# Properties

 Label 690.2.q.a Level $690$ Weight $2$ Character orbit 690.q Analytic conductor $5.510$ Analytic rank $0$ Dimension $160$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 690.q (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.50967773947$$ Analytic rank: $$0$$ Dimension: $$160$$ Relative dimension: $$16$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$160q + 16q^{4} - 16q^{5} - 2q^{6} + 42q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$160q + 16q^{4} - 16q^{5} - 2q^{6} + 42q^{9} - 12q^{11} - 12q^{14} - 16q^{16} - 8q^{18} + 16q^{20} + 62q^{21} + 4q^{23} + 2q^{24} - 16q^{25} + 42q^{27} - 2q^{30} - 4q^{31} + 16q^{33} + 2q^{36} + 72q^{38} - 124q^{39} + 44q^{41} + 44q^{43} + 12q^{44} - 2q^{45} + 4q^{46} + 70q^{49} - 2q^{51} - 52q^{53} + 92q^{54} + 10q^{55} - 54q^{56} - 38q^{57} - 36q^{58} - 44q^{61} - 220q^{63} + 16q^{64} - 34q^{66} - 44q^{67} + 22q^{69} - 12q^{70} - 36q^{72} - 28q^{73} - 24q^{74} - 88q^{77} - 54q^{78} - 44q^{79} - 16q^{80} - 66q^{81} - 28q^{82} + 4q^{83} - 18q^{84} + 158q^{86} - 64q^{87} + 80q^{89} - 8q^{90} - 4q^{92} + 4q^{93} + 24q^{94} - 2q^{96} - 88q^{98} + 190q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −0.540641 0.841254i −1.73087 + 0.0639912i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.989611 + 1.42150i 0.918508 0.795892i 0.989821 0.142315i 2.99181 0.221521i 0.755750 + 0.654861i
11.2 −0.540641 0.841254i −1.61107 + 0.635963i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.40602 + 1.01149i −3.41511 + 2.95921i 0.989821 0.142315i 2.19110 2.04916i 0.755750 + 0.654861i
11.3 −0.540641 0.841254i −1.10251 1.33584i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.527715 + 1.64970i −3.34609 + 2.89940i 0.989821 0.142315i −0.568927 + 2.94556i 0.755750 + 0.654861i
11.4 −0.540641 0.841254i −0.717428 + 1.57648i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.71409 0.248772i 3.34933 2.90221i 0.989821 0.142315i −1.97059 2.26203i 0.755750 + 0.654861i
11.5 −0.540641 0.841254i 0.0741266 1.73046i −0.415415 + 0.909632i −0.959493 + 0.281733i −1.49583 + 0.873200i 1.57600 1.36561i 0.989821 0.142315i −2.98901 0.256547i 0.755750 + 0.654861i
11.6 −0.540641 0.841254i 1.20125 + 1.24780i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.400274 1.68516i −1.03609 + 0.897775i 0.989821 0.142315i −0.114014 + 2.99783i 0.755750 + 0.654861i
11.7 −0.540641 0.841254i 1.47579 0.906664i −0.415415 + 0.909632i −0.959493 + 0.281733i −1.56061 0.751335i −1.93677 + 1.67822i 0.989821 0.142315i 1.35592 2.67609i 0.755750 + 0.654861i
11.8 −0.540641 0.841254i 1.50109 + 0.864143i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.0845842 1.72998i 0.920755 0.797839i 0.989821 0.142315i 1.50651 + 2.59431i 0.755750 + 0.654861i
