Properties

Label 531.2.i.c
Level $531$
Weight $2$
Character orbit 531.i
Analytic conductor $4.240$
Analytic rank $0$
Dimension $140$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 531.i (of order \(29\), degree \(28\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.24005634733\)
Analytic rank: \(0\)
Dimension: \(140\)
Relative dimension: \(5\) over \(\Q(\zeta_{29})\)
Twist minimal: no (minimal twist has level 177)
Sato-Tate group: $\mathrm{SU}(2)[C_{29}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 140q + q^{2} - 9q^{4} + 2q^{5} - 2q^{7} + 9q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 140q + q^{2} - 9q^{4} + 2q^{5} - 2q^{7} + 9q^{8} + 88q^{10} + 14q^{11} - 12q^{13} + q^{14} - 41q^{16} + 16q^{17} - 10q^{19} + 32q^{20} - 26q^{22} + 22q^{23} + 27q^{25} + 56q^{26} - 50q^{28} + 24q^{29} - 24q^{31} - 106q^{32} - 54q^{34} + 70q^{35} - 28q^{37} + 80q^{38} - 50q^{40} + 40q^{41} + 4q^{43} + 104q^{44} - 28q^{46} - 31q^{47} - q^{49} - 39q^{50} + 62q^{52} - 4q^{53} + 5q^{55} - 96q^{56} + 128q^{58} + q^{59} - 16q^{61} - 223q^{62} + 97q^{64} - 121q^{65} - 12q^{67} - 10q^{68} - 2q^{70} + 22q^{71} + 179q^{73} + 38q^{74} + 112q^{76} + 62q^{77} - 84q^{79} - 204q^{80} - 152q^{82} + 88q^{83} - 118q^{85} + 118q^{86} + 18q^{88} + 86q^{89} + 78q^{91} + 174q^{92} - 164q^{94} - 218q^{95} - 84q^{97} - 129q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.31462 + 2.47964i 0 −3.29802 4.86421i −1.27307 + 0.280224i 0 1.14406 0.869689i 10.8169 1.17641i 0 0.978753 3.52515i
19.2 −0.655840 + 1.23705i 0 0.0222185 + 0.0327698i −2.32817 + 0.512469i 0 −1.36308 + 1.03619i −2.83899 + 0.308758i 0 0.892959 3.21615i
19.3 −0.0292574 + 0.0551852i 0 1.12018 + 1.65215i 2.41343 0.531236i 0 −2.91820 + 2.21836i −0.248138 + 0.0269866i 0 −0.0412942 + 0.148728i
19.4 0.454038 0.856407i 0 0.595092 + 0.877695i −3.60206 + 0.792874i 0 1.56865 1.19245i 2.94914 0.320738i 0 −0.956451 + 3.44483i
19.5 1.07727 2.03195i 0 −1.84595 2.72257i 3.23491 0.712057i 0 1.31099 0.996592i −2.94796 + 0.320610i 0 2.03802 7.34027i
28.1 −1.31462 2.47964i 0 −3.29802 + 4.86421i −1.27307 0.280224i 0 1.14406 + 0.869689i 10.8169 + 1.17641i 0 0.978753 + 3.52515i
28.2 −0.655840 1.23705i 0 0.0222185 0.0327698i −2.32817 0.512469i 0 −1.36308 1.03619i −2.83899 0.308758i 0 0.892959 + 3.21615i
28.3 −0.0292574 0.0551852i 0 1.12018 1.65215i 2.41343 + 0.531236i 0 −2.91820 2.21836i −0.248138 0.0269866i 0 −0.0412942 0.148728i
28.4 0.454038 + 0.856407i 0 0.595092 0.877695i −3.60206 0.792874i 0 1.56865 + 1.19245i 2.94914 + 0.320738i 0 −0.956451 3.44483i
28.5 1.07727 + 2.03195i 0 −1.84595 + 2.72257i 3.23491 + 0.712057i 0 1.31099 + 0.996592i −2.94796 0.320610i 0 2.03802 + 7.34027i
46.1 −1.65338 1.94651i 0 −0.731672 + 4.46300i −0.865278 + 1.63209i 0 −2.96873 + 0.322869i 5.52030 3.32145i 0 4.60751 1.01419i
