# Properties

 Label 690.2.w.b Level $690$ Weight $2$ Character orbit 690.w Analytic conductor $5.510$ Analytic rank $0$ Dimension $240$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 690.w (of order $$44$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.50967773947$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$12$$ over $$\Q(\zeta_{44})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$240q + 24q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$240q + 24q^{6} - 44q^{10} + 24q^{16} - 44q^{21} + 96q^{23} + 16q^{25} + 16q^{26} + 44q^{28} - 16q^{31} + 44q^{33} + 16q^{35} - 24q^{36} + 44q^{37} - 88q^{43} + 8q^{46} + 96q^{47} - 24q^{50} - 24q^{55} + 44q^{57} - 16q^{58} + 88q^{61} + 56q^{62} - 88q^{65} + 132q^{67} - 56q^{70} + 16q^{71} + 48q^{73} + 24q^{81} - 24q^{82} + 44q^{85} - 16q^{87} + 44q^{88} - 124q^{92} + 32q^{93} + 20q^{95} - 24q^{96} - 56q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i −1.95274 + 1.08941i 0.142315 0.989821i 1.32241 2.42181i −0.977147 + 0.212565i −0.909632 0.415415i 1.87005 1.22594i
7.2 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i −1.89440 1.18796i 0.142315 0.989821i 0.244741 0.448210i −0.977147 + 0.212565i −0.909632 0.415415i 1.97432 + 1.04979i
7.3 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i −0.619349 2.14858i 0.142315 0.989821i −1.99507 + 3.65370i −0.977147 + 0.212565i −0.909632 0.415415i 0.771049 + 2.09892i
7.4 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i 1.62532 + 1.53569i 0.142315 0.989821i 1.83156 3.35425i −0.977147 + 0.212565i −0.909632 0.415415i −1.73074 1.41582i
7.5 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i 1.87213 1.22275i 0.142315 0.989821i −0.241465 + 0.442209i −0.977147 + 0.212565i −0.909632 0.415415i −1.78013 + 1.35319i
7.6 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i 1.98312 + 1.03307i 0.142315 0.989821i −0.653533 + 1.19686i −0.977147 + 0.212565i −0.909632 0.415415i −2.05177 0.888964i
7.7 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i −2.18918 + 0.455502i 0.142315 0.989821i 0.172948 0.316731i 0.977147 0.212565i −0.909632 0.415415i −2.15111 + 0.610516i
7.8 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i −1.40926 1.73608i 0.142315 0.989821i 1.26240 2.31192i 0.977147 0.212565i −0.909632 0.415415i −1.52952 1.63113i
7.9 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i 0.0285953 + 2.23589i 0.142315 0.989821i −1.81898 + 3.33121i 0.977147 0.212565i −0.909632 0.415415i 0.188029 + 2.22815i
