Newspace parameters
Level: | \( N \) | \(=\) | \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 966.s (of order \(22\), degree \(10\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.71354883526\) |
Analytic rank: | \(0\) |
Dimension: | \(160\) |
Relative dimension: | \(16\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
97.1 | −0.959493 | − | 0.281733i | −0.755750 | − | 0.654861i | 0.841254 | + | 0.540641i | −0.480377 | + | 3.34109i | 0.540641 | + | 0.841254i | −0.131654 | + | 2.64247i | −0.654861 | − | 0.755750i | 0.142315 | + | 0.989821i | 1.40221 | − | 3.07042i |
97.2 | −0.959493 | − | 0.281733i | −0.755750 | − | 0.654861i | 0.841254 | + | 0.540641i | −0.341716 | + | 2.37669i | 0.540641 | + | 0.841254i | −0.0945662 | − | 2.64406i | −0.654861 | − | 0.755750i | 0.142315 | + | 0.989821i | 0.997464 | − | 2.18414i |
97.3 | −0.959493 | − | 0.281733i | −0.755750 | − | 0.654861i | 0.841254 | + | 0.540641i | −0.288108 | + | 2.00384i | 0.540641 | + | 0.841254i | 2.63618 | + | 0.224848i | −0.654861 | − | 0.755750i | 0.142315 | + | 0.989821i | 0.840984 | − | 1.84150i |
97.4 | −0.959493 | − | 0.281733i | −0.755750 | − | 0.654861i | 0.841254 | + | 0.540641i | 0.0153446 | − | 0.106724i | 0.540641 | + | 0.841254i | −1.90691 | − | 1.83403i | −0.654861 | − | 0.755750i | 0.142315 | + | 0.989821i | −0.0447907 | + | 0.0980779i |
97.5 | −0.959493 | − | 0.281733i | −0.755750 | − | 0.654861i | 0.841254 | + | 0.540641i | 0.0416256 | − | 0.289512i | 0.540641 | + | 0.841254i | −0.445694 | + | 2.60794i | −0.654861 | − | 0.755750i | 0.142315 | + | 0.989821i | −0.121504 | + | 0.266058i |
97.6 | −0.959493 | − | 0.281733i | −0.755750 | − | 0.654861i | 0.841254 | + | 0.540641i | 0.126102 | − | 0.877058i | 0.540641 | + | 0.841254i | 2.30089 | − | 1.30610i | −0.654861 | − | 0.755750i | 0.142315 | + | 0.989821i | −0.368090 | + | 0.806004i |
97.7 | −0.959493 | − | 0.281733i | −0.755750 | − | 0.654861i | 0.841254 | + | 0.540641i | 0.286709 | − | 1.99411i | 0.540641 | + | 0.841254i | −2.61261 | − | 0.417427i | −0.654861 | − | 0.755750i | 0.142315 | + | 0.989821i | −0.836900 | + | 1.83256i |
97.8 | −0.959493 | − | 0.281733i | −0.755750 | − | 0.654861i | 0.841254 | + | 0.540641i | 0.486537 | − | 3.38394i | 0.540641 | + | 0.841254i | 2.17047 | + | 1.51296i | −0.654861 | − | 0.755750i | 0.142315 | + | 0.989821i | −1.42020 | + | 3.10979i |
97.9 | −0.959493 | − | 0.281733i | 0.755750 | + | 0.654861i | 0.841254 | + | 0.540641i | −0.486537 | + | 3.38394i | −0.540641 | − | 0.841254i | 0.277936 | − | 2.63111i | −0.654861 | − | 0.755750i | 0.142315 | + | 0.989821i | 1.42020 | − | 3.10979i |
97.10 | −0.959493 | − | 0.281733i | 0.755750 | + | 0.654861i | 0.841254 | + | 0.540641i | −0.286709 | + | 1.99411i | −0.540641 | − | 0.841254i | −1.39543 | + | 2.24784i | −0.654861 | − | 0.755750i | 0.142315 | + | 0.989821i | 0.836900 | − | 1.83256i |