46.2 −1.15330 1.35777i 0 −0.189873 + 1.15817i 1.46043 2.75467i 0 1.72018 0.187081i −1.26142 + 0.758972i 0 −5.42452 + 1.19403i
46.3 −0.398193 0.468789i 0 0.262358 1.60032i −0.780962 + 1.47305i 0 −2.98728 + 0.324887i −1.90875 + 1.14846i 0 1.00152 0.220452i
46.4 0.814427 + 0.958818i 0 0.0675241 0.411879i −0.588484 + 1.11000i 0 2.16620 0.235588i 2.60580 1.56786i 0 −1.54356 + 0.339764i
46.5 1.74306 + 2.05209i 0 −0.849247 + 5.18018i 1.70562 3.21713i 0 3.51311 0.382074i −7.49637 + 4.51042i 0 9.57483 2.10758i
64.1 −2.28036 0.768343i 0 3.01751 + 2.29385i −0.739682 + 1.85646i 0 −0.807669 + 0.373667i −2.41776 3.56592i 0 3.11314 3.66507i
64.2 −0.827908 0.278955i 0 −0.984570 0.748451i −0.302390 + 0.758941i 0 3.54386 1.63956i 1.58690 + 2.34050i 0 0.462061 0.543981i
64.3 −0.288668 0.0972636i 0 −1.51832 1.15419i 1.30154 3.26662i 0 −3.34578 + 1.54792i 0.667919 + 0.985107i 0 −0.693437 + 0.816377i
64.4 1.73089 + 0.583204i 0 1.06366 + 0.808574i 0.835093 2.09593i 0 2.04844 0.947709i −0.680501 1.00366i 0 2.66781 3.14078i
64.5 2.61370 + 0.880659i 0 4.46370 + 3.39321i −0.422705 + 1.06091i 0 −3.24336 + 1.50054i 5.58292 + 8.23419i 0 −2.03912 + 2.40064i
See next 80 embeddings (of 140 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 523.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.c even 29 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 531.2.i.c 140
3.b odd 2 1 177.2.e.a 140
59.c even 29 1 inner 531.2.i.c 140
177.h odd 58 1 177.2.e.a 140
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.e.a 140 3.b odd 2 1
177.2.e.a 140 177.h odd 58 1
531.2.i.c 140 1.a even 1 1 trivial
531.2.i.c 140 59.c even 29 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(17\!\cdots\!76\)\( T_{2}^{109} + \)\(15\!\cdots\!73\)\( T_{2}^{108} + \)\(54\!\cdots\!81\)\( T_{2}^{107} + \)\(26\!\cdots\!18\)\( T_{2}^{106} - \)\(19\!\cdots\!24\)\( T_{2}^{105} - \)\(44\!\cdots\!52\)\( T_{2}^{104} - \)\(95\!\cdots\!64\)\( T_{2}^{103} - \)\(52\!\cdots\!48\)\( T_{2}^{102} - \)\(85\!\cdots\!70\)\( T_{2}^{101} + \)\(21\!\cdots\!05\)\( T_{2}^{100} - \)\(40\!\cdots\!45\)\( T_{2}^{99} + \)\(13\!\cdots\!06\)\( T_{2}^{98} + \)\(26\!\cdots\!97\)\( T_{2}^{97} + \)\(21\!\cdots\!47\)\( T_{2}^{96} - \)\(57\!\cdots\!05\)\( T_{2}^{95} + \)\(84\!\cdots\!26\)\( T_{2}^{94} - \)\(28\!\cdots\!67\)\( T_{2}^{93} + \)\(35\!\cdots\!42\)\( T_{2}^{92} + \)\(33\!\cdots\!80\)\( T_{2}^{91} + \)\(43\!\cdots\!99\)\( T_{2}^{90} + \)\(18\!\cdots\!41\)\( T_{2}^{89} + \)\(11\!\cdots\!44\)\( T_{2}^{88} - \)\(66\!\cdots\!53\)\( T_{2}^{87} - \)\(48\!\cdots\!26\)\( T_{2}^{86} + \)\(46\!\cdots\!87\)\( T_{2}^{85} + \)\(48\!\cdots\!56\)\( T_{2}^{84} - \)\(83\!\cdots\!66\)\( T_{2}^{83} + \)\(62\!\cdots\!14\)\( T_{2}^{82} + \)\(44\!\cdots\!26\)\( T_{2}^{81} + \)\(91\!\cdots\!50\)\( T_{2}^{80} - \)\(49\!\cdots\!08\)\( T_{2}^{79} + \)\(13\!\cdots\!30\)\( T_{2}^{78} + \)\(89\!\cdots\!48\)\( T_{2}^{77} + \)\(16\!