Properties

Label 507.2.p.a
Level $507$
Weight $2$
Character orbit 507.p
Analytic conductor $4.048$
Analytic rank $0$
Dimension $168$
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.p (of order \(26\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(14\) over \(\Q(\zeta_{26})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{26}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 168q - 14q^{3} + 12q^{4} - 14q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 168q - 14q^{3} + 12q^{4} - 14q^{9} - 4q^{10} + 12q^{12} + 13q^{13} + 2q^{14} - 8q^{16} - 4q^{17} - 72q^{22} + 48q^{23} - 44q^{25} - 39q^{26} - 14q^{27} + 45q^{29} - 4q^{30} - 26q^{31} + 130q^{32} + 13q^{33} - 65q^{34} - 35q^{35} + 12q^{36} + 61q^{38} + 12q^{40} - 63q^{42} + 72q^{43} - 39q^{44} - 8q^{48} - 68q^{49} - 52q^{50} - 4q^{51} + 65q^{52} - q^{53} + 53q^{55} - 14q^{56} - 13q^{57} - 26q^{58} - 104q^{59} + 117q^{60} + 12q^{61} + 49q^{62} - 32q^{64} - 52q^{65} - 46q^{66} + 26q^{67} - 84q^{68} - 4q^{69} - 39q^{71} - 52q^{73} + 29q^{74} + 8q^{75} - 130q^{76} + 60q^{77} + 65q^{78} + 14q^{79} - 14q^{81} + 45q^{82} + 78q^{83} - 13q^{85} - 13q^{86} - 46q^{87} - 26q^{88} - 4q^{90} - 208q^{91} + 82q^{92} - 39q^{93} + 25q^{94} - 66q^{95} + 65q^{96} + 26q^{97} - 104q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −2.41243 0.914913i 0.568065 + 0.822984i 3.48572 + 3.08808i −3.08975 0.375164i −0.617456 2.50512i −0.149094 0.284076i −3.18567 6.06979i −0.354605 + 0.935016i 7.11057 + 3.73191i
25.2 −2.23311 0.846906i 0.568065 + 0.822984i 2.77250 + 2.45622i 0.980994 + 0.119114i −0.571560 2.31891i 1.47491 + 2.81021i −1.89129 3.60355i −0.354605 + 0.935016i −2.08979 1.09680i
25.3 −1.83220 0.694863i 0.568065 + 0.822984i 1.37712 + 1.22002i −0.128032 0.0155459i −0.468949 1.90260i −2.04557 3.89750i 0.145878 + 0.277948i −0.354605 + 0.935016i 0.223779 + 0.117448i
25.4 −1.24255 0.471239i 0.568065 + 0.822984i −0.175147 0.155167i −1.73849 0.211091i −0.318029 1.29030i −0.0913225 0.174001i 1.37966 + 2.62872i −0.354605 + 0.935016i 2.06070 + 1.08154i
25.5 −0.911757 0.345784i 0.568065 + 0.822984i −0.785288 0.695704i 2.27932 + 0.276760i −0.233362 0.946788i 1.78194 + 3.39521i 1.38175 + 2.63271i −0.354605 + 0.935016i −1.98249 1.04049i
25.6 −0.642861 0.243805i 0.568065 + 0.822984i −1.14319 1.01278i 4.00656 + 0.486484i −0.164539 0.667561i −0.644348 1.22770i 1.12702 + 2.14737i −0.354605 + 0.935016i −2.45705 1.28956i
25.7 −0.239472 0.0908198i 0.568065 + 0.822984i −1.44792 1.28275i −3.85869 0.468530i −0.0612925 0.248673i 0.285060 + 0.543137i 0.468284 + 0.892241i −0.354605 + 0.935016i 0.881497 + 0.462645i
