# Properties

 Label 690.2.q.b 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.

 $$\operatorname{Tr}(f)(q) =$$ $$160q + 16q^{4} + 16q^{5} - 2q^{6} - 46q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$160q + 16q^{4} + 16q^{5} - 2q^{6} - 46q^{9} + 12q^{11} + 12q^{14} - 16q^{16} - 8q^{18} - 16q^{20} + 70q^{21} - 4q^{23} + 2q^{24} - 16q^{25} + 42q^{27} + 2q^{30} - 4q^{31} - 16q^{33} + 2q^{36} - 72q^{38} + 140q^{39} - 44q^{41} + 44q^{43} - 12q^{44} + 2q^{45} + 4q^{46} + 70q^{49} + 2q^{51} + 52q^{53} - 62q^{54} + 10q^{55} + 54q^{56} - 94q^{57} - 36q^{58} - 44q^{61} + 16q^{64} - 54q^{66} - 44q^{67} - 30q^{69} - 12q^{70} - 36q^{72} - 28q^{73} + 24q^{74} + 88q^{77} - 54q^{78} - 44q^{79} + 16q^{80} - 66q^{81} - 28q^{82} - 4q^{83} - 4q^{84} - 158q^{86} + 156q^{87} - 80q^{89} + 8q^{90} + 4q^{92} + 4q^{93} + 24q^{94} - 2q^{96} + 88q^{98} - 58q^{99} + O(q^{100})$$

## 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 −1.68349 0.407274i −0.415415 + 0.909632i 0.959493 0.281733i 0.567541 + 1.63643i 0.0904807 0.0784019i 0.989821 0.142315i 2.66826 + 1.37128i −0.755750 0.654861i
11.2 −0.540641 0.841254i −1.23589 + 1.21350i −0.415415 + 0.909632i 0.959493 0.281733i 1.68903 + 0.383629i −1.57741 + 1.36683i 0.989821 0.142315i 0.0548407 2.99950i −0.755750 0.654861i
11.3 −0.540641 0.841254i −1.01936 1.40032i −0.415415 + 0.909632i 0.959493 0.281733i −0.626917 + 1.61461i 2.43170 2.10708i 0.989821 0.142315i −0.921804 + 2.85487i −0.755750 0.654861i
11.4 −0.540641 0.841254i −0.329943 + 1.70033i −0.415415 + 0.909632i 0.959493 0.281733i 1.60879 0.641705i 0.461235 0.399662i 0.989821 0.142315i −2.78228 1.12203i −0.755750 0.654861i
11.5 −0.540641 0.841254i −0.0752308 1.73042i −0.415415 + 0.909632i 0.959493 0.281733i −1.41505 + 0.998822i −2.94981 + 2.55602i 0.989821 0.142315i −2.98868 + 0.260361i −0.755750 0.654861i
11.6 −0.540641 0.841254i 0.869657 + 1.49790i −0.415415 + 0.909632i 0.959493 0.281733i 0.789940 1.54143i 3.70222 3.20799i 0.989821 0.142315i −1.48739 + 2.60531i −0.755750 0.654861i
11.7 −0.540641 0.841254i 0.989774 1.42139i −0.415415 + 0.909632i 0.959493 0.281733i −1.73086 0.0641895i −0.315364 + 0.273265i 0.989821 0.142315i −1.04070 2.81371i −0.755750 0.654861i
11.8 −0.540641 0.841254i 1.72873 0.107190i −0.415415 + 0.909632i 0.959493 0.281733i −1.02480 1.39635i 1.12640 0.976034i 0.989821 0.142315i 2.97702 0.370607i −0.755750 0.654861i
11.9 0.540641 + 0.841254i −1.68374 + 0.406235i −0.415415 + 0.909632i 0.959493 0.281733i −1.25204 1.19682i 0.920755 0.797839i −0.989821 + 0.142315i 2.66995 1.36799i 0.755750 + 0.654861i
11.10 0.540641 + 0.841254i −1.50413 + 0.858826i −0.415415 + 0.909632i 0.959493 0.281733i −1.53569 0.801041i −1.03609 + 0.897775i −0.989821 + 0.142315i 1.52484 2.58358i 0.755750 + 0.654861i
11.11 0.540641 + 0.841254i −1.16057 1.28572i −0.415415 + 0.909632i 0.959493 0.281733i 0.454160 1.67145i −1.93677 + 1.67822i −0.989821 + 0.142315i −0.306134 + 2.98434i 0.755750 + 0.654861i
11.12 0.540641 + 0.841254i 0.244221 + 1.71475i −0.415415 + 0.909632i 0.959493 0.281733i −1.31050 + 1.13251i 3.34933 2.90221i −0.989821 + 0.142315i −2.88071 + 0.837554i 0.755750 + 0.654861i
11.13 0.540641 + 0.841254i 0.416404 1.68125i −0.415415 + 0.909632i 0.959493 0.281733i 1.63948 0.558652i 1.57600 1.36561i −0.989821 + 0.142315i −2.65322 1.40016i 0.755750 + 0.654861i
11.14 0.540641 + 0.841254i 1.36664 + 1.06409i −0.415415 + 0.909632i 0.959493 0.281733i −0.156311 + 1.72498i −3.41511 + 2.95921i −0.989821 + 0.142315i 0.735409 + 2.90847i 0.755750 + 0.654861i
11.15 0.540641 + 0.841254i 1.43420 0.971113i −0.415415 + 0.909632i 0.959493 0.281733i 1.59234 + 0.681505i −3.34609 + 2.89940i −0.989821 + 0.142315i 1.11388 2.78555i 0.755750 + 0.654861i
11.16 0.540641 + 0.841254i 1.64273 + 0.549041i −0.415415 + 0.909632i 0.959493 0.281733i 0.426243 + 1.67878i 0.918508 0.795892i −0.989821 + 0.142315i 2.39711 + 1.80385i 0.755750 + 0.654861i
191.1 −0.755750 0.654861i −1.58079 0.707884i 0.142315 + 0.989821i −0.415415 0.909632i 0.731117 + 1.57018i 0.325415 + 1.10826i 0.540641 0.841254i 1.99780 + 2.23803i −0.281733 + 0.959493i
191.2 −0.755750 0.654861i −1.30022 + 1.14430i 0.142315 + 0.989821i −0.415415 0.909632i 1.73200 0.0133460i 0.519422 + 1.76899i 0.540641 0.841254i 0.381134 2.97569i −0.281733 + 0.959493i
191.3 −0.755750 0.654861i −0.982602 + 1.42636i 0.142315 + 0.989821i −0.415415 0.909632i 1.67667 0.434500i −0.819457 2.79081i 0.540641 0.841254i −1.06898 2.80308i −0.281733 + 0.959493i
191.4 −0.755750 0.654861i −0.340196 1.69831i 0.142315 + 0.989821i −0.415415 0.909632i −0.855055 + 1.50628i 0.680617 + 2.31797i 0.540641 0.841254i −2.76853 + 1.15552i −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.b yes 160
3.b odd 2 1 690.2.q.a 160
23.d odd 22 1 690.2.q.a 160
69.g even 22 1 inner 690.2.q.b yes 160

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

## 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 $$S_{2}^{\mathrm{new}}(690, [\chi])$$.