# Properties

 Label 690.2.n.a Level $690$ Weight $2$ Character orbit 690.n 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.n (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.50967773947$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$240q - 24q^{2} + 2q^{3} - 24q^{4} + 2q^{6} - 24q^{8} - 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$240q - 24q^{2} + 2q^{3} - 24q^{4} + 2q^{6} - 24q^{8} - 6q^{9} - 9q^{12} + 11q^{15} - 24q^{16} - 6q^{18} - 4q^{23} + 2q^{24} - 12q^{25} + 2q^{27} + 22q^{30} + 28q^{31} - 24q^{32} - 36q^{35} - 6q^{36} - 4q^{46} + 104q^{47} - 9q^{48} + 70q^{49} + 54q^{50} - 9q^{54} - 26q^{55} - 44q^{57} - 11q^{60} + 44q^{61} + 28q^{62} - 121q^{63} - 24q^{64} + 44q^{65} + 44q^{66} - 102q^{69} - 36q^{70} + 16q^{72} - 82q^{75} + 8q^{77} - 44q^{79} + 74q^{81} - 11q^{84} + 22q^{85} - 93q^{87} - 4q^{92} + 172q^{93} + 16q^{94} + 26q^{95} + 2q^{96} + 4q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1 −0.142315 0.989821i −1.68612 + 0.396228i −0.959493 + 0.281733i 2.06421 + 0.859668i 0.632155 + 1.61257i 0.955977 0.614369i 0.415415 + 0.909632i 2.68601 1.33618i 0.557150 2.16554i
89.2 −0.142315 0.989821i −1.67059 + 0.457309i −0.959493 + 0.281733i 1.26417 1.84441i 0.690404 + 1.58850i −1.66558 + 1.07040i 0.415415 + 0.909632i 2.58174 1.52795i −2.00555 0.988819i
89.3 −0.142315 0.989821i −1.66535 + 0.476026i −0.959493 + 0.281733i −2.23496 + 0.0702463i 0.708185 + 1.58066i 3.21136 2.06381i 0.415415 + 0.909632i 2.54680 1.58550i 0.387600 + 2.20222i
89.4 −0.142315 0.989821i −1.58165 0.705954i −0.959493 + 0.281733i −1.76486 + 1.37305i −0.473676 + 1.66602i −3.04110 + 1.95439i 0.415415 + 0.909632i 2.00326 + 2.23315i 1.61024 + 1.55149i
89.5 −0.142315 0.989821i −1.42695 0.981738i −0.959493 + 0.281733i −0.584056 2.15844i −0.768669 + 1.55214i −1.28454 + 0.825522i 0.415415 + 0.909632i 1.07238 + 2.80178i −2.05335 + 0.885290i
89.6 −0.142315 0.989821i −1.34594 + 1.09016i −0.959493 + 0.281733i −2.02703 0.944001i 1.27061 + 1.17710i −1.08447 + 0.696948i 0.415415 + 0.909632i 0.623112 2.93458i −0.645916 + 2.14075i
89.7 −0.142315 0.989821i −1.16282 1.28368i −0.959493 + 0.281733i 2.23565 0.0433117i −1.10513 + 1.33367i 1.95813 1.25842i 0.415415 + 0.909632i −0.295684 + 2.98539i −0.361037 2.20673i
89.8 −0.142315 0.989821i −0.943600 + 1.45245i −0.959493 + 0.281733i −0.0269160 + 2.23591i 1.57196 + 0.727290i −2.74772 + 1.76585i 0.415415 + 0.909632i −1.21924 2.74107i 2.21698 0.291561i
89.9 −0.142315 0.989821i −0.763868 1.55451i −0.959493 + 0.281733i −1.25961 1.84754i −1.42998 + 0.977323i 3.27194 2.10275i 0.415415 + 0.909632i −1.83301 + 2.37488i −1.64947 + 1.50972i
89.10 −0.142315 0.989821i −0.479764 + 1.66428i −0.959493 + 0.281733i 2.20932 0.344845i 1.71562 + 0.238029i 2.74772 1.76585i 0.415415 + 0.909632i −2.53965 1.59692i −0.655753 2.13775i
89.11 −0.142315 0.989821i −0.397135 1.68591i −0.959493 + 0.281733i 2.02447 + 0.949486i −1.61223 + 0.633022i −4.08214 + 2.62343i 0.415415 + 0.909632i −2.68457 + 1.33906i 0.651709 2.13899i
89.12 −0.142315 0.989821i 0.0512012 1.73129i −0.959493 + 0.281733i 0.182307 + 2.22862i −1.72096 + 0.195709i 1.96759 1.26449i 0.415415 + 0.909632i −2.99476 0.177289i 2.17999 0.497618i