7.10 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i 0.855176 + 2.06608i 0.142315 0.989821i 1.43032 2.61943i 0.977147 0.212565i −0.909632 0.415415i 1.00039 + 1.99981i
7.11 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i 1.40256 1.74150i 0.142315 0.989821i 1.87882 3.44079i 0.977147 0.212565i −0.909632 0.415415i 1.27475 1.83712i
7.12 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i 1.93237 1.12513i 0.142315 0.989821i −0.968092 + 1.77293i 0.977147 0.212565i −0.909632 0.415415i 1.84719 1.26012i
37.1 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i −2.21203 0.326962i −0.841254 0.540641i 0.672927 + 1.80419i −0.0713392 0.997452i 0.989821 + 0.142315i 1.78476 + 1.34708i
37.2 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i −1.79792 1.32947i −0.841254 0.540641i −1.02771 2.75539i −0.0713392 0.997452i 0.989821 + 0.142315i 0.940853 + 2.02850i
37.3 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i 0.217321 2.22548i −0.841254 0.540641i 0.154967 + 0.415484i −0.0713392 0.997452i 0.989821 + 0.142315i −1.25730 + 1.84911i
37.4 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i 1.06201 + 1.96778i −0.841254 0.540641i −0.702461 1.88337i −0.0713392 0.997452i 0.989821 + 0.142315i 0.0109547 2.23604i
37.5 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i 2.06085 + 0.867694i −0.841254 0.540641i −1.60657 4.30737i −0.0713392 0.997452i 0.989821 + 0.142315i −1.39292 1.74922i
37.6 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i 2.18586 0.471182i −0.841254 0.540641i 1.33355 + 3.57540i −0.0713392 0.997452i 0.989821 + 0.142315i −2.14430 0.634025i
37.7 0.877679 + 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i −2.09572 0.779713i −0.841254 0.540641i −0.448960 1.20371i 0.0713392 + 0.997452i 0.989821 + 0.142315i −1.46569 1.68871i
37.8 0.877679 + 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i −1.84527 1.26292i −0.841254 0.540641i 1.33335 + 3.57484i 0.0713392 + 0.997452i 0.989821 + 0.142315i −1.01431 1.99278i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 613.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.d odd 22 1 inner
115.l even 44 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.w.b 240
5.c odd 4 1 inner 690.2.w.b 240
23.d odd 22 1 inner 690.2.w.b 240
115.l even 44 1 inner 690.2.w.b 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.w.b 240 1.a even 1 1 trivial
690.2.w.b 240 5.c odd 4 1 inner
690.2.w.b 240 23.d odd 22 1 inner