25.8 0.00155211 0.000588639i 0.568065 + 0.822984i −1.49702 1.32624i −0.897149 0.108934i 0.000397260 0.00161175i −0.178306 0.339734i −0.00308573 0.00587937i −0.354605 + 0.935016i −0.00132835 0.000697174i
25.9 0.725144 + 0.275011i 0.568065 + 0.822984i −1.04682 0.927401i 1.67905 + 0.203873i 0.185599 + 0.753005i −1.40603 2.67897i −1.22487 2.33380i −0.354605 + 0.935016i 1.16148 + 0.609593i
25.10 0.759337 + 0.287979i 0.568065 + 0.822984i −1.00336 0.888900i 0.883771 + 0.107309i 0.194351 + 0.788513i 2.17066 + 4.13585i −1.26072 2.40210i −0.354605 + 0.935016i 0.640177 + 0.335991i
25.11 1.35653 + 0.514462i 0.568065 + 0.822984i 0.0784691 + 0.0695176i −3.71157 0.450666i 0.347200 + 1.40865i −1.65187 3.14737i −1.27776 2.43458i −0.354605 + 0.935016i −4.80299 2.52080i
25.12 1.96188 + 0.744043i 0.568065 + 0.822984i 1.79835 + 1.59320i 1.22751 + 0.149046i 0.502139 + 2.03726i −0.253099 0.482239i 0.392545 + 0.747932i −0.354605 + 0.935016i 2.29732 + 1.20573i
25.13 2.15817 + 0.818487i 0.568065 + 0.822984i 2.49077 + 2.20663i −1.10490 0.134159i 0.552380 + 2.24110i 1.27631 + 2.43180i 1.42410 + 2.71339i −0.354605 + 0.935016i −2.27476 1.19389i
25.14 2.55177 + 0.967759i 0.568065 + 0.822984i 4.07796 + 3.61275i 1.13985 + 0.138403i 0.653121 + 2.64982i −2.09908 3.99947i 4.37317 + 8.33237i −0.354605 + 0.935016i 2.77471 + 1.45628i
64.1 −2.32286 0.282046i −0.748511 + 0.663123i 3.37425 + 0.831678i −0.0918625 0.175029i 1.92572 1.32923i −0.0642964 0.0243844i −3.22762 1.22407i 0.120537 0.992709i 0.164018 + 0.432479i
64.2 −2.29315 0.278439i −0.748511 + 0.663123i 3.23914 + 0.798376i 1.23177 + 2.34695i 1.90109 1.31223i −4.14543 1.57215i −2.88578 1.09443i 0.120537 0.992709i −2.17116 5.72488i
64.3 −2.27015 0.275646i −0.748511 + 0.663123i 3.13571 + 0.772883i −1.62219 3.09083i 1.88202 1.29906i 2.36774 + 0.897966i −2.62905 0.997067i 0.120537 0.992709i 2.83064 + 7.46378i
64.4 −1.39250 0.169080i −0.748511 + 0.663123i −0.0314082 0.00774143i −1.23718 2.35725i 1.15442 0.796842i −2.90606 1.10212i 2.66558 + 1.01092i 0.120537 0.992709i 1.32421 + 3.49165i
64.5 −1.16970 0.142027i −0.748511 + 0.663123i −0.593855 0.146372i 1.79511 + 3.42030i 0.969716 0.669346i 2.92200 + 1.10817i 2.87729 + 1.09121i 0.120537 0.992709i −1.61397 4.25568i
64.6 −0.915602 0.111174i −0.748511 + 0.663123i −1.11592 0.275048i 0.391488 + 0.745917i 0.759060 0.523942i −1.12887 0.428124i 2.71594 + 1.03002i 0.120537 0.992709i −0.275520 0.726487i
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 493.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.h even 26 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.p.a 168
169.h even 26 1 inner 507.2.p.a 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.p.a 168 1.a even 1 1 trivial