11.9 0.540641 + 0.841254i −1.62851 0.589888i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.384191 1.68890i 1.12640 0.976034i −0.989821 + 0.142315i 2.30406 + 1.92127i −0.755750 0.654861i
11.10 0.540641 + 0.841254i −1.25644 + 1.19221i −0.415415 + 0.909632i −0.959493 + 0.281733i −1.68223 0.412423i 3.70222 3.20799i −0.989821 + 0.142315i 0.157263 2.99588i −0.755750 0.654861i
11.11 0.540641 + 0.841254i −0.549229 1.64266i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.08496 1.35013i −0.315364 + 0.273265i −0.989821 + 0.142315i −2.39669 + 1.80440i −0.755750 0.654861i
11.12 0.540641 + 0.841254i −0.162462 + 1.72441i −0.415415 + 0.909632i −0.959493 + 0.281733i −1.53850 + 0.795618i 0.461235 0.399662i −0.989821 + 0.142315i −2.94721 0.560302i −0.755750 0.654861i
11.13 0.540641 + 0.841254i 0.559698 1.63913i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.68152 0.415331i −2.94981 + 2.55602i −0.989821 + 0.142315i −2.37348 1.83483i −0.755750 0.654861i
11.14 0.540641 + 0.841254i 0.843944 + 1.51253i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.816154 + 1.52771i −1.57741 + 1.36683i −0.989821 + 0.142315i −1.57552 + 2.55299i −0.755750 0.654861i
11.15 0.540641 + 0.841254i 1.37259 1.05641i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.63079 + 0.583554i 2.43170 2.10708i −0.989821 + 0.142315i 0.767988 2.90003i −0.755750 0.654861i
11.16 0.540641 + 0.841254i 1.73004 + 0.0835168i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.865069 + 1.50055i 0.0904807 0.0784019i −0.989821 + 0.142315i 2.98605 + 0.288974i −0.755750 0.654861i
191.1 −0.755750 0.654861i −1.72862 + 0.108947i 0.142315 + 0.989821i 0.415415 + 0.909632i 1.37775 + 1.04967i −0.501114 1.70664i 0.540641 0.841254i 2.97626 0.376654i 0.281733 0.959493i
191.2 −0.755750 0.654861i −1.23307 1.21636i 0.142315 + 0.989821i 0.415415 + 0.909632i 0.135347 + 1.72675i 0.0399458 + 0.136043i 0.540641 0.841254i 0.0409354 + 2.99972i 0.281733 0.959493i
191.3 −0.755750 0.654861i −0.906618 + 1.47582i 0.142315 + 0.989821i 0.415415 + 0.909632i 1.65163 0.521642i −0.302591 1.03053i 0.540641 0.841254i −1.35609 2.67601i 0.281733 0.959493i
191.4 −0.755750 0.654861i 0.359391 1.69435i 0.142315 + 0.989821i 0.415415 + 0.909632i −1.38118 + 1.04516i 0.816450 + 2.78057i 0.540641 0.841254i −2.74168 1.21787i 0.281733 0.959493i
See next 80 embeddings (of 160 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 641.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
69.g even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.q.a 160
3.b odd 2 1 690.2.q.b yes 160
23.d odd 22 1 690.2.q.b yes 160
69.g even 22 1 inner 690.2.q.a 160

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.q.a 160 1.a even 1 1 trivial