89.13 −0.142315 0.989821i 0.0575173 + 1.73110i −0.959493 + 0.281733i −1.22287 1.87206i 1.70529 0.303292i 1.08447 0.696948i 0.415415 + 0.909632i −2.99338 + 0.199136i −1.67897 + 1.47684i
89.14 −0.142315 0.989821i 0.277794 1.70963i −0.959493 + 0.281733i −1.79300 + 1.33609i −1.73176 0.0316608i −0.483302 + 0.310599i 0.415415 + 0.909632i −2.84566 0.949849i 1.57766 + 1.58461i
89.15 −0.142315 0.989821i 0.730817 + 1.57032i −0.959493 + 0.281733i −0.248537 2.22221i 1.45033 0.946859i −3.21136 + 2.06381i 0.415415 + 0.909632i −1.93181 + 2.29523i −2.16422 + 0.562261i
89.16 −0.142315 0.989821i 0.748393 + 1.56202i −0.959493 + 0.281733i −1.64573 + 1.51379i 1.43961 0.963074i 1.66558 1.07040i 0.415415 + 0.909632i −1.87982 + 2.33801i 1.73260 + 1.41354i
89.17 −0.142315 0.989821i 0.804725 + 1.53376i −0.959493 + 0.281733i 1.14469 + 1.92086i 1.40362 1.01481i −0.955977 + 0.614369i 0.415415 + 0.909632i −1.70483 + 2.46851i 1.73840 1.40640i
89.18 −0.142315 0.989821i 1.11013 1.32951i −0.959493 + 0.281733i 1.06732 1.96490i −1.47397 0.909626i 0.483302 0.310599i 0.415415 + 0.909632i −0.535201 2.95187i −2.09680 0.776821i
89.19 −0.142315 0.989821i 1.27489 1.17245i −0.959493 + 0.281733i 2.23188 0.136714i −1.34195 1.09506i −1.96759 + 1.26449i 0.415415 + 0.909632i 0.250714 2.98951i −0.452953 2.18971i
89.20 −0.142315 0.989821i 1.53419 0.803900i −0.959493 + 0.281733i 1.22793 + 1.86874i −1.01406 1.40417i 4.08214 2.62343i 0.415415 + 0.909632i 1.70749 2.46667i 1.67496 1.48138i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 659.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
23.d odd 22 1 inner
345.n 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.n.a 240
3.b odd 2 1 690.2.n.b yes 240
5.b even 2 1 690.2.n.b yes 240
15.d odd 2 1 inner 690.2.n.a 240
23.d odd 22 1 inner 690.2.n.a 240
69.g even 22 1 690.2.n.b yes 240
115.i odd 22 1 690.2.n.b yes 240
345.n even 22 1 inner 690.2.n.a 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.n.a 240 1.a even 1 1 trivial
690.2.n.a 240 15.d odd 2 1 inner
690.2.n.a 240 23.d odd 22 1 inner
690.2.n.a 240 345.n even 22 1 inner
690.2.n.b yes 240 3.b odd 2 1
690.2.n.b yes 240 5.b even 2 1
690.2.n.b yes 240 69.g even 22 1
690.2.n.b yes 240 115.i odd 22 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$19\!\cdots\!51$$$$T_{17}^{106} +$$$$15\!\cdots\!27$$$$T_{17}^{105} +$$$$96\!\cdots\!76$$$$T_{17}^{104} -$$$$61\!\cdots\!89$$$$T_{17}^{103} -$$$$51\!\cdots\!64$$$$T_{17}^{102} +$$$$18\!\cdots\!44$$$$T_{17}^{101} +$$$$29\!\cdots\!06$$$$T_{17}^{100} -$$$$94\!\cdots\!68$$$$T_{17}^{99} -$$$$99\!\cdots\!68$$$$T_{17}^{98} +$$$$27\!\cdots\!78$$$$T_{17}^{97} +$$$$33\!\cdots\!44$$$$T_{17}^{96} -$$$$59\!\cdots\!28$$$$T_{17}^{95} -$$$$12\!\cdots\!73$$$$T_{17}^{94} +$$$$15\!\cdots\!25$$$$T_{17}^{93} +$$$$45\!\cdots\!91$$$$T_{17}^{92} -$$$$11\!\cdots\!98$$$$T_{17}^{91} -$$$$97\!\cdots\!90$$$$T_{17}^{90} +$$$$22\!\cdots\!98$$$$T_{17}^{89} +$$$$23\!\cdots\!08$$$$T_{17}^{88} +$$$$16\!\cdots\!86$$$$T_{17}^{87} -$$$$12\!\cdots\!08$$$$T_{17}^{86} +$$$$14\!