# Properties

 Label 690.2.n.b 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.

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

## 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 −1.73163 0.0381696i −0.959493 + 0.281733i −2.23565 + 0.0433117i −0.208656 1.71944i 1.95813 1.25842i −0.415415 0.909632i 2.99709 + 0.132191i −0.361037 2.20673i
89.2 0.142315 + 0.989821i −1.67640 0.435517i −0.959493 + 0.281733i 0.584056 + 2.15844i 0.192507 1.72132i −1.28454 + 0.825522i −0.415415 0.909632i 2.62065 + 1.46020i −2.05335 + 0.885290i
89.3 0.142315 + 0.989821i −1.67505 + 0.440696i −0.959493 + 0.281733i 1.25961 + 1.84754i −0.674594 1.59528i 3.27194 2.10275i −0.415415 0.909632i 2.61157 1.47637i −1.64947 + 1.50972i
89.4 0.142315 + 0.989821i −1.56929 0.733032i −0.959493 + 0.281733i 1.76486 1.37305i 0.502238 1.65764i −3.04110 + 1.95439i −0.415415 0.909632i 1.92533 + 2.30068i 1.61024 + 1.55149i
89.5 0.142315 + 0.989821i −1.53419 + 0.803900i −0.959493 + 0.281733i −2.02447 0.949486i −1.01406 1.40417i −4.08214 + 2.62343i −0.415415 0.909632i 1.70749 2.46667i 0.651709 2.13899i
89.6 0.142315 + 0.989821i −1.27489 + 1.17245i −0.959493 + 0.281733i −0.182307 2.22862i −1.34195 1.09506i 1.96759 1.26449i −0.415415 0.909632i 0.250714 2.98951i 2.17999 0.497618i
89.7 0.142315 + 0.989821i −1.11013 + 1.32951i −0.959493 + 0.281733i 1.79300 1.33609i −1.47397 0.909626i −0.483302 + 0.310599i −0.415415 0.909632i −0.535201 2.95187i 1.57766 + 1.58461i
89.8 0.142315 + 0.989821i −0.804725 1.53376i −0.959493 + 0.281733i −2.06421 0.859668i 1.40362 1.01481i 0.955977 0.614369i −0.415415 0.909632i −1.70483 + 2.46851i 0.557150 2.16554i
89.9 0.142315 + 0.989821i −0.748393 1.56202i −0.959493 + 0.281733i −1.26417 + 1.84441i 1.43961 0.963074i −1.66558 + 1.07040i −0.415415 0.909632i −1.87982 + 2.33801i −2.00555 0.988819i
89.10 0.142315 + 0.989821i −0.730817 1.57032i −0.959493 + 0.281733i 2.23496 0.0702463i 1.45033 0.946859i 3.21136 2.06381i −0.415415 0.909632i −1.93181 + 2.29523i 0.387600 + 2.20222i
89.11 0.142315 + 0.989821i −0.277794 + 1.70963i −0.959493 + 0.281733i −1.06732 + 1.96490i −1.73176 0.0316608i 0.483302 0.310599i −0.415415 0.909632i −2.84566 0.949849i −2.09680 0.776821i
89.12 0.142315 + 0.989821i −0.0575173 1.73110i −0.959493 + 0.281733i 2.02703 + 0.944001i 1.70529 0.303292i −1.08447 + 0.696948i −0.415415 0.909632i −2.99338 + 0.199136i −0.645916 + 2.14075i
89.13 0.142315 + 0.989821i −0.0512012 + 1.73129i −0.959493 + 0.281733i −2.23188 + 0.136714i −1.72096 + 0.195709i −1.96759 + 1.26449i −0.415415 0.909632i −2.99476 0.177289i −0.452953 2.18971i
89.14 0.142315 + 0.989821i 0.397135 + 1.68591i −0.959493 + 0.281733i −1.22793 1.86874i −1.61223 + 0.633022i 4.08214 2.62343i −0.415415 0.909632i −2.68457 + 1.33906i 1.67496 1.48138i
89.15 0.142315 + 0.989821i 0.479764 1.66428i −0.959493 + 0.281733i 0.0269160 2.23591i 1.71562 + 0.238029i −2.74772 + 1.76585i −0.415415 0.909632i −2.53965 1.59692i 2.21698 0.291561i
89.16 0.142315 + 0.989821i 0.763868 + 1.55451i −0.959493 + 0.281733i 2.00799 + 0.983855i −1.42998 + 0.977323i −3.27194 + 2.10275i −0.415415 0.909632i −1.83301 + 2.37488i −0.688074 + 2.12757i
89.17 0.142315 + 0.989821i 0.943600 1.45245i −0.959493 + 0.281733i −2.20932 + 0.344845i 1.57196 + 0.727290i 2.74772 1.76585i −0.415415 0.909632i −1.21924 2.74107i −0.655753 2.13775i
89.18 0.142315 + 0.989821i 1.16282 + 1.28368i −0.959493 + 0.281733i −0.275295 2.21906i −1.10513 + 1.33367i −1.95813 + 1.25842i −0.415415 0.909632i −0.295684 + 2.98539i 2.15729 0.588298i
89.19 0.142315 + 0.989821i 1.34594 1.09016i −0.959493 + 0.281733i 1.22287 + 1.87206i 1.27061 + 1.17710i 1.08447 0.696948i −0.415415 0.909632i 0.623112 2.93458i −1.67897 + 1.47684i
89.20 0.142315 + 0.989821i 1.42695 + 0.981738i −0.959493 + 0.281733i 2.21959 + 0.270933i −0.768669 + 1.55214i 1.28454 0.825522i −0.415415 0.909632i 1.07238 + 2.80178i 0.0477058 + 2.23556i
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.b yes 240
3.b odd 2 1 690.2.n.a 240
5.b even 2 1 690.2.n.a 240
15.d odd 2 1 inner 690.2.n.b yes 240
23.d odd 22 1 inner 690.2.n.b yes 240
69.g even 22 1 690.2.n.a 240
115.i odd 22 1 690.2.n.a 240
345.n even 22 1 inner 690.2.n.b yes 240

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

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