690.2.q.a 160 69.g even 22 1 inner
690.2.q.b yes 160 3.b odd 2 1
690.2.q.b yes 160 23.d odd 22 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$12\!\cdots\!53$$$$T_{11}^{146} +$$$$58\!\cdots\!96$$$$T_{11}^{145} +$$$$46\!\cdots\!24$$$$T_{11}^{144} +$$$$22\!\cdots\!34$$$$T_{11}^{143} +$$$$17\!\cdots\!00$$$$T_{11}^{142} +$$$$93\!\cdots\!20$$$$T_{11}^{141} +$$$$67\!\cdots\!69$$$$T_{11}^{140} +$$$$32\!\cdots\!36$$$$T_{11}^{139} +$$$$22\!\cdots\!60$$$$T_{11}^{138} +$$$$10\!\cdots\!30$$$$T_{11}^{137} +$$$$70\!\cdots\!33$$$$T_{11}^{136} +$$$$34\!\cdots\!98$$$$T_{11}^{135} +$$$$22\!\cdots\!11$$$$T_{11}^{134} +$$$$11\!\cdots\!24$$$$T_{11}^{133} +$$$$73\!\cdots\!78$$$$T_{11}^{132} +$$$$37\!\cdots\!18$$$$T_{11}^{131} +$$$$22\!\cdots\!79$$$$T_{11}^{130} +$$$$11\!\cdots\!60$$$$T_{11}^{129} +$$$$67\!\cdots\!11$$$$T_{11}^{128} +$$$$34\!\cdots\!42$$$$T_{11}^{127} +$$$$19\!\cdots\!04$$$$T_{11}^{126} +$$$$94\!\cdots\!26$$$$T_{11}^{125} +$$$$51\!\cdots\!59$$$$T_{11}^{124} +$$$$25\!\cdots\!18$$$$T_{11}^{123} +$$$$13\!\cdots\!68$$$$T_{11}^{122} +$$$$63\!\cdots\!32$$$$T_{11}^{121} +$$$$32\!\cdots\!65$$$$T_{11}^{120} +$$$$15\!\cdots\!40$$$$T_{11}^{119} +$$$$76\!\cdots\!90$$$$T_{11}^{118} +$$$$36\!\cdots\!26$$$$T_{11}^{117} +$$$$18\!\cdots\!88$$$$T_{11}^{116} +$$$$86\!\cdots\!86$$$$T_{11}^{115} +$$$$42\!\cdots\!61$$$$T_{11}^{114} +$$$$20\!\cdots\!76$$$$T_{11}^{113} +$$$$96\!\cdots\!49$$$$T_{11}^{112} +$$$$43\!\cdots\!46$$$$T_{11}^{111} +$$$$19\!\cdots\!69$$$$T_{11}^{110} +$$$$85\!\cdots\!30$$$$T_{11}^{109} +$$$$36\!\cdots\!35$$$$T_{11}^{108} +$$$$14\!\cdots\!32$$$$T_{11}^{107} +$$$$60\!\cdots\!42$$$$T_{11}^{106} +$$$$23\!\cdots\!88$$$$T_{11}^{105} +$$$$90\!\cdots\!90$$$$T_{11}^{104} +$$$$33\!\cdots\!28$$$$T_{11}^{103} +$$$$12\!\cdots\!09$$$$T_{11}^{102} +$$$$41\!\cdots\!74$$$$T_{11}^{101} +$$$$14\!\cdots\!43$$$$T_{11}^{100} +$$$$47\!\cdots\!96$$$$T_{11}^{99} +$$$$15\!\cdots\!94$$$$T_{11}^{98} +$$$$47\!\cdots\!16$$$$T_{11}^{97} +$$$$14\!\cdots\!44$$$$T_{11}^{96} +$$$$42\!\cdots\!48$$$$T_{11}^{95} +$$$$13\!\cdots\!94$$$$T_{11}^{94} +$$$$34\!\cdots\!64$$$$T_{11}^{93} +$$$$10\!\cdots\!53$$$$T_{11}^{92} +$$$$26\!\cdots\!20$$$$T_{11}^{91} +$$$$80\!\cdots\!88$$$$T_{11}^{90} +$$$$18\!\cdots\!48$$$$T_{11}^{89} +$$$$61\!\cdots\!15$$$$T_{11}^{88} +$$$$13\!\cdots\!64$$$$T_{11}^{87} +$$$$51\!\cdots\!46$$$$T_{11}^{86} +$$$$11\!\cdots\!34$$$$T_{11}^{85} +$$$$44\!\cdots\!10$$$$T_{11}^{84} +$$$$94\!\cdots\!30$$$$T_{11}^{83} +$$$$38\!\cdots\!78$$$$T_{11}^{82} +$$$$70\!\cdots\!78$$$$T_{11}^{81} +$$$$26\!\cdots\!90$$$$T_{11}^{80} +$$$$31\!\cdots\!62$$$$T_{11}^{79} +$$$$13\!\cdots\!01$$$$T_{11}^{78} +$$$$24\!\cdots\!78$$$$T_{11}^{77} +$$$$72\!\cdots\!65$$$$T_{11}^{76} +$$$$20\!