690.2.w.b 240 115.l even 44 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$10\!\cdots\!56$$$$T_{7}^{221} -$$$$71\!\cdots\!54$$$$T_{7}^{220} +$$$$81\!\cdots\!76$$$$T_{7}^{219} +$$$$30\!\cdots\!44$$$$T_{7}^{218} -$$$$57\!\cdots\!36$$$$T_{7}^{217} +$$$$41\!\cdots\!81$$$$T_{7}^{216} -$$$$69\!\cdots\!40$$$$T_{7}^{215} -$$$$23\!\cdots\!28$$$$T_{7}^{214} +$$$$40\!\cdots\!92$$$$T_{7}^{213} -$$$$24\!\cdots\!56$$$$T_{7}^{212} +$$$$51\!\cdots\!04$$$$T_{7}^{211} +$$$$16\!\cdots\!88$$$$T_{7}^{210} -$$$$25\!\cdots\!44$$$$T_{7}^{209} +$$$$13\!\cdots\!63$$$$T_{7}^{208} -$$$$29\!\cdots\!40$$$$T_{7}^{207} -$$$$79\!\cdots\!16$$$$T_{7}^{206} +$$$$11\!\cdots\!44$$$$T_{7}^{205} -$$$$57\!\cdots\!46$$$$T_{7}^{204} +$$$$10\!\cdots\!76$$$$T_{7}^{203} +$$$$34\!\cdots\!48$$$$T_{7}^{202} -$$$$43\!\cdots\!28$$$$T_{7}^{201} +$$$$21\!\cdots\!20$$$$T_{7}^{200} -$$$$39\!\cdots\!64$$$$T_{7}^{199} -$$$$12\!\cdots\!44$$$$T_{7}^{198} +$$$$14\!\cdots\!16$$$$T_{7}^{197} -$$$$64\!\cdots\!90$$$$T_{7}^{196} +$$$$98\!\cdots\!12$$$$T_{7}^{195} +$$$$45\!\cdots\!28$$$$T_{7}^{194} -$$$$45\!\cdots\!44$$$$T_{7}^{193} +$$$$19\!\cdots\!79$$$$T_{7}^{192} -$$$$25\!\cdots\!00$$$$T_{7}^{191} -$$$$13\!\cdots\!32$$$$T_{7}^{190} +$$$$11\!\cdots\!24$$$$T_{7}^{189} -$$$$43\!\cdots\!84$$$$T_{7}^{188} +$$$$28\!\cdots\!96$$$$T_{7}^{187} +$$$$38\!\cdots\!12$$$$T_{7}^{186} -$$$$26\!\cdots\!72$$$$T_{7}^{185} +$$$$91\!\cdots\!47$$$$T_{7}^{184} -$$$$37\!\cdots\!08$$$$T_{7}^{183} -$$$$79\!\cdots\!16$$$$T_{7}^{182} +$$$$52\!\cdots\!92$$$$T_{7}^{181} -$$$$17\!\cdots\!28$$$$T_{7}^{180} -$$$$67\!\cdots\!08$$$$T_{7}^{179} +$$$$17\!\cdots\!68$$$$T_{7}^{178} -$$$$90\!\cdots\!52$$$$T_{7}^{177} +$$$$24\!\cdots\!58$$$$T_{7}^{176} +$$$$31\!\cdots\!76$$$$T_{7}^{175} -$$$$26\!\cdots\!20$$$$T_{7}^{174} +$$$$14\!\cdots\!44$$$$T_{7}^{173} -$$$$37\!\cdots\!31$$$$T_{7}^{172} -$$$$77\!\cdots\!32$$$$T_{7}^{171} +$$$$46\!\cdots\!36$$$$T_{7}^{170} -$$$$16\!\cdots\!52$$$$T_{7}^{169} +$$$$34\!\cdots\!93$$$$T_{7}^{168} +$$$$17\!\cdots\!36$$$$T_{7}^{167} -$$$$41\!\cdots\!04$$$$T_{7}^{166} +$$$$21\!\cdots\!52$$$$T_{7}^{165} -$$$$15\!\cdots\!17$$$$T_{7}^{164} -$$$$24\!\cdots\!48$$$$T_{7}^{163} +$$$$46\!\cdots\!08$$$$T_{7}^{162} -$$$$37\!\cdots\!52$$$$T_{7}^{161} +$$$$22\!\cdots\!91$$$$T_{7}^{160} +$$$$27\!\cdots\!04$$$$T_{7}^{159} +$$$$15\!\cdots\!20$$$$T_{7}^{158} +$$$$23\!\cdots\!24$$$$T_{7}^{157} +$$$$76\!\cdots\!80$$$$T_{7}^{156} -$$$$15\!\cdots\!28$$$$T_{7}^{155} +$$$$47\!\cdots\!32$$$$T_{7}^{154} +$$$$84\!\cdots\!84$$$$T_{7}^{153} +$$$$21\!\cdots\!66$$$$T_{7}^{152} +$$$$12\!\cdots\!64$$$$T_{7}^{151} +$$$$32\!\cdots\!16$$$$T_{7}^{150} +$$$$14\!\cdots\!92$$$$T_{7}^{149} +$$$$72\!\cdots\!24$$$$T_{7}^{148} +$$$$32\!\cdots\!16$$$$T_{7}^{147} -$$$$11\!\cdots\!20$$$$T_{7}^{146} +$$$$31\!