 17q^{9} - 6q^{10} - 8q^{11} - 21q^{12} + 54q^{13} - 30q^{14} - 6q^{15} - 45q^{16} - 18q^{17} - q^{18} - 20q^{19} - 58q^{20} - 8q^{21} + 44q^{22} + 40q^{23} - 9q^{24} + 7q^{25} - 2q^{26} - 17q^{27} - 40q^{28} + 11q^{29} - 6q^{30} + 2q^{31} + 61q^{32} + 5q^{33} - q^{34} + 11q^{35} - 21q^{36} - 34q^{37} + 17q^{38} - 11q^{39} - 31q^{40} - 58q^{41} + 35q^{42} + 32q^{43} - 41q^{44} - 6q^{45} + 76q^{46} - 36q^{47} - 45q^{48} + 9q^{49} - 35q^{50} - 18q^{51} - 24q^{52} + 66q^{53} - q^{54} + 7q^{55} - 114q^{56} - 7q^{57} - 60q^{58} + 40q^{59} + 59q^{60} - 54q^{61} - 31q^{62} - 8q^{63} + 75q^{64} - 26q^{65} + 18q^{66} + 2q^{67} + 26q^{68} - 12q^{69} - 56q^{70} - 37q^{71} - 9q^{72} + 70q^{73} + 174q^{74} - 45q^{75} - 26q^{76} + 24q^{78} - 66q^{79} + 126q^{80} - 17q^{81} - 17q^{82} - 2q^{83} - 40q^{84} + 54q^{85} + 61q^{86} + 24q^{87} + 94q^{88} - 114q^{89} - 6q^{90} + 104q^{91} - 78q^{92} + 67q^{93} - 63q^{94} - 70q^{95} - 4q^{96} + 36q^{97} - 65q^{98} + 18q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1 −2.61147 0.643669i 0.120537 0.992709i 4.63454 + 2.43240i 2.12219 3.07452i −0.953754 + 2.51484i 1.40227 + 1.24231i −6.51088 5.76814i −0.970942 0.239316i −7.52100 + 6.66303i
40.2 −2.55822 0.630545i 0.120537 0.992709i 4.37599 + 2.29670i −1.41367 + 2.04805i −0.934307 + 2.46356i −3.49265 3.09422i −5.80225 5.14035i −0.970942 0.239316i 4.90787 4.34799i
40.3 −2.29214 0.564962i 0.120537 0.992709i 3.16382 + 1.66050i −1.62559 + 2.35507i −0.837130 + 2.20733i 3.18940 + 2.82556i −2.77972 2.46262i −0.970942 0.239316i 5.05660 4.47975i
40.4 −1.94540 0.479497i 0.120537 0.992709i 1.78373 + 0.936175i 1.40868 2.04083i −0.710492 + 1.87341i −0.476086 0.421776i −0.0217150 0.0192378i −0.970942 0.239316i −3.71901 + 3.29475i
40.5 −1.68275 0.414759i 0.120537 0.992709i 0.888694 + 0.466423i 0.797976 1.15607i −0.614568 + 1.62048i −1.62398 1.43872i 1.29250 + 1.14506i −0.970942 0.239316i −1.82228 + 1.61440i
40.6 −1.18717 0.292610i 0.120537 0.992709i −0.447170 0.234693i −0.808035 + 1.17064i −0.433574 + 1.14324i −0.0218939 0.0193963i 2.29259 + 2.03106i −0.970942 0.239316i 1.30181 1.15330i
40.7 −1.09223 0.269211i 0.120537 0.992709i −0.650419 0.341366i −2.21140 + 3.20377i −0.398902 + 1.05182i −0.833262 0.738206i 2.30254 + 2.03987i −0.970942 0.239316i 3.27785 2.90392i
