# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(588, [\chi])$$.