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Properties

Label 966.2.s.a
Level $966$
Weight $2$
Character orbit 966.s
Analytic conductor $7.714$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $4$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 160q - 16q^{2} - 16q^{4} - 16q^{8} + 16q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 160q - 16q^{2} - 16q^{4} - 16q^{8} + 16q^{9} - 22q^{14} - 16q^{16} + 16q^{18} - 36q^{23} + 96q^{25} + 22q^{28} - 20q^{29} - 16q^{32} - 50q^{35} + 16q^{36} + 22q^{37} + 4q^{39} - 110q^{43} + 8q^{46} - 36q^{50} + 22q^{51} - 88q^{53} + 22q^{57} + 24q^{58} - 16q^{64} - 72q^{70} + 48q^{71} + 16q^{72} - 22q^{74} + 24q^{77} + 4q^{78} + 88q^{79} - 16q^{81} + 22q^{84} + 76q^{85} + 44q^{86} - 44q^{88} + 8q^{92} - 14q^{95} + 22q^{98} - 22q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 937.16
Significant digits:
Format:

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 \(S_{2}^{\mathrm{new}}(966, [\chi])\).