40.8 −0.374061 0.0921977i 0.120537 0.992709i −1.63949 0.860471i 0.205107 0.297149i −0.136614 + 0.360220i −0.762777 0.675762i 1.11067 + 0.983969i −0.970942 0.239316i −0.104119 + 0.0922415i
40.9 0.203567 + 0.0501748i 0.120537 0.992709i −1.73199 0.909018i −0.892774 + 1.29341i 0.0743463 0.196035i 2.37726 + 2.10607i −0.620832 0.550009i −0.970942 0.239316i −0.246636 + 0.218500i
40.10 0.479187 + 0.118109i 0.120537 0.992709i −1.55524 0.816254i 2.38960 3.46193i 0.175008 0.461457i −0.109460 0.0969731i −1.38767 1.22937i −0.970942 0.239316i 1.55395 1.37668i
40.11 0.781588 + 0.192644i 0.120537 0.992709i −1.19714 0.628310i −1.69825 + 2.46034i 0.285450 0.752669i 1.79485 + 1.59010i −2.01970 1.78930i −0.970942 0.239316i −1.80130 + 1.59582i
40.12 1.31905 + 0.325116i 0.120537 0.992709i −0.136731 0.0717618i 0.705120 1.02154i 0.481738 1.27024i −2.12000 1.87816i −2.19076 1.94084i −0.970942 0.239316i 1.26221 1.11822i
40.13 1.37667 + 0.339319i 0.120537 0.992709i 0.00917504 + 0.00481543i −2.23921 + 3.24406i 0.502784 1.32573i −3.33005 2.95017i −2.11159 1.87070i −0.970942 0.239316i −4.18343 + 3.70620i
40.14 1.45113 + 0.357672i 0.120537 0.992709i 0.206949 + 0.108615i 0.974467 1.41176i 0.529980 1.39744i 2.67610 + 2.37082i −1.97593 1.75052i −0.970942 0.239316i 1.91903 1.70011i
40.15 2.11067 + 0.520234i 0.120537 0.992709i 2.41339 + 1.26664i 1.14528 1.65922i 0.770855 2.03258i −0.139189 0.123310i 1.18063 + 1.04595i −0.970942 0.239316i 3.28049 2.90626i
40.16 2.38165 + 0.587024i 0.120537 0.992709i 3.55674 + 1.86672i −1.58170 + 2.29149i 0.869820 2.29353i 2.70346 + 2.39506i 3.70302 + 3.28059i −0.970942 0.239316i −5.11221 + 4.52903i
40.17 2.66897 + 0.657842i 0.120537 0.992709i 4.91973 + 2.58207i 0.113107 0.163863i 0.974754 2.57022i −1.95577 1.73266i 7.31692 + 6.48223i −0.970942 0.239316i 0.409674 0.362939i
79.1 −2.42650 1.27353i −0.970942 0.239316i 3.12991 + 4.53445i −0.0390694 0.103018i 2.05122 + 1.81722i −0.184062 1.51589i −1.15934 9.54803i 0.885456 + 0.464723i −0.0363936 + 0.299728i
79.2 −1.97300 1.03551i −0.970942 0.239316i 1.68431 + 2.44014i 1.49468 + 3.94114i 1.66785 + 1.47759i −0.165780 1.36532i −0.259180 2.13454i 0.885456 + 0.464723i 1.13209 9.32359i
79.3 −1.80313 0.946355i −0.970942 0.239316i 1.21956 + 1.76683i −0.526754 1.38893i 1.52426 + 1.35037i −0.593546 4.88829i −0.0360490 0.296891i 0.885456 + 0.464723i −0.364621 + 3.00293i