97.11 | −0.959493 | − | 0.281733i | 0.755750 | + | 0.654861i | 0.841254 | + | 0.540641i | −0.126102 | + | 0.877058i | −0.540641 | − | 0.841254i | 2.49385 | − | 0.883586i | −0.654861 | − | 0.755750i | 0.142315 | + | 0.989821i | 0.368090 | − | 0.806004i |
97.12 | −0.959493 | − | 0.281733i | 0.755750 | + | 0.654861i | 0.841254 | + | 0.540641i | −0.0416256 | + | 0.289512i | −0.540641 | − | 0.841254i | −2.26282 | − | 1.37100i | −0.654861 | − | 0.755750i | 0.142315 | + | 0.989821i | 0.121504 | − | 0.266058i |
97.13 | −0.959493 | − | 0.281733i | 0.755750 | + | 0.654861i | 0.841254 | + | 0.540641i | −0.0153446 | + | 0.106724i | −0.540641 | − | 0.841254i | 0.137306 | + | 2.64219i | −0.654861 | − | 0.755750i | 0.142315 | + | 0.989821i | 0.0447907 | − | 0.0980779i |
97.14 | −0.959493 | − | 0.281733i | 0.755750 | + | 0.654861i | 0.841254 | + | 0.540641i | 0.288108 | − | 2.00384i | −0.540641 | − | 0.841254i | 1.55640 | − | 2.13954i | −0.654861 | − | 0.755750i | 0.142315 | + | 0.989821i | −0.840984 | + | 1.84150i |
97.15 | −0.959493 | − | 0.281733i | 0.755750 | + | 0.654861i | 0.841254 | + | 0.540641i | 0.341716 | − | 2.37669i | −0.540641 | − | 0.841254i | 1.93632 | + | 1.80296i | −0.654861 | − | 0.755750i | 0.142315 | + | 0.989821i | −0.997464 | + | 2.18414i |
97.16 | −0.959493 | − | 0.281733i | 0.755750 | + | 0.654861i | 0.841254 | + | 0.540641i | 0.480377 | − | 3.34109i | −0.540641 | − | 0.841254i | −2.08326 | − | 1.63096i | −0.654861 | − | 0.755750i | 0.142315 | + | 0.989821i | −1.40221 | + | 3.07042i |
181.1 | −0.142315 | − | 0.989821i | −0.909632 | − | 0.415415i | −0.959493 | + | 0.281733i | −2.74900 | + | 3.17251i | −0.281733 | + | 0.959493i | 1.98395 | + | 1.75041i | 0.415415 | + | 0.909632i | 0.654861 | + | 0.755750i | 3.53144 | + | 2.26952i |
181.2 | −0.142315 | − | 0.989821i | −0.909632 | − | 0.415415i | −0.959493 | + | 0.281733i | −1.32721 | + | 1.53168i | −0.281733 | + | 0.959493i | −2.59428 | + | 0.519336i | 0.415415 | + | 0.909632i | 0.654861 | + | 0.755750i | 1.70498 | + | 1.09572i |
181.3 | −0.142315 | − | 0.989821i | −0.909632 | − | 0.415415i | −0.959493 | + | 0.281733i | −1.30557 | + | 1.50670i | −0.281733 | + | 0.959493i | 2.10508 | − | 1.60270i | 0.415415 | + | 0.909632i | 0.654861 | + | 0.755750i | 1.67717 | + | 1.07785i |
181.4 | −0.142315 | − | 0.989821i | −0.909632 | − | 0.415415i | −0.959493 | + | 0.281733i | −0.296662 | + | 0.342366i | −0.281733 | + | 0.959493i | 2.35940 | − | 1.19718i | 0.415415 | + | 0.909632i | 0.654861 | + | 0.755750i | 0.381101 | + | 0.244919i |
See next 80 embeddings (of 160 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
23.d | odd | 22 | 1 | inner |
161.k | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 966.2.s.a | ✓ | 160 |
7.b | odd | 2 | 1 | inner | 966.2.s.a | ✓ | 160 |
23.d | odd | 22 | 1 | inner | 966.2.s.a | ✓ | 160 |
161.k | even | 22 | 1 | inner | 966.2.s.a | ✓ | 160 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
