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 $4$

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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:

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