See next 80 embeddings (of 204 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 469.17
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.m.b 204
169.g even 13 1 inner 507.2.m.b 204
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.m.b 204 1.a even 1 1 trivial
507.2.m.b 204 169.g even 13 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(22\!\cdots\!01\)\( T_{2}^{180} + \)\(34\!\cdots\!55\)\( T_{2}^{179} + \)\(17\!\cdots\!65\)\( T_{2}^{178} + \)\(26\!\cdots\!68\)\( T_{2}^{177} + \)\(12\!\cdots\!22\)\( T_{2}^{176} + \)\(20\!\cdots\!08\)\( T_{2}^{175} + \)\(93\!\cdots\!47\)\( T_{2}^{174} + \)\(14\!\cdots\!61\)\( T_{2}^{173} + \)\(65\!\cdots\!14\)\( T_{2}^{172} + \)\(10\!\cdots\!15\)\( T_{2}^{171} + \)\(44\!\cdots\!00\)\( T_{2}^{170} + \)\(68\!\cdots\!04\)\( T_{2}^{169} + \)\(29\!\cdots\!17\)\( T_{2}^{168} + \)\(44\!\cdots\!30\)\( T_{2}^{167} + \)\(19\!\cdots\!61\)\( T_{2}^{166} + \)\(27\!\cdots\!35\)\( T_{2}^{165} + \)\(12\!\cdots\!04\)\( T_{2}^{164} + \)\(17\!\cdots\!03\)\( T_{2}^{163} + \)\(76\!\cdots\!58\)\( T_{2}^{162} + \)\(10\!\cdots\!95\)\( T_{2}^{161} + \)\(46\!\cdots\!20\)\( T_{2}^{160} + \)\(61\!\cdots\!33\)\( T_{2}^{159} + \)\(27\!\cdots\!91\)\( T_{2}^{158} + \)\(36\!\cdots\!70\)\( T_{2}^{157} + \)\(15\!\cdots\!89\)\( T_{2}^{156} + \)\(20\!\cdots\!99\)\( T_{2}^{155} + \)\(90\!\cdots\!65\)\( T_{2}^{154} + \)\(11\!\cdots\!81\)\( T_{2}^{153} + \)\(49\!\cdots\!51\)\( T_{2}^{152} + \)\(62\!\cdots\!68\)\( T_{2}^{151} + \)\(26\!\cdots\!94\)\( T_{2}^{150} + \)\(32\!\cdots\!79\)\( T_{2}^{149} + \)\(14\!\cdots\!92\)\( T_{2}^{148} + \)\(16\!\cdots\!96\)\( T_{2}^{147} + \)\(71\!\cdots\!13\)\( T_{2}^{146} + \)\(78\!\cdots\!26\)\( T_{2}^{145} + \)\(35\!\cdots\!21\)\( T_{2}^{144} + \)\(36\!\cdots\!75\)\( T_{2}^{143} + \)\(17\!\cdots\!82\)\( T_{2}^{142} + \)\(16\!\cdots\!65\)\( T_{2}^{141} + \)\(79\!\cdots\!71\)\( T_{2}^{140} + \)\(75\!\cdots\!15\)\( T_{2}^{139} + \)\(36\!\cdots\!71\)\( T_{2}^{138} + \)\(33\!\cdots\!68\)\( T_{2}^{137} + \)\(16\!\cdots\!92\)\( T_{2}^{136} + \)\(14\!\cdots\!24\)\( T_{2}^{135} + \)\(71\!\cdots\!90\)\( T_{2}^{134} + \)\(63\!\cdots\!74\)\( T_{2}^{133} + \)\(30\!\cdots\!31\)\( T_{2}^{132} + \)\(26\!\cdots\!88\)\( T_{2}^{131} + \)\(12\!\cdots\!73\)\( T_{2}^{130} + \)\(10\!\cdots\!03\)\( T_{2}^{129} + \)\(49\!\cdots\!52\)\( T_{2}^{128} + \)\(38\!\cdots\!65\)\( T_{2}^{127} + \)\(18\!