966.2.s.a | ✓ | 160 | 1.a | even | 1 | 1 | trivial |
966.2.s.a | ✓ | 160 | 7.b | odd | 2 | 1 | inner |
966.2.s.a | ✓ | 160 | 23.d | odd | 22 | 1 | inner |
966.2.s.a | ✓ | 160 | 161.k | even | 22 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \(26\!\cdots\!47\)\( T_{5}^{140} + \)\(37\!\cdots\!11\)\( T_{5}^{138} + \)\(48\!\cdots\!02\)\( T_{5}^{136} + \)\(67\!\cdots\!03\)\( T_{5}^{134} + \)\(81\!\cdots\!49\)\( T_{5}^{132} + \)\(99\!\cdots\!25\)\( T_{5}^{130} + \)\(15\!\cdots\!73\)\( T_{5}^{128} + \)\(21\!\cdots\!43\)\( T_{5}^{126} + \)\(20\!\cdots\!96\)\( T_{5}^{124} + \)\(14\!\cdots\!88\)\( T_{5}^{122} + \)\(13\!\cdots\!41\)\( T_{5}^{120} + \)\(16\!\cdots\!58\)\( T_{5}^{118} + \)\(18\!\cdots\!08\)\( T_{5}^{116} + \)\(16\!\cdots\!16\)\( T_{5}^{114} + \)\(12\!\cdots\!95\)\( T_{5}^{112} + \)\(94\!\cdots\!88\)\( T_{5}^{110} + \)\(74\!\cdots\!05\)\( T_{5}^{108} + \)\(60\!\cdots\!52\)\( T_{5}^{106} + \)\(46\!\cdots\!29\)\( T_{5}^{104} + \)\(34\!\cdots\!32\)\( T_{5}^{102} + \)\(24\!\cdots\!28\)\( T_{5}^{100} + \)\(16\!\cdots\!11\)\( T_{5}^{98} + \)\(10\!\cdots\!68\)\( T_{5}^{96} + \)\(64\!\cdots\!11\)\( T_{5}^{94} + \)\(39\!\cdots\!21\)\( T_{5}^{92} + \)\(23\!\cdots\!83\)\( T_{5}^{90} + \)\(13\!\cdots\!04\)\( T_{5}^{88} + \)\(77\!\cdots\!05\)\( T_{5}^{86} + \)\(41\!\cdots\!04\)\( T_{5}^{84} + \)\(20\!\cdots\!04\)\( T_{5}^{82} + \)\(10\!\cdots\!53\)\( T_{5}^{80} + \)\(45\!\cdots\!28\)\( T_{5}^{78} + \)\(19\!\cdots\!15\)\( T_{5}^{76} + \)\(82\!\cdots\!63\)\( T_{5}^{74} + \)\(33\!\cdots\!19\)\( T_{5}^{72} + \)\(13\!\cdots\!86\)\( T_{5}^{70} + \)\(53\!\cdots\!65\)\( T_{5}^{68} + \)\(19\!\cdots\!60\)\( T_{5}^{66} + \)\(63\!\cdots\!76\)\( T_{5}^{64} + \)\(18\!\cdots\!24\)\( T_{5}^{62} + \)\(46\!\cdots\!42\)\( T_{5}^{60} + \)\(10\!\cdots\!86\)\( T_{5}^{58} + \)\(19\!\cdots\!66\)\( T_{5}^{56} + \)\(34\!\cdots\!56\)\( T_{5}^{54} + \)\(54\!\cdots\!64\)\( T_{5}^{52} + \)\(82\!\cdots\!36\)\( T_{5}^{50} + \)\(12\!\cdots\!58\)\( T_{5}^{48} + \)\(16\!\cdots\!92\)\( T_{5}^{46} + \)\(24\!\cdots\!54\)\( T_{5}^{44} + \)\(31\!\cdots\!92\)\( T_{5}^{42} + \)\(52\!\cdots\!05\)\( T_{5}^{40} + \)\(36\!\cdots\!12\)\( T_{5}^{38} + \)\(96\!\cdots\!68\)\( T_{5}^{36} - \)\(83\!\cdots\!42\)\( T_{5}^{34} + \)\(13\!\cdots\!66\)\( T_{5}^{32} + \)\(57\!\cdots\!66\)\( T_{5}^{30} + \)\(52\!\cdots\!57\)\( T_{5}^{28} + \)\(12\!\cdots\!59\)\( T_{5}^{26} + \)\(57\!\cdots\!63\)\( T_{5}^{24} + \)\(14\!\cdots\!26\)\( T_{5}^{22} + \)\(55\!\cdots\!57\)\( T_{5}^{20} + \)\(20\!\cdots\!36\)\( T_{5}^{18} + \)\(62\!\cdots\!36\)\( T_{5}^{16} + \)\(11\!\cdots\!36\)\( T_{5}^{14} + \)\(16\!\cdots\!40\)\( T_{5}^{12} + \)\(14\!\cdots\!92\)\( T_{5}^{10} + \)\(10\!\cdots\!84\)\( T_{5}^{8} + \)\(90\!\cdots\!72\)\( T_{5}^{6} + \)\(49\!\cdots\!40\)\( T_{5}^{4} + \)\(81\!\cdots\!60\)\( T_{5}^{2} + \)\(43\!\cdots\!36\)\( \)">\(T_{5}^{160} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).