Properties

Label 588.2.x.b
Level $588$
Weight $2$
Character orbit 588.x
Analytic conductor $4.695$
Analytic rank $0$
Dimension $168$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.x (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.69520363885\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(28\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 168 q + 28 q^{3} - 2 q^{7} + 6 q^{8} - 28 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 168 q + 28 q^{3} - 2 q^{7} + 6 q^{8} - 28 q^{9} - 20 q^{10} + 14 q^{14} - 20 q^{16} - 12 q^{19} + 25 q^{20} + 2 q^{21} - 6 q^{22} - 27 q^{24} + 32 q^{25} - 6 q^{26} + 28 q^{27} + 6 q^{28} - 8 q^{30} + 4 q^{31} - 45 q^{32} - 44 q^{34} + 12 q^{35} - 10 q^{37} - 35 q^{38} - 14 q^{39} + 40 q^{40} + 7 q^{42} + 20 q^{44} + 28 q^{46} + 8 q^{47} - 8 q^{48} - 8 q^{49} + 114 q^{50} - 20 q^{52} - 8 q^{53} + 23 q^{56} + 12 q^{57} - 6 q^{58} - 20 q^{59} + 10 q^{60} - 14 q^{61} + 16 q^{62} + 12 q^{63} - 42 q^{64} - 8 q^{65} + 6 q^{66} + 16 q^{68} + 19 q^{70} - 28 q^{71} - 15 q^{72} + 22 q^{74} - 18 q^{75} - 49 q^{76} + 8 q^{77} + 6 q^{78} - 26 q^{80} - 28 q^{81} - 12 q^{82} - 10 q^{83} - 27 q^{84} - 24 q^{85} - 34 q^{86} + 94 q^{88} - 20 q^{90} + 16 q^{91} + 7 q^{92} - 4 q^{93} + 11 q^{94} + 10 q^{96} - 150 q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1 −1.41421 + 0.00366928i 0.900969 + 0.433884i 1.99997 0.0103783i 0.477768 0.992096i −1.27575 0.610296i 2.27188 + 1.35593i −2.82834 + 0.0220155i 0.623490 + 0.781831i −0.672024 + 1.40478i
55.2 −1.40198 + 0.185596i 0.900969 + 0.433884i 1.93111 0.520405i −1.40445 + 2.91638i −1.34367 0.441081i 0.602515 2.57623i −2.61079 + 1.08801i 0.623490 + 0.781831i 1.42775 4.34937i
55.3 −1.37146 + 0.345091i 0.900969 + 0.433884i 1.76182 0.946559i 1.63442 3.39390i −1.38538 0.284140i −0.364664 2.62050i −2.08963 + 1.90616i 0.623490 + 0.781831i −1.07034 + 5.21863i
55.4 −1.36844 0.356880i 0.900969 + 0.433884i 1.74527 + 0.976739i −0.0842420 + 0.174930i −1.07808 0.915283i −2.61856 0.378316i −2.03973 1.95946i 0.623490 + 0.781831i 0.177709 0.209318i
55.5 −1.25082 + 0.659893i 0.900969 + 0.433884i 1.12908 1.65081i −0.617993 + 1.28328i −1.41326 + 0.0518343i −2.62009 + 0.367586i −0.322916 + 2.80993i 0.623490 + 0.781831i −0.0738292 2.01295i
55.6 −1.14486 0.830241i 0.900969 + 0.433884i 0.621399 + 1.90102i −0.591080 + 1.22739i −0.671253 1.24476i 2.62768 0.308728i 0.866889 2.69230i 0.623490 + 0.781831i 1.69573 0.914448i
55.7 −1.08805 0.903405i 0.900969 + 0.433884i 0.367718 + 1.96591i 1.27172 2.64076i −0.588329 1.28603i −0.874558 2.49703i 1.37591 2.47121i 0.623490 + 0.781831i −3.76938 + 1.72441i
55.8 −0.974763 + 1.02462i 0.900969 + 0.433884i −0.0996732 1.99751i 1.10901 2.30289i −1.32280 + 0.500213i 0.953739 + 2.46787i 2.14384 + 1.84498i 0.623490 + 0.781831i 1.27855 + 3.38109i
