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Properties

Label 966.2.s.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 160q + 16q^{2} - 16q^{4} + 16q^{8} + 16q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 160q + 16q^{2} - 16q^{4} + 16q^{8} + 16q^{9} + 22q^{14} - 16q^{16} - 16q^{18} - 52q^{23} - 48q^{25} + 22q^{28} - 12q^{29} + 16q^{32} + 22q^{35} + 16q^{36} + 22q^{37} - 4q^{39} + 154q^{43} + 8q^{46} + 4q^{50} + 22q^{51} + 88q^{53} + 22q^{57} + 56q^{58} - 16q^{64} + 16q^{71} - 16q^{72} + 22q^{74} - 72q^{77} + 4q^{78} + 88q^{79} - 16q^{81} - 22q^{84} - 68q^{85} - 44q^{86} + 44q^{88} - 8q^{92} - 8q^{93} - 66q^{95} + 66q^{98} - 66q^{99} + O(q^{100}) \)

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 −0.518172 + 3.60397i −0.540641 − 0.841254i 2.64368 + 0.104630i 0.654861 + 0.755750i 0.142315 + 0.989821i −1.51254 + 3.31199i
97.2 0.959493 + 0.281733i −0.755750 − 0.654861i 0.841254 + 0.540641i −0.332974 + 2.31589i −0.540641 − 0.841254i −2.61372 + 0.410464i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.971947 + 2.12827i
97.3 0.959493 + 0.281733i −0.755750 − 0.654861i 0.841254 + 0.540641i −0.141036 + 0.980930i −0.540641 − 0.841254i −0.385560 − 2.61751i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.411683 + 0.901461i
97.4 0.959493 + 0.281733i −0.755750 − 0.654861i 0.841254 + 0.540641i −0.108989 + 0.758036i −0.540641 − 0.841254i 1.08396 − 2.41351i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.318138 + 0.696624i
97.5 0.959493 + 0.281733i −0.755750 − 0.654861i 0.841254 + 0.540641i −0.0778857 + 0.541707i −0.540641 − 0.841254i 2.00180 + 1.72997i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.227347 + 0.497821i
97.6 0.959493 + 0.281733i −0.755750 − 0.654861i 0.841254 + 0.540641i 0.258147 − 1.79545i −0.540641 − 0.841254i −1.04979 + 2.42857i 0.654861 + 0.755750i 0.142315 + 0.989821i 0.753527 − 1.64999i
97.7 0.959493 + 0.281733i −0.755750 − 0.654861i 0.841254 + 0.540641i 0.445696 − 3.09989i −0.540641 − 0.841254i −2.43154 − 1.04288i 0.654861 + 0.755750i 0.142315 + 0.989821i 1.30098 − 2.84875i
97.8 0.959493 + 0.281733i −0.755750 − 0.654861i 0.841254 + 0.540641i 0.629097 − 4.37546i −0.540641 − 0.841254i 2.43657 − 1.03108i 0.654861 + 0.755750i 0.142315 + 0.989821i 1.83632 − 4.02099i
97.9 0.959493 + 0.281733i 0.755750 + 0.654861i 0.841254 + 0.540641i −0.629097 + 4.37546i 0.540641 + 0.841254i 2.37485 − 1.16622i 0.654861 + 0.755750i 0.142315 + 0.989821i −1.83632 + 4.02099i
97.10 0.959493 + 0.281733i 0.755750 + 0.654861i 0.841254 + 0.540641i −0.445696 + 3.09989i 0.540641 + 0.841254i −0.804171 + 2.52058i 0.654861 + 0.755750i 0.142315 + 0.989821i −1.30098 + 2.84875i
97.11 0.959493 + 0.281733i 0.755750 + 0.654861i 0.841254 + 0.540641i −0.258147 + 1.79545i 0.540641 + 0.841254i −2.52286 − 0.796992i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.753527 + 1.64999i
97.12 0.959493 + 0.281733i 0.755750 + 0.654861i 0.841254 + 0.540641i 0.0778857 − 0.541707i 0.540641 + 0.841254i 0.00348138 − 2.64575i 0.654861 + 0.755750i 0.142315 + 0.989821i 0.227347 − 0.497821i
97.13 0.959493 + 0.281733i 0.755750 + 0.654861i 0.841254 + 0.540641i 0.108989 − 0.758036i 0.540641 + 0.841254i 2.53385 + 0.761316i 0.654861 + 0.755750i 0.142315 + 0.989821i 0.318138 − 0.696624i
97.14 0.959493 + 0.281733i 0.755750 + 0.654861i 0.841254 + 0.540641i 0.141036 − 0.980930i 0.540641 + 0.841254i 1.72569 + 2.00549i 0.654861 + 0.755750i 0.142315 + 0.989821i 0.411683 − 0.901461i
97.15 0.959493 + 0.281733i 0.755750 + 0.654861i 0.841254 + 0.540641i 0.332974 − 2.31589i 0.540641 + 0.841254i −2.02183 + 1.70652i 0.654861 + 0.755750i 0.142315 + 0.989821i 0.971947 − 2.12827i