\cdots\!22\)\( T_{2}^{76} - \)\(16\!\cdots\!85\)\( T_{2}^{75} + \)\(71\!\cdots\!71\)\( T_{2}^{74} - \)\(86\!\cdots\!21\)\( T_{2}^{73} + \)\(13\!\cdots\!80\)\( T_{2}^{72} - \)\(23\!\cdots\!50\)\( T_{2}^{71} + \)\(11\!\cdots\!94\)\( T_{2}^{70} - \)\(19\!\cdots\!59\)\( T_{2}^{69} + \)\(43\!\cdots\!96\)\( T_{2}^{68} - \)\(99\!\cdots\!74\)\( T_{2}^{67} + \)\(16\!\cdots\!48\)\( T_{2}^{66} - \)\(30\!\cdots\!27\)\( T_{2}^{65} + \)\(55\!\cdots\!80\)\( T_{2}^{64} - \)\(90\!\cdots\!16\)\( T_{2}^{63} + \)\(14\!\cdots\!36\)\( T_{2}^{62} - \)\(22\!\cdots\!43\)\( T_{2}^{61} + \)\(31\!\cdots\!29\)\( T_{2}^{60} - \)\(39\!\cdots\!90\)\( T_{2}^{59} + \)\(49\!\cdots\!38\)\( T_{2}^{58} - \)\(53\!\cdots\!38\)\( T_{2}^{57} + \)\(51\!\cdots\!10\)\( T_{2}^{56} - \)\(47\!\cdots\!65\)\( T_{2}^{55} + \)\(48\!\cdots\!67\)\( T_{2}^{54} - \)\(38\!\cdots\!07\)\( T_{2}^{53} + \)\(46\!\cdots\!46\)\( T_{2}^{52} - \)\(53\!\cdots\!75\)\( T_{2}^{51} + \)\(57\!\cdots\!11\)\( T_{2}^{50} - \)\(56\!\cdots\!93\)\( T_{2}^{49} + \)\(59\!\cdots\!87\)\( T_{2}^{48} - \)\(42\!\cdots\!58\)\( T_{2}^{47} + \)\(25\!\cdots\!24\)\( T_{2}^{46} - \)\(13\!\cdots\!04\)\( T_{2}^{45} + \)\(10\!\cdots\!55\)\( T_{2}^{44} + \)\(16\!\cdots\!60\)\( T_{2}^{43} + \)\(38\!\cdots\!51\)\( T_{2}^{42} + \)\(40\!\cdots\!88\)\( T_{2}^{41} + \)\(98\!\cdots\!00\)\( T_{2}^{40} + \)\(97\!\cdots\!56\)\( T_{2}^{39} + \)\(16\!\cdots\!63\)\( T_{2}^{38} + \)\(19\!\cdots\!51\)\( T_{2}^{37} + \)\(25\!\cdots\!35\)\( T_{2}^{36} + \)\(31\!\cdots\!78\)\( T_{2}^{35} + \)\(34\!\cdots\!38\)\( T_{2}^{34} + \)\(35\!\cdots\!25\)\( T_{2}^{33} + \)\(33\!\cdots\!11\)\( T_{2}^{32} + \)\(29\!\cdots\!36\)\( T_{2}^{31} + \)\(24\!\cdots\!65\)\( T_{2}^{30} + \)\(19\!\cdots\!67\)\( T_{2}^{29} + \)\(15\!\cdots\!93\)\( T_{2}^{28} + \)\(12\!\cdots\!74\)\( T_{2}^{27} + \)\(93\!\cdots\!61\)\( T_{2}^{26} + \)\(71\!\cdots\!64\)\( T_{2}^{25} + \)\(53\!\cdots\!00\)\( T_{2}^{24} + \)\(39\!\cdots\!03\)\( T_{2}^{23} + \)\(28\!\cdots\!93\)\( T_{2}^{22} + \)\(19\!\cdots\!77\)\( T_{2}^{21} + \)\(12\!\cdots\!47\)\( T_{2}^{20} + \)\(72\!\cdots\!14\)\( T_{2}^{19} + \)\(40\!\cdots\!84\)\( T_{2}^{18} + \)\(20\!\cdots\!56\)\( T_{2}^{17} + \)\(93\!\cdots\!05\)\( T_{2}^{16} + \)\(38\!\cdots\!68\)\( T_{2}^{15} + \)\(14\!\cdots\!80\)\( T_{2}^{14} + \)\(46\!\cdots\!77\)\( T_{2}^{13} + \)\(13\!\cdots\!12\)\( T_{2}^{12} + \)\(35\!\cdots\!97\)\( T_{2}^{11} + \)\(79\!\cdots\!45\)\( T_{2}^{10} + \)\(15\!\cdots\!98\)\( T_{2}^{9} + \)\(24\!\cdots\!23\)\( T_{2}^{8} + \)\(33\!\cdots\!67\)\( T_{2}^{7} + \)\(37\!\cdots\!26\)\( T_{2}^{6} + \)\(33\!\cdots\!63\)\( T_{2}^{5} + \)\(24\!\cdots\!88\)\( T_{2}^{4} + \)\(13\!\cdots\!02\)\( T_{2}^{3} + \)\(57\!\cdots\!02\)\( T_{2}^{2} + \)\(16\!\cdots\!73\)\( T_{2} + 23694752761 \)">\(T_{2}^{140} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(531, [\chi])\).