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