55.9 −0.778541 + 1.18062i 0.900969 + 0.433884i −0.787748 1.83833i −1.73990 + 3.61294i −1.21369 + 0.725909i 2.50782 + 0.843102i 2.78367 + 0.501181i 0.623490 + 0.781831i −2.91094 4.86699i
55.10 −0.611563 1.27514i 0.900969 + 0.433884i −1.25198 + 1.55966i −0.776660 + 1.61275i 0.00226428 1.41421i −0.448708 + 2.60742i 2.75446 + 0.642622i 0.623490 + 0.781831i 2.53146 + 0.00405309i
55.11 −0.446111 + 1.34201i 0.900969 + 0.433884i −1.60197 1.19737i −0.0202251 + 0.0419978i −0.984208 + 1.01555i −2.50508 + 0.851222i 2.32154 1.61569i 0.623490 + 0.781831i −0.0473388 0.0458779i
55.12 −0.315691 1.37853i 0.900969 + 0.433884i −1.80068 + 0.870378i 0.236353 0.490793i 0.313693 1.37898i −2.32580 + 1.26121i 1.76830 + 2.20751i 0.623490 + 0.781831i −0.751186 0.170881i
55.13 −0.302651 + 1.38145i 0.900969 + 0.433884i −1.81681 0.836193i 0.0432081 0.0897226i −0.872067 + 1.11333i −0.103018 2.64374i 1.70502 2.25675i 0.623490 + 0.781831i 0.110870 + 0.0868444i
55.14 −0.225403 1.39614i 0.900969 + 0.433884i −1.89839 + 0.629386i 1.60689 3.33674i 0.402679 1.35567i 2.64502 + 0.0624089i 1.30661 + 2.50854i 0.623490 + 0.781831i −5.02073 1.49132i
55.15 0.148148 1.40643i 0.900969 + 0.433884i −1.95610 0.416719i −1.14386 + 2.37526i 0.743705 1.20287i 1.90517 1.83585i −0.875880 + 2.68939i 0.623490 + 0.781831i 3.17118 + 1.96066i
55.16 0.357271 1.36834i 0.900969 + 0.433884i −1.74472 0.977737i 1.10712 2.29896i 0.915591 1.07782i −2.45319 0.990882i −1.96121 + 2.03805i 0.623490 + 0.781831i −2.75022 2.33627i
55.17 0.421544 + 1.34993i 0.900969 + 0.433884i −1.64460 + 1.13811i 1.52760 3.17209i −0.205913 + 1.39914i 1.99561 1.73711i −2.22963 1.74033i 0.623490 + 0.781831i 4.92603 + 0.724967i
55.18 0.511445 + 1.31849i 0.900969 + 0.433884i −1.47685 + 1.34867i −1.72652 + 3.58517i −0.111277 + 1.40983i −1.74761 1.98642i −2.53354 1.25744i 0.623490 + 0.781831i −5.61004 0.442796i
55.19 0.696560 + 1.23077i 0.900969 + 0.433884i −1.02961 + 1.71462i −0.997385 + 2.07109i 0.0935667 + 1.41111i 0.739943 + 2.54017i −2.82749 0.0728789i 0.623490 + 0.781831i −3.24378 + 0.215085i
55.20 0.761615 1.19161i 0.900969 + 0.433884i −0.839884 1.81510i −1.00022 + 2.07699i 1.20321 0.743154i −0.0377719 + 2.64548i −2.80257 0.381593i 0.623490 + 0.781831i 1.71318 + 2.77375i
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
196.j even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.x.b yes 168
4.b odd 2 1 588.2.x.a 168
49.f odd 14 1 588.2.x.a 168
196.j even 14 1 inner 588.2.x.b yes 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.x.a 168 4.b odd 2 1
588.2.x.a 168 49.f odd 14 1
588.2.x.b yes 168 1.a even 1 1 trivial