507.2.p.a 168 169.h even 26 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(13\!\cdots\!75\)\( T_{2}^{142} + 792267957273 T_{2}^{141} + \)\(78\!\cdots\!03\)\( T_{2}^{140} - \)\(45\!\cdots\!43\)\( T_{2}^{139} - \)\(44\!\cdots\!42\)\( T_{2}^{138} + \)\(25\!\cdots\!11\)\( T_{2}^{137} + \)\(23\!\cdots\!01\)\( T_{2}^{136} - \)\(13\!\cdots\!41\)\( T_{2}^{135} - \)\(12\!\cdots\!20\)\( T_{2}^{134} + \)\(74\!\cdots\!80\)\( T_{2}^{133} + \)\(63\!\cdots\!70\)\( T_{2}^{132} - \)\(39\!\cdots\!64\)\( T_{2}^{131} - \)\(31\!\cdots\!30\)\( T_{2}^{130} + \)\(19\!\cdots\!71\)\( T_{2}^{129} + \)\(15\!\cdots\!03\)\( T_{2}^{128} - \)\(86\!\cdots\!69\)\( T_{2}^{127} - \)\(76\!\cdots\!71\)\( T_{2}^{126} + \)\(36\!\cdots\!67\)\( T_{2}^{125} + \)\(36\!\cdots\!52\)\( T_{2}^{124} - \)\(14\!\cdots\!24\)\( T_{2}^{123} - \)\(16\!\cdots\!66\)\( T_{2}^{122} + \)\(58\!\cdots\!70\)\( T_{2}^{121} + \)\(76\!\cdots\!64\)\( T_{2}^{120} - \)\(24\!\cdots\!91\)\( T_{2}^{119} - \)\(33\!\cdots\!48\)\( T_{2}^{118} + \)\(10\!\cdots\!26\)\( T_{2}^{117} + \)\(13\!\cdots\!82\)\( T_{2}^{116} - \)\(30\!\cdots\!79\)\( T_{2}^{115} - \)\(56\!\cdots\!25\)\( T_{2}^{114} + \)\(89\!\cdots\!22\)\( T_{2}^{113} + \)\(23\!\cdots\!93\)\( T_{2}^{112} - \)\(26\!\cdots\!09\)\( T_{2}^{111} - \)\(93\!\cdots\!18\)\( T_{2}^{110} + \)\(76\!\cdots\!06\)\( T_{2}^{109} + \)\(36\!\cdots\!42\)\( T_{2}^{108} + \)\(26\!\cdots\!53\)\( T_{2}^{107} - \)\(13\!\cdots\!23\)\( T_{2}^{106} - \)\(19\!\cdots\!85\)\( T_{2}^{105} + \)\(51\!\cdots\!27\)\( T_{2}^{104} + \)\(12\!\cdots\!73\)\( T_{2}^{103} - \)\(18\!\cdots\!11\)\( T_{2}^{102} - \)\(58\!\cdots\!94\)\( T_{2}^{101} + \)\(62\!\cdots\!22\)\( T_{2}^{100} + \)\(25\!\cdots\!19\)\( T_{2}^{99} - \)\(19\!\cdots\!22\)\( T_{2}^{98} - \)\(11\!\cdots\!45\)\( T_{2}^{97} + \)\(59\!\cdots\!70\)\( T_{2}^{96} + \)\(44\!\cdots\!19\)\( T_{2}^{95} - \)\(16\!\cdots\!83\)\( T_{2}^{94} - \)\(15\!\cdots\!17\)\( T_{2}^{93} + \)\(46\!\cdots\!51\)\( T_{2}^{92} + \)\(52\!\cdots\!80\)\( T_{2}^{91} - \)\(12\!\cdots\!23\)\( T_{2}^{90} - \)\(15\!\cdots\!15\)\( T_{2}^{89} + \)\(30\!\cdots\!72\)\( T_{2}^{88} + \)\(40\!\cdots\!88\)\( T_{2}^{87} - \)\(74\!\cdots\!24\)\( T_{2}^{86} - \)\(10\!\cdots\!12\)\( T_{2}^{85} + \)\(17\!\cdots\!78\)\( T_{2}^{84} + \)\(23\!\cdots\!54\)\( T_{2}^{83} - \)\(39\!\cdots\!23\)\( T_{2}^{82} - \)\(48\!\cdots\!76\)\( T_{2}^{81} + \)\(83\!\cdots\!42\)\( T_{2}^{80} + \)\(91\!\cdots\!58\)\( T_{2}^{79} - \)\(17\!\cdots\!44\)\( T_{2}^{78} - \)\(15\!\cdots\!43\)\( T_{2}^{77} + \)\(34\!\cdots\!16\)\( T_{2}^{76} + \)\(25\!\cdots\!75\)\( T_{2}^{75} - \)\(64\!\cdots\!62\)\( T_{2}^{74} - \)\(37\!\cdots\!34\)\( T_{2}^{73} + \)\(11\!