\cdots\!55\)\( T_{2}^{126} + \)\(13\!\cdots\!56\)\( T_{2}^{125} + \)\(69\!\cdots\!22\)\( T_{2}^{124} + \)\(40\!\cdots\!22\)\( T_{2}^{123} + \)\(24\!\cdots\!11\)\( T_{2}^{122} + \)\(10\!\cdots\!58\)\( T_{2}^{121} + \)\(82\!\cdots\!74\)\( T_{2}^{120} + \)\(23\!\cdots\!59\)\( T_{2}^{119} + \)\(26\!\cdots\!79\)\( T_{2}^{118} + \)\(27\!\cdots\!16\)\( T_{2}^{117} + \)\(82\!\cdots\!49\)\( T_{2}^{116} - \)\(75\!\cdots\!46\)\( T_{2}^{115} + \)\(24\!\cdots\!60\)\( T_{2}^{114} - \)\(72\!\cdots\!86\)\( T_{2}^{113} + \)\(73\!\cdots\!71\)\( T_{2}^{112} - \)\(34\!\cdots\!41\)\( T_{2}^{111} + \)\(21\!\cdots\!06\)\( T_{2}^{110} - \)\(13\!\cdots\!68\)\( T_{2}^{109} + \)\(60\!\cdots\!74\)\( T_{2}^{108} - \)\(45\!\cdots\!31\)\( T_{2}^{107} + \)\(16\!\cdots\!68\)\( T_{2}^{106} - \)\(14\!\cdots\!27\)\( T_{2}^{105} + \)\(46\!\cdots\!74\)\( T_{2}^{104} - \)\(45\!\cdots\!50\)\( T_{2}^{103} + \)\(12\!\cdots\!56\)\( T_{2}^{102} - \)\(13\!\cdots\!33\)\( T_{2}^{101} + \)\(33\!\cdots\!72\)\( T_{2}^{100} - \)\(37\!\cdots\!17\)\( T_{2}^{99} + \)\(87\!\cdots\!70\)\( T_{2}^{98} - \)\(10\!\cdots\!28\)\( T_{2}^{97} + \)\(22\!\cdots\!51\)\( T_{2}^{96} - \)\(28\!\cdots\!34\)\( T_{2}^{95} + \)\(55\!\cdots\!78\)\( T_{2}^{94} - \)\(69\!\cdots\!13\)\( T_{2}^{93} + \)\(12\!\cdots\!89\)\( T_{2}^{92} - \)\(15\!\cdots\!77\)\( T_{2}^{91} + \)\(27\!\cdots\!02\)\( T_{2}^{90} - \)\(32\!\cdots\!13\)\( T_{2}^{89} + \)\(51\!\cdots\!37\)\( T_{2}^{88} - \)\(53\!\cdots\!46\)\( T_{2}^{87} + \)\(81\!\cdots\!89\)\( T_{2}^{86} - \)\(71\!\cdots\!02\)\( T_{2}^{85} + \)\(10\!\cdots\!68\)\( T_{2}^{84} - \)\(75\!\cdots\!78\)\( T_{2}^{83} + \)\(13\!\cdots\!84\)\( T_{2}^{82} - \)\(85\!\cdots\!31\)\( T_{2}^{81} + \)\(23\!\cdots\!85\)\( T_{2}^{80} - \)\(15\!\cdots\!27\)\( T_{2}^{79} + \)\(46\!\cdots\!34\)\( T_{2}^{78} - \)\(36\!\cdots\!59\)\( T_{2}^{77} + \)\(85\!\cdots\!34\)\( T_{2}^{76} - \)\(63\!\cdots\!06\)\( T_{2}^{75} + \)\(13\!\cdots\!04\)\( T_{2}^{74} - \)\(82\!\cdots\!47\)\( T_{2}^{73} + \)\(18\!\cdots\!31\)\( T_{2}^{72} - \)\(71\!\cdots\!96\)\( T_{2}^{71} + \)\(20\!\cdots\!51\)\( T_{2}^{70} - \)\(32\!\cdots\!77\)\( T_{2}^{69} + \)\(23\!\cdots\!62\)\( T_{2}^{68} + \)\(56\!\cdots\!11\)\( T_{2}^{67} + \)\(34\!\cdots\!16\)\( T_{2}^{66} + \)\(57\!\cdots\!87\)\( T_{2}^{65} + \)\(56\!\cdots\!16\)\( T_{2}^{64} + \)\(48\!\cdots\!86\)\( T_{2}^{63} + \)\(78\!