588.2.x.b yes 168 196.j even 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(60\!\cdots\!46\)\( T_{11}^{154} + \)\(46\!\cdots\!52\)\( T_{11}^{153} + \)\(22\!\cdots\!34\)\( T_{11}^{152} - \)\(20\!\cdots\!88\)\( T_{11}^{151} - \)\(75\!\cdots\!26\)\( T_{11}^{150} + \)\(82\!\cdots\!36\)\( T_{11}^{149} + \)\(23\!\cdots\!22\)\( T_{11}^{148} - \)\(30\!\cdots\!12\)\( T_{11}^{147} - \)\(63\!\cdots\!74\)\( T_{11}^{146} + \)\(10\!\cdots\!92\)\( T_{11}^{145} + \)\(15\!\cdots\!46\)\( T_{11}^{144} - \)\(34\!\cdots\!36\)\( T_{11}^{143} - \)\(30\!\cdots\!80\)\( T_{11}^{142} + \)\(10\!\cdots\!24\)\( T_{11}^{141} + \)\(41\!\cdots\!44\)\( T_{11}^{140} - \)\(30\!\cdots\!80\)\( T_{11}^{139} + \)\(48\!\cdots\!68\)\( T_{11}^{138} + \)\(81\!\cdots\!00\)\( T_{11}^{137} - \)\(31\!\cdots\!23\)\( T_{11}^{136} - \)\(20\!\cdots\!56\)\( T_{11}^{135} + \)\(14\!\cdots\!82\)\( T_{11}^{134} + \)\(50\!\cdots\!60\)\( T_{11}^{133} - \)\(51\!\cdots\!33\)\( T_{11}^{132} - \)\(11\!\cdots\!92\)\( T_{11}^{131} + \)\(14\!\cdots\!86\)\( T_{11}^{130} + \)\(24\!\cdots\!92\)\( T_{11}^{129} - \)\(38\!\cdots\!23\)\( T_{11}^{128} - \)\(49\!\cdots\!68\)\( T_{11}^{127} + \)\(90\!\cdots\!54\)\( T_{11}^{126} + \)\(97\!\cdots\!20\)\( T_{11}^{125} - \)\(19\!\cdots\!61\)\( T_{11}^{124} - \)\(18\!\cdots\!20\)\( T_{11}^{123} + \)\(39\!\cdots\!10\)\( T_{11}^{122} + \)\(32\!\cdots\!32\)\( T_{11}^{121} - \)\(74\!\cdots\!19\)\( T_{11}^{120} - \)\(55\!\cdots\!84\)\( T_{11}^{119} + \)\(13\!\cdots\!42\)\( T_{11}^{118} + \)\(90\!\cdots\!32\)\( T_{11}^{117} - \)\(22\!\cdots\!62\)\( T_{11}^{116} - \)\(14\!\cdots\!16\)\( T_{11}^{115} + \)\(36\!\cdots\!18\)\( T_{11}^{114} + \)\(21\!\cdots\!80\)\( T_{11}^{113} - \)\(55\!\cdots\!99\)\( T_{11}^{112} - \)\(32\!\cdots\!60\)\( T_{11}^{111} + \)\(80\!\cdots\!44\)\( T_{11}^{110} + \)\(45\!\cdots\!24\)\( T_{11}^{109} - \)\(11\!\cdots\!24\)\( T_{11}^{108} - \)\(62\!\cdots\!52\)\( T_{11}^{107} + \)\(14\!\cdots\!84\)\( T_{11}^{106} + \)\(83\!\cdots\!76\)\( T_{11}^{105} - \)\(17\!\cdots\!31\)\( T_{11}^{104} - \)\(10\!\cdots\!64\)\( T_{11}^{103} + \)\(20\!\cdots\!86\)\( T_{11}^{102} + \)\(13\!\cdots\!52\)\( T_{11}^{101} - \)\(22\!\cdots\!15\)\( T_{11}^{100} - \)\(15\!\cdots\!40\)\( T_{11}^{99} + \)\(22\!\cdots\!32\)\( T_{11}^{98} + \)\(18\!\cdots\!80\)\( T_{11}^{97} - \)\(20\!\cdots\!00\)\( T_{11}^{96} - \)\(19\!\cdots\!08\)\( T_{11}^{95} + \)\(15\!\cdots\!90\)\( T_{11}^{94} + \)\(20\!\cdots\!44\)\( T_{11}^{93} - \)\(92\!\cdots\!11\)\( T_{11}^{92} - \)\(20\!\cdots\!00\)\( T_{11}^{91} + \)\(15\!\cdots\!66\)\( T_{11}^{90} + \)\(19\!\cdots\!60\)\( T_{11}^{89} + \)\(62\!\cdots\!38\)\( T_{11}^{88} - \)\(17\!\cdots\!28\)\( T_{11}^{87} - \)\(13\!\cdots\!64\)\( T_{11}^{86} + \)\(14\!\cdots\!68\)\( T_{11}^{85} + \)\(17\!\cdots\!60\)\( T_{11}^{84} - \)\(10\!\cdots\!24\)\( T_{11}^{83} - \)\(19\!\cdots\!84\)\( T_{11}^{82} + \)\(68\!\cdots\!84\)\( T_{11}^{81} + \)\(18\!\cdots\!41\)\( T_{11}^{80} - \)\(37\!\cdots\!48\)\( T_{11}^{79} - \)\(15\!\cdots\!24\)\( T_{11}^{78} + \)\(16\!\cdots\!56\)\( T_{11}^{77} + \)\(10\!\cdots\!69\)\( T_{11}^{76} - \)\(35\!\cdots\!64\)\( T_{11}^{75} - \)\(68\!