\cdots\!24$$$$T_{11}^{75} +$$$$75\!\cdots\!90$$$$T_{11}^{74} +$$$$91\!\cdots\!32$$$$T_{11}^{73} +$$$$65\!\cdots\!07$$$$T_{11}^{72} +$$$$65\!\cdots\!56$$$$T_{11}^{71} +$$$$32\!\cdots\!83$$$$T_{11}^{70} +$$$$94\!\cdots\!26$$$$T_{11}^{69} +$$$$11\!\cdots\!38$$$$T_{11}^{68} -$$$$28\!\cdots\!56$$$$T_{11}^{67} +$$$$64\!\cdots\!62$$$$T_{11}^{66} +$$$$54\!\cdots\!86$$$$T_{11}^{65} +$$$$44\!\cdots\!93$$$$T_{11}^{64} +$$$$51\!\cdots\!26$$$$T_{11}^{63} +$$$$20\!\cdots\!52$$$$T_{11}^{62} +$$$$17\!\cdots\!74$$$$T_{11}^{61} +$$$$71\!\cdots\!02$$$$T_{11}^{60} +$$$$93\!\cdots\!22$$$$T_{11}^{59} +$$$$36\!\cdots\!59$$$$T_{11}^{58} +$$$$67\!\cdots\!56$$$$T_{11}^{57} +$$$$19\!\cdots\!35$$$$T_{11}^{56} +$$$$28\!\cdots\!58$$$$T_{11}^{55} +$$$$62\!\cdots\!80$$$$T_{11}^{54} +$$$$79\!\cdots\!56$$$$T_{11}^{53} +$$$$12\!\cdots\!39$$$$T_{11}^{52} +$$$$15\!\cdots\!36$$$$T_{11}^{51} +$$$$49\!\cdots\!54$$$$T_{11}^{50} +$$$$54\!\cdots\!76$$$$T_{11}^{49} +$$$$17\!\cdots\!09$$$$T_{11}^{48} +$$$$31\!\cdots\!68$$$$T_{11}^{47} +$$$$45\!\cdots\!89$$$$T_{11}^{46} +$$$$70\!\cdots\!04$$$$T_{11}^{45} +$$$$17\!\cdots\!13$$$$T_{11}^{44} +$$$$15\!\cdots\!46$$$$T_{11}^{43} +$$$$38\!\cdots\!48$$$$T_{11}^{42} +$$$$77\!\cdots\!50$$$$T_{11}^{41} +$$$$98\!\cdots\!33$$$$T_{11}^{40} +$$$$15\!\cdots\!34$$$$T_{11}^{39} +$$$$28\!\cdots\!03$$$$T_{11}^{38} +$$$$23\!\cdots\!50$$$$T_{11}^{37} +$$$$29\!\cdots\!01$$$$T_{11}^{36} +$$$$37\!\cdots\!40$$$$T_{11}^{35} +$$$$21\!\cdots\!50$$$$T_{11}^{34} +$$$$11\!\cdots\!34$$$$T_{11}^{33} +$$$$27\!\cdots\!28$$$$T_{11}^{32} -$$$$11\!\cdots\!34$$$$T_{11}^{31} +$$$$12\!\cdots\!51$$$$T_{11}^{30} -$$$$49\!\cdots\!98$$$$T_{11}^{29} +$$$$50\!\cdots\!25$$$$T_{11}^{28} -$$$$25\!\cdots\!04$$$$T_{11}^{27} +$$$$11\!\cdots\!17$$$$T_{11}^{26} -$$$$46\!\cdots\!46$$$$T_{11}^{25} +$$$$58\!\cdots\!73$$$$T_{11}^{24} -$$$$26\!\cdots\!24$$$$T_{11}^{23} +$$$$13\!\cdots\!87$$$$T_{11}^{22} -$$$$64\!\cdots\!26$$$$T_{11}^{21} +$$$$29\!\cdots\!01$$$$T_{11}^{20} -$$$$44\!\cdots\!76$$$$T_{11}^{19} +$$$$17\!\cdots\!76$$$$T_{11}^{18} +$$$$12\!\cdots\!80$$$$T_{11}^{17} +$$$$63\!\cdots\!08$$$$T_{11}^{16} +$$$$49\!\cdots\!48$$$$T_{11}^{15} +$$$$10\!\cdots\!80$$$$T_{11}^{14} +$$$$26\!\cdots\!12$$$$T_{11}^{13} +$$$$11\!\cdots\!12$$$$T_{11}^{12} +$$$$11\!\cdots\!36$$$$T_{11}^{11} +$$$$51\!\cdots\!60$$$$T_{11}^{10} +$$$$84\!\cdots\!52$$$$T_{11}^{9} +$$$$73\!\cdots\!64$$$$T_{11}^{8} +$$$$73\!\cdots\!04$$$$T_{11}^{7} +$$$$91\!\cdots\!16$$$$T_{11}^{6} -$$$$39\!\cdots\!04$$$$T_{11}^{5} +$$$$68\!\cdots\!36$$$$T_{11}^{4} -$$$$41\!\cdots\!56$$$$T_{11}^{3} +$$$$91\!\cdots\!24$$$$T_{11}^{2} -$$$$98\!\cdots\!20$$$$T_{11} +$$$$12\!\cdots\!44$$">$$T_{11}^{160} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(690, [\chi])$$.