\cdots\!88$$$$T_{17}^{85} +$$$$39\!\cdots\!78$$$$T_{17}^{84} -$$$$12\!\cdots\!03$$$$T_{17}^{83} -$$$$74\!\cdots\!42$$$$T_{17}^{82} +$$$$40\!\cdots\!52$$$$T_{17}^{81} +$$$$98\!\cdots\!41$$$$T_{17}^{80} -$$$$11\!\cdots\!36$$$$T_{17}^{79} +$$$$24\!\cdots\!16$$$$T_{17}^{78} -$$$$23\!\cdots\!29$$$$T_{17}^{77} +$$$$77\!\cdots\!13$$$$T_{17}^{76} +$$$$16\!\cdots\!07$$$$T_{17}^{75} -$$$$14\!\cdots\!89$$$$T_{17}^{74} -$$$$17\!\cdots\!30$$$$T_{17}^{73} +$$$$60\!\cdots\!76$$$$T_{17}^{72} -$$$$28\!\cdots\!25$$$$T_{17}^{71} +$$$$39\!\cdots\!24$$$$T_{17}^{70} +$$$$11\!\cdots\!49$$$$T_{17}^{69} -$$$$72\!\cdots\!25$$$$T_{17}^{68} +$$$$28\!\cdots\!13$$$$T_{17}^{67} -$$$$12\!\cdots\!96$$$$T_{17}^{66} +$$$$34\!\cdots\!22$$$$T_{17}^{65} -$$$$47\!\cdots\!65$$$$T_{17}^{64} +$$$$35\!\cdots\!04$$$$T_{17}^{63} -$$$$31\!\cdots\!09$$$$T_{17}^{62} +$$$$72\!\cdots\!50$$$$T_{17}^{61} +$$$$37\!\cdots\!82$$$$T_{17}^{60} -$$$$31\!\cdots\!72$$$$T_{17}^{59} +$$$$10\!\cdots\!91$$$$T_{17}^{58} -$$$$24\!\cdots\!42$$$$T_{17}^{57} +$$$$43\!\cdots\!32$$$$T_{17}^{56} -$$$$24\!\cdots\!33$$$$T_{17}^{55} -$$$$55\!\cdots\!95$$$$T_{17}^{54} +$$$$39\!\cdots\!78$$$$T_{17}^{53} -$$$$15\!\cdots\!93$$$$T_{17}^{52} +$$$$36\!\cdots\!39$$$$T_{17}^{51} -$$$$56\!\cdots\!82$$$$T_{17}^{50} +$$$$10\!\cdots\!15$$$$T_{17}^{49} +$$$$38\!\cdots\!96$$$$T_{17}^{48} -$$$$22\!\cdots\!78$$$$T_{17}^{47} +$$$$86\!\cdots\!97$$$$T_{17}^{46} -$$$$26\!\cdots\!57$$$$T_{17}^{45} +$$$$68\!\cdots\!57$$$$T_{17}^{44} -$$$$15\!\cdots\!98$$$$T_{17}^{43} +$$$$35\!\cdots\!29$$$$T_{17}^{42} -$$$$76\!\cdots\!92$$$$T_{17}^{41} +$$$$15\!\cdots\!72$$$$T_{17}^{40} -$$$$31\!\cdots\!57$$$$T_{17}^{39} +$$$$60\!\cdots\!65$$$$T_{17}^{38} -$$$$10\!\cdots\!58$$$$T_{17}^{37} +$$$$17\!\cdots\!35$$$$T_{17}^{36} -$$$$28\!\cdots\!32$$$$T_{17}^{35} +$$$$52\!\cdots\!97$$$$T_{17}^{34} -$$$$10\!\cdots\!21$$$$T_{17}^{33} +$$$$20\!\cdots\!63$$$$T_{17}^{32} -$$$$33\!\cdots\!42$$$$T_{17}^{31} +$$$$44\!\cdots\!93$$$$T_{17}^{30} -$$$$51\!\cdots\!10$$$$T_{17}^{29} +$$$$55\!\cdots\!10$$$$T_{17}^{28} -$$$$59\!\cdots\!76$$$$T_{17}^{27} +$$$$58\!\cdots\!68$$$$T_{17}^{26} -$$$$47\!\cdots\!60$$$$T_{17}^{25} +$$$$36\!\cdots\!52$$$$T_{17}^{24} -$$$$39\!\cdots\!08$$$$T_{17}^{23} +$$$$51\!\cdots\!60$$$$T_{17}^{22} -$$$$54\!\cdots\!52$$$$T_{17}^{21} +$$$$43\!\cdots\!64$$$$T_{17}^{20} -$$$$34\!\cdots\!20$$$$T_{17}^{19} +$$$$34\!\cdots\!72$$$$T_{17}^{18} -$$$$27\!\cdots\!84$$$$T_{17}^{17} +$$$$11\!\cdots\!92$$$$T_{17}^{16} +$$$$17\!\cdots\!60$$$$T_{17}^{15} -$$$$26\!\cdots\!76$$$$T_{17}^{14} -$$$$81\!\cdots\!08$$$$T_{17}^{13} +$$$$12\!\cdots\!00$$$$T_{17}^{12} -$$$$52\!\cdots\!96$$$$T_{17}^{11} -$$$$19\!\cdots\!84$$$$T_{17}^{10} +$$$$69\!\cdots\!88$$$$T_{17}^{9} +$$$$20\!\cdots\!92$$$$T_{17}^{8} -$$$$11\!\cdots\!16$$$$T_{17}^{7} +$$$$76\!\cdots\!36$$$$T_{17}^{6} +$$$$36\!\cdots\!88$$$$T_{17}^{5} -$$$$12\!\cdots\!12$$$$T_{17}^{4} -$$$$16\!\cdots\!44$$$$T_{17}^{3} +$$$$57\!\cdots\!72$$$$T_{17}^{2} -$$$$39\!\cdots\!56$$$$T_{17} +$$$$23\!\cdots\!84$$">$$T_{17}^{120} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(690, [\chi])$$.