\cdots\!72$$$$T_{7}^{145} +$$$$31\!\cdots\!15$$$$T_{7}^{144} +$$$$76\!\cdots\!48$$$$T_{7}^{143} +$$$$17\!\cdots\!16$$$$T_{7}^{142} -$$$$79\!\cdots\!76$$$$T_{7}^{141} -$$$$35\!\cdots\!78$$$$T_{7}^{140} -$$$$15\!\cdots\!88$$$$T_{7}^{139} -$$$$54\!\cdots\!60$$$$T_{7}^{138} -$$$$12\!\cdots\!12$$$$T_{7}^{137} +$$$$74\!\cdots\!03$$$$T_{7}^{136} +$$$$13\!\cdots\!96$$$$T_{7}^{135} +$$$$62\!\cdots\!00$$$$T_{7}^{134} +$$$$17\!\cdots\!92$$$$T_{7}^{133} +$$$$32\!\cdots\!65$$$$T_{7}^{132} +$$$$23\!\cdots\!28$$$$T_{7}^{131} -$$$$11\!\cdots\!84$$$$T_{7}^{130} -$$$$75\!\cdots\!08$$$$T_{7}^{129} -$$$$27\!\cdots\!45$$$$T_{7}^{128} -$$$$66\!\cdots\!12$$$$T_{7}^{127} -$$$$88\!\cdots\!04$$$$T_{7}^{126} +$$$$30\!\cdots\!24$$$$T_{7}^{125} +$$$$43\!\cdots\!04$$$$T_{7}^{124} +$$$$11\!\cdots\!16$$$$T_{7}^{123} +$$$$24\!\cdots\!80$$$$T_{7}^{122} +$$$$58\!\cdots\!52$$$$T_{7}^{121} +$$$$13\!\cdots\!82$$$$T_{7}^{120} +$$$$18\!\cdots\!48$$$$T_{7}^{119} +$$$$19\!\cdots\!36$$$$T_{7}^{118} +$$$$11\!\cdots\!84$$$$T_{7}^{117} +$$$$59\!\cdots\!30$$$$T_{7}^{116} +$$$$13\!\cdots\!28$$$$T_{7}^{115} +$$$$92\!\cdots\!44$$$$T_{7}^{114} -$$$$29\!\cdots\!84$$$$T_{7}^{113} -$$$$40\!\cdots\!18$$$$T_{7}^{112} +$$$$12\!\cdots\!40$$$$T_{7}^{111} +$$$$40\!\cdots\!52$$$$T_{7}^{110} +$$$$11\!\cdots\!28$$$$T_{7}^{109} -$$$$85\!\cdots\!42$$$$T_{7}^{108} -$$$$99\!\cdots\!20$$$$T_{7}^{107} +$$$$60\!\cdots\!68$$$$T_{7}^{106} +$$$$86\!\cdots\!48$$$$T_{7}^{105} -$$$$12\!\cdots\!75$$$$T_{7}^{104} -$$$$60\!\cdots\!00$$$$T_{7}^{103} -$$$$54\!\cdots\!64$$$$T_{7}^{102} +$$$$89\!\cdots\!76$$$$T_{7}^{101} +$$$$39\!\cdots\!11$$$$T_{7}^{100} +$$$$60\!\cdots\!40$$$$T_{7}^{99} +$$$$44\!\cdots\!56$$$$T_{7}^{98} -$$$$12\!\cdots\!76$$$$T_{7}^{97} -$$$$58\!\cdots\!43$$$$T_{7}^{96} -$$$$11\!\cdots\!76$$$$T_{7}^{95} -$$$$31\!\cdots\!32$$$$T_{7}^{94} +$$$$35\!\cdots\!28$$$$T_{7}^{93} +$$$$85\!\cdots\!54$$$$T_{7}^{92} +$$$$47\!\cdots\!80$$$$T_{7}^{91} -$$$$14\!\cdots\!16$$$$T_{7}^{90} -$$$$29\!\cdots\!72$$$$T_{7}^{89} +$$$$91\!\cdots\!80$$$$T_{7}^{88} +$$$$93\!\cdots\!68$$$$T_{7}^{87} +$$$$60\!\cdots\!84$$$$T_{7}^{86} -$$$$26\!\cdots\!16$$$$T_{7}^{85} -$$$$52\!\cdots\!43$$$$T_{7}^{84} +$$$$30\!\cdots\!48$$$$T_{7}^{83} +$$$$22\!\cdots\!72$$$$T_{7}^{82} +$$$$15\!\cdots\!88$$$$T_{7}^{81} -$$$$79\!\cdots\!93$$$$T_{7}^{80} -$$$$22\!\cdots\!52$$$$T_{7}^{79} -$$$$15\!\cdots\!72$$$$T_{7}^{78} +$$$$49\!\cdots\!52$$$$T_{7}^{77} +$$$$16\!\cdots\!30$$$$T_{7}^{76} +$$$$18\!\cdots\!96$$$$T_{7}^{75} -$$$$53\!\cdots\!76$$$$T_{7}^{74} -$$$$59\!\cdots\!60$$$$T_{7}^{73} -$$$$10\!