\cdots\!58\)\( T_{2}^{72} + \)\(50\!\cdots\!16\)\( T_{2}^{71} - \)\(18\!\cdots\!09\)\( T_{2}^{70} - \)\(63\!\cdots\!76\)\( T_{2}^{69} + \)\(29\!\cdots\!02\)\( T_{2}^{68} + \)\(85\!\cdots\!33\)\( T_{2}^{67} - \)\(43\!\cdots\!47\)\( T_{2}^{66} - \)\(12\!\cdots\!28\)\( T_{2}^{65} + \)\(57\!\cdots\!70\)\( T_{2}^{64} + \)\(22\!\cdots\!06\)\( T_{2}^{63} - \)\(65\!\cdots\!35\)\( T_{2}^{62} - \)\(37\!\cdots\!39\)\( T_{2}^{61} + \)\(57\!\cdots\!29\)\( T_{2}^{60} + \)\(48\!\cdots\!61\)\( T_{2}^{59} - \)\(30\!\cdots\!84\)\( T_{2}^{58} - \)\(48\!\cdots\!19\)\( T_{2}^{57} - \)\(64\!\cdots\!44\)\( T_{2}^{56} + \)\(23\!\cdots\!26\)\( T_{2}^{55} + \)\(26\!\cdots\!47\)\( T_{2}^{54} + \)\(18\!\cdots\!11\)\( T_{2}^{53} + \)\(40\!\cdots\!15\)\( T_{2}^{52} - \)\(16\!\cdots\!58\)\( T_{2}^{51} - \)\(30\!\cdots\!50\)\( T_{2}^{50} - \)\(13\!\cdots\!26\)\( T_{2}^{49} + \)\(19\!\cdots\!58\)\( T_{2}^{48} + \)\(22\!\cdots\!46\)\( T_{2}^{47} + \)\(39\!\cdots\!31\)\( T_{2}^{46} - \)\(86\!\cdots\!45\)\( T_{2}^{45} - \)\(48\!\cdots\!26\)\( T_{2}^{44} - \)\(87\!\cdots\!60\)\( T_{2}^{43} + \)\(14\!\cdots\!91\)\( T_{2}^{42} + \)\(13\!\cdots\!61\)\( T_{2}^{41} + \)\(52\!\cdots\!14\)\( T_{2}^{40} + \)\(16\!\cdots\!40\)\( T_{2}^{39} - \)\(47\!\cdots\!92\)\( T_{2}^{38} - \)\(56\!\cdots\!07\)\( T_{2}^{37} + \)\(19\!\cdots\!66\)\( T_{2}^{36} + \)\(34\!\cdots\!15\)\( T_{2}^{35} - \)\(17\!\cdots\!80\)\( T_{2}^{34} - \)\(63\!\cdots\!85\)\( T_{2}^{33} + \)\(44\!\cdots\!27\)\( T_{2}^{32} + \)\(54\!\cdots\!31\)\( T_{2}^{31} + \)\(28\!\cdots\!88\)\( T_{2}^{30} + \)\(13\!\cdots\!29\)\( T_{2}^{29} + \)\(74\!\cdots\!79\)\( T_{2}^{28} + \)\(28\!\cdots\!37\)\( T_{2}^{27} + \)\(24\!\cdots\!19\)\( T_{2}^{26} + \)\(41\!\cdots\!60\)\( T_{2}^{25} + \)\(53\!\cdots\!57\)\( T_{2}^{24} + \)\(44\!\cdots\!74\)\( T_{2}^{23} + \)\(32\!\cdots\!98\)\( T_{2}^{22} + \)\(19\!\cdots\!35\)\( T_{2}^{21} + \)\(10\!\cdots\!83\)\( T_{2}^{20} + \)\(51\!\cdots\!31\)\( T_{2}^{19} + \)\(21\!\cdots\!60\)\( T_{2}^{18} + \)\(79\!\cdots\!94\)\( T_{2}^{17} + \)\(25\!\cdots\!57\)\( T_{2}^{16} + \)\(73\!\cdots\!02\)\( T_{2}^{15} + \)\(17\!\cdots\!91\)\( T_{2}^{14} + \)\(30\!\cdots\!13\)\( T_{2}^{13} + \)\(36\!\cdots\!09\)\( T_{2}^{12} + \)\(12\!\cdots\!85\)\( T_{2}^{11} - \)\(41\!\cdots\!22\)\( T_{2}^{10} - \)\(80\!\cdots\!88\)\( T_{2}^{9} - \)\(50\!\cdots\!87\)\( T_{2}^{8} + \)\(25\!\cdots\!64\)\( T_{2}^{7} + \)\(58\!\cdots\!86\)\( T_{2}^{6} + \)\(16\!\cdots\!16\)\( T_{2}^{5} - \)\(10\!\cdots\!67\)\( T_{2}^{4} - \)\(52\!\cdots\!35\)\( T_{2}^{3} + 182488358700 T_{2}^{2} - 528737625 T_{2} + 455625 \)">\(T_{2}^{168} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\).