\cdots\!40\)\( T_{11}^{74} - \)\(19\!\cdots\!64\)\( T_{11}^{73} + \)\(38\!\cdots\!90\)\( T_{11}^{72} + \)\(32\!\cdots\!60\)\( T_{11}^{71} - \)\(19\!\cdots\!62\)\( T_{11}^{70} - \)\(27\!\cdots\!36\)\( T_{11}^{69} + \)\(85\!\cdots\!52\)\( T_{11}^{68} + \)\(19\!\cdots\!36\)\( T_{11}^{67} - \)\(31\!\cdots\!88\)\( T_{11}^{66} - \)\(11\!\cdots\!40\)\( T_{11}^{65} + \)\(79\!\cdots\!23\)\( T_{11}^{64} + \)\(58\!\cdots\!84\)\( T_{11}^{63} + \)\(40\!\cdots\!26\)\( T_{11}^{62} - \)\(25\!\cdots\!24\)\( T_{11}^{61} - \)\(18\!\cdots\!78\)\( T_{11}^{60} + \)\(86\!\cdots\!48\)\( T_{11}^{59} + \)\(12\!\cdots\!56\)\( T_{11}^{58} - \)\(23\!\cdots\!24\)\( T_{11}^{57} - \)\(59\!\cdots\!70\)\( T_{11}^{56} + \)\(49\!\cdots\!12\)\( T_{11}^{55} + \)\(22\!\cdots\!92\)\( T_{11}^{54} - \)\(61\!\cdots\!48\)\( T_{11}^{53} - \)\(81\!\cdots\!41\)\( T_{11}^{52} - \)\(23\!\cdots\!92\)\( T_{11}^{51} + \)\(27\!\cdots\!56\)\( T_{11}^{50} + \)\(28\!\cdots\!64\)\( T_{11}^{49} - \)\(67\!\cdots\!60\)\( T_{11}^{48} - \)\(15\!\cdots\!12\)\( T_{11}^{47} + \)\(54\!\cdots\!92\)\( T_{11}^{46} + \)\(49\!\cdots\!88\)\( T_{11}^{45} + \)\(42\!\cdots\!73\)\( T_{11}^{44} - \)\(66\!\cdots\!72\)\( T_{11}^{43} - \)\(16\!\cdots\!42\)\( T_{11}^{42} - \)\(72\!\cdots\!36\)\( T_{11}^{41} + \)\(15\!\cdots\!81\)\( T_{11}^{40} + \)\(24\!\cdots\!00\)\( T_{11}^{39} + \)\(10\!\cdots\!38\)\( T_{11}^{38} + \)\(55\!\cdots\!32\)\( T_{11}^{37} + \)\(22\!\cdots\!85\)\( T_{11}^{36} + \)\(21\!\cdots\!88\)\( T_{11}^{35} - \)\(16\!\cdots\!56\)\( T_{11}^{34} - \)\(35\!\cdots\!88\)\( T_{11}^{33} - \)\(27\!\cdots\!72\)\( T_{11}^{32} + \)\(30\!\cdots\!80\)\( T_{11}^{31} + \)\(43\!\cdots\!04\)\( T_{11}^{30} - \)\(26\!\cdots\!24\)\( T_{11}^{29} - \)\(30\!\cdots\!80\)\( T_{11}^{28} + \)\(24\!\cdots\!16\)\( T_{11}^{27} + \)\(44\!\cdots\!72\)\( T_{11}^{26} - \)\(21\!\cdots\!92\)\( T_{11}^{25} - \)\(43\!\cdots\!00\)\( T_{11}^{24} + \)\(18\!\cdots\!40\)\( T_{11}^{23} + \)\(63\!\cdots\!28\)\( T_{11}^{22} - \)\(10\!\cdots\!52\)\( T_{11}^{21} - \)\(42\!\cdots\!76\)\( T_{11}^{20} + \)\(47\!\cdots\!00\)\( T_{11}^{19} + \)\(19\!\cdots\!48\)\( T_{11}^{18} - \)\(13\!\cdots\!20\)\( T_{11}^{17} - \)\(70\!\cdots\!08\)\( T_{11}^{16} + \)\(22\!\cdots\!92\)\( T_{11}^{15} + \)\(21\!\cdots\!76\)\( T_{11}^{14} - \)\(61\!\cdots\!36\)\( T_{11}^{13} - \)\(33\!\cdots\!20\)\( T_{11}^{12} + \)\(24\!\cdots\!64\)\( T_{11}^{11} + \)\(10\!\cdots\!72\)\( T_{11}^{10} - \)\(44\!\cdots\!28\)\( T_{11}^{9} - \)\(18\!\cdots\!40\)\( T_{11}^{8} + \)\(49\!\cdots\!76\)\( T_{11}^{7} + \)\(18\!\cdots\!44\)\( T_{11}^{6} - \)\(41\!\cdots\!16\)\( T_{11}^{5} - \)\(11\!\cdots\!76\)\( T_{11}^{4} + \)\(27\!\cdots\!84\)\( T_{11}^{3} + \)\(38\!\cdots\!36\)\( T_{11}^{2} - \)\(11\!\cdots\!24\)\( T_{11} + \)\(89\!\cdots\!56\)\( \)">\(T_{11}^{168} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\).