\cdots\!29$$$$T_{7}^{72} -$$$$62\!\cdots\!84$$$$T_{7}^{71} +$$$$10\!\cdots\!28$$$$T_{7}^{70} +$$$$30\!\cdots\!12$$$$T_{7}^{69} +$$$$31\!\cdots\!00$$$$T_{7}^{68} -$$$$37\!\cdots\!20$$$$T_{7}^{67} -$$$$56\!\cdots\!44$$$$T_{7}^{66} -$$$$76\!\cdots\!24$$$$T_{7}^{65} -$$$$21\!\cdots\!82$$$$T_{7}^{64} +$$$$87\!\cdots\!28$$$$T_{7}^{63} +$$$$17\!\cdots\!40$$$$T_{7}^{62} +$$$$15\!\cdots\!84$$$$T_{7}^{61} +$$$$53\!\cdots\!53$$$$T_{7}^{60} -$$$$55\!\cdots\!40$$$$T_{7}^{59} -$$$$96\!\cdots\!32$$$$T_{7}^{58} -$$$$67\!\cdots\!88$$$$T_{7}^{57} -$$$$20\!\cdots\!36$$$$T_{7}^{56} +$$$$78\!\cdots\!16$$$$T_{7}^{55} -$$$$50\!\cdots\!44$$$$T_{7}^{54} -$$$$11\!\cdots\!20$$$$T_{7}^{53} -$$$$41\!\cdots\!35$$$$T_{7}^{52} +$$$$77\!\cdots\!88$$$$T_{7}^{51} +$$$$11\!\cdots\!16$$$$T_{7}^{50} +$$$$70\!\cdots\!20$$$$T_{7}^{49} +$$$$12\!\cdots\!05$$$$T_{7}^{48} +$$$$17\!\cdots\!32$$$$T_{7}^{47} +$$$$18\!\cdots\!80$$$$T_{7}^{46} +$$$$30\!\cdots\!56$$$$T_{7}^{45} +$$$$10\!\cdots\!58$$$$T_{7}^{44} -$$$$20\!\cdots\!76$$$$T_{7}^{43} -$$$$19\!\cdots\!36$$$$T_{7}^{42} -$$$$18\!\cdots\!32$$$$T_{7}^{41} -$$$$54\!\cdots\!35$$$$T_{7}^{40} +$$$$65\!\cdots\!76$$$$T_{7}^{39} +$$$$22\!\cdots\!20$$$$T_{7}^{38} +$$$$37\!\cdots\!00$$$$T_{7}^{37} +$$$$18\!\cdots\!88$$$$T_{7}^{36} -$$$$45\!\cdots\!68$$$$T_{7}^{35} +$$$$40\!\cdots\!76$$$$T_{7}^{34} -$$$$17\!\cdots\!20$$$$T_{7}^{33} -$$$$16\!\cdots\!76$$$$T_{7}^{32} +$$$$99\!\cdots\!44$$$$T_{7}^{31} -$$$$71\!\cdots\!16$$$$T_{7}^{30} +$$$$43\!\cdots\!64$$$$T_{7}^{29} +$$$$15\!\cdots\!72$$$$T_{7}^{28} -$$$$21\!\cdots\!64$$$$T_{7}^{27} -$$$$71\!\cdots\!60$$$$T_{7}^{26} -$$$$73\!\cdots\!64$$$$T_{7}^{25} -$$$$54\!\cdots\!48$$$$T_{7}^{24} +$$$$17\!\cdots\!44$$$$T_{7}^{23} -$$$$11\!\cdots\!96$$$$T_{7}^{22} +$$$$17\!\cdots\!04$$$$T_{7}^{21} +$$$$58\!\cdots\!48$$$$T_{7}^{20} -$$$$20\!\cdots\!96$$$$T_{7}^{19} +$$$$25\!\cdots\!24$$$$T_{7}^{18} +$$$$21\!\cdots\!40$$$$T_{7}^{17} -$$$$19\!\cdots\!96$$$$T_{7}^{16} -$$$$31\!\cdots\!64$$$$T_{7}^{15} +$$$$16\!\cdots\!76$$$$T_{7}^{14} -$$$$68\!\cdots\!92$$$$T_{7}^{13} +$$$$11\!\cdots\!88$$$$T_{7}^{12} +$$$$98\!\cdots\!32$$$$T_{7}^{11} -$$$$53\!\cdots\!56$$$$T_{7}^{10} +$$$$44\!\cdots\!56$$$$T_{7}^{9} +$$$$16\!\cdots\!08$$$$T_{7}^{8} -$$$$18\!\cdots\!00$$$$T_{7}^{7} +$$$$90\!\cdots\!68$$$$T_{7}^{6} -$$$$35\!\cdots\!44$$$$T_{7}^{5} +$$$$11\!\cdots\!48$$$$T_{7}^{4} -$$$$27\!\cdots\!16$$$$T_{7}^{3} +$$$$51\!\cdots\!28$$$$T_{7}^{2} -$$$$75\!\cdots\!76$$$$T_{7} +$$$$55\!\cdots\!96$$">$$T_{7}^{240} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(690, [\chi])$$.