\cdots\!43\)\( T_{2}^{62} + \)\(11\!\cdots\!31\)\( T_{2}^{61} + \)\(92\!\cdots\!62\)\( T_{2}^{60} + \)\(21\!\cdots\!09\)\( T_{2}^{59} + \)\(10\!\cdots\!20\)\( T_{2}^{58} + \)\(29\!\cdots\!79\)\( T_{2}^{57} + \)\(10\!\cdots\!50\)\( T_{2}^{56} + \)\(31\!\cdots\!22\)\( T_{2}^{55} + \)\(96\!\cdots\!83\)\( T_{2}^{54} + \)\(27\!\cdots\!31\)\( T_{2}^{53} + \)\(75\!\cdots\!05\)\( T_{2}^{52} + \)\(19\!\cdots\!48\)\( T_{2}^{51} + \)\(49\!\cdots\!81\)\( T_{2}^{50} + \)\(90\!\cdots\!19\)\( T_{2}^{49} + \)\(27\!\cdots\!71\)\( T_{2}^{48} + \)\(89\!\cdots\!58\)\( T_{2}^{47} + \)\(13\!\cdots\!22\)\( T_{2}^{46} - \)\(90\!\cdots\!92\)\( T_{2}^{45} + \)\(71\!\cdots\!86\)\( T_{2}^{44} + \)\(34\!\cdots\!77\)\( T_{2}^{43} + \)\(54\!\cdots\!28\)\( T_{2}^{42} + \)\(22\!\cdots\!24\)\( T_{2}^{41} + \)\(44\!\cdots\!76\)\( T_{2}^{40} + \)\(26\!\cdots\!54\)\( T_{2}^{39} + \)\(29\!\cdots\!52\)\( T_{2}^{38} + \)\(15\!\cdots\!17\)\( T_{2}^{37} + \)\(13\!\cdots\!74\)\( T_{2}^{36} + \)\(59\!\cdots\!06\)\( T_{2}^{35} + \)\(37\!\cdots\!50\)\( T_{2}^{34} + \)\(10\!\cdots\!08\)\( T_{2}^{33} + \)\(52\!\cdots\!73\)\( T_{2}^{32} - \)\(55\!\cdots\!05\)\( T_{2}^{31} - \)\(38\!\cdots\!36\)\( T_{2}^{30} - \)\(61\!\cdots\!38\)\( T_{2}^{29} - \)\(17\!\cdots\!98\)\( T_{2}^{28} - \)\(23\!\cdots\!19\)\( T_{2}^{27} + \)\(66\!\cdots\!76\)\( T_{2}^{26} + \)\(60\!\cdots\!04\)\( T_{2}^{25} + \)\(36\!\cdots\!80\)\( T_{2}^{24} + \)\(16\!\cdots\!95\)\( T_{2}^{23} + \)\(60\!\cdots\!48\)\( T_{2}^{22} + \)\(15\!\cdots\!17\)\( T_{2}^{21} + \)\(27\!\cdots\!38\)\( T_{2}^{20} + \)\(83\!\cdots\!33\)\( T_{2}^{19} + \)\(10\!\cdots\!36\)\( T_{2}^{18} + \)\(11\!\cdots\!88\)\( T_{2}^{17} + \)\(78\!\cdots\!96\)\( T_{2}^{16} + \)\(24\!\cdots\!74\)\( T_{2}^{15} + \)\(46\!\cdots\!61\)\( T_{2}^{14} - \)\(12\!\cdots\!87\)\( T_{2}^{13} - \)\(18\!\cdots\!14\)\( T_{2}^{12} - \)\(56\!\cdots\!54\)\( T_{2}^{11} - \)\(66\!\cdots\!45\)\( T_{2}^{10} + \)\(16\!\cdots\!72\)\( T_{2}^{9} + \)\(30\!\cdots\!46\)\( T_{2}^{8} + \)\(76\!\cdots\!67\)\( T_{2}^{7} + \)\(12\!\cdots\!41\)\( T_{2}^{6} + \)\(14\!\cdots\!11\)\( T_{2}^{5} + \)\(12\!\cdots\!75\)\( T_{2}^{4} + \)\(67\!\cdots\!69\)\( T_{2}^{3} + \)\(23\!\cdots\!69\)\( T_{2}^{2} + \)\(48\!\cdots\!44\)\( T_{2} + \)\(43\!\cdots\!89\)\( \)">\(T_{2}^{204} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\).