97.16 0.959493 + 0.281733i 0.755750 + 0.654861i 0.841254 + 0.540641i 0.518172 − 3.60397i 0.540641 + 0.841254i 1.65217 − 2.06648i 0.654861 + 0.755750i 0.142315 + 0.989821i 1.51254 − 3.31199i
181.1 0.142315 + 0.989821i −0.909632 − 0.415415i −0.959493 + 0.281733i −2.34664 + 2.70817i 0.281733 − 0.959493i −2.57403 + 0.611844i −0.415415 − 0.909632i 0.654861 + 0.755750i −3.01456 − 1.93734i
181.2 0.142315 + 0.989821i −0.909632 − 0.415415i −0.959493 + 0.281733i −2.30682 + 2.66222i 0.281733 − 0.959493i 1.33240 + 2.28576i −0.415415 − 0.909632i 0.654861 + 0.755750i −2.96342 − 1.90447i
181.3 0.142315 + 0.989821i −0.909632 − 0.415415i −0.959493 + 0.281733i −0.762866 + 0.880394i 0.281733 − 0.959493i 2.62112 + 0.360195i −0.415415 − 0.909632i 0.654861 + 0.755750i −0.980000 − 0.629808i
181.4 0.142315 + 0.989821i −0.909632 − 0.415415i −0.959493 + 0.281733i −0.278516 + 0.321425i 0.281733 − 0.959493i −0.505900 − 2.59693i −0.415415 − 0.909632i 0.654861 + 0.755750i −0.357790 − 0.229938i
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.b ✓ 160
7.b odd 2 1 inner 966.2.s.b ✓ 160
23.d odd 22 1 inner 966.2.s.b ✓ 160
161.k even 22 1 inner 966.2.s.b ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.s.b ✓ 160 1.a even 1 1 trivial
966.2.s.b ✓ 160 7.b odd 2 1 inner
966.2.s.b ✓ 160 23.d odd 22 1 inner
966.2.s.b ✓ 160 161.k even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(17\!\cdots\!00\)\( T_{5}^{142} + \)\(23\!\cdots\!51\)\( T_{5}^{140} + \)\(32\!\cdots\!35\)\( T_{5}^{138} + \)\(42\!\cdots\!90\)\( T_{5}^{136} + \)\(55\!\cdots\!67\)\( T_{5}^{134} + \)\(69\!\cdots\!33\)\( T_{5}^{132} + \)\(82\!\cdots\!21\)\( T_{5}^{130} + \)\(92\!\cdots\!57\)\( T_{5}^{128} + \)\(97\!\cdots\!19\)\( T_{5}^{126} + \)\(10\!\cdots\!56\)\( T_{5}^{124} + \)\(10\!\cdots\!40\)\( T_{5}^{122} + \)\(12\!\cdots\!01\)\( T_{5}^{120} + \)\(12\!\cdots\!90\)\( T_{5}^{118} + \)\(12\!\cdots\!20\)\( T_{5}^{116} + \)\(10\!\cdots\!80\)\( T_{5}^{114} + \)\(84\!\cdots\!87\)\( T_{5}^{112} + \)\(78\!\cdots\!92\)\( T_{5}^{110} + \)\(84\!\cdots\!93\)\( T_{5}^{108} + \)\(83\!\cdots\!08\)\( T_{5}^{106} + \)\(65\!\cdots\!57\)\( T_{5}^{104} + \)\(40\!\cdots\!08\)\( T_{5}^{102} + \)\(26\!\cdots\!92\)\( T_{5}^{100} + \)\(22\!\cdots\!15\)\( T_{5}^{98} + \)\(23\!\cdots\!68\)\( T_{5}^{96} + \)\(17\!\cdots\!15\)\( T_{5}^{94} + \)\(89\!\cdots\!77\)\( T_{5}^{92} + \)\(47\!\cdots\!03\)\( T_{5}^{90} + \)\(27\!\cdots\!52\)\( T_{5}^{88} + \)\(13\!\cdots\!45\)\( T_{5}^{86} + \)\(60\!\cdots\!48\)\( T_{5}^{84} + \)\(29\!\cdots\!68\)\( T_{5}^{82} + \)\(14\!\cdots\!05\)\( T_{5}^{80} + \)\(61\!\cdots\!56\)\( T_{5}^{78} + \)\(24\!\cdots\!51\)\( T_{5}^{76} + \)\(98\!\cdots\!91\)\( T_{5}^{74} + \)\(34\!\cdots\!83\)\( T_{5}^{72} + \)\(10\!\cdots\!70\)\( T_{5}^{70} + \)\(30\!\cdots\!69\)\( T_{5}^{68} + \)\(80\!\cdots\!16\)\( T_{5}^{66} + \)\(18\!\cdots\!32\)\( T_{5}^{64} + \)\(35\!\cdots\!80\)\( T_{5}^{62} + \)\(58\!\cdots\!46\)\( T_{5}^{60} + \)\(83\!\cdots\!62\)\( T_{5}^{58} + \)\(11\!\cdots\!66\)\( T_{5}^{56} + \)\(15\!\cdots\!68\)\( T_{5}^{54} + \)\(24\!\cdots\!48\)\( T_{5}^{52} + \)\(35\!\cdots\!32\)\( T_{5}^{50} + \)\(47\!\cdots\!70\)\( T_{5}^{48} + \)\(54\!\cdots\!56\)\( T_{5}^{46} + \)\(54\!\cdots\!14\)\( T_{5}^{44} + \)\(47\!\cdots\!32\)\( T_{5}^{42} + \)\(36\!\cdots\!25\)\( T_{5}^{40} + \)\(22\!\cdots\!92\)\( T_{5}^{38} + \)\(11\!\cdots\!28\)\( T_{5}^{36} + \)\(39\!\cdots\!02\)\( T_{5}^{34} + \)\(97\!\cdots\!42\)\( T_{5}^{32} + \)\(17\!\cdots\!66\)\( T_{5}^{30} + \)\(30\!\cdots\!69\)\( T_{5}^{28} + \)\(48\!\cdots\!23\)\( T_{5}^{26} + \)\(36\!\cdots\!39\)\( T_{5}^{24} + \)\(34\!\cdots\!62\)\( T_{5}^{22} + \)\(28\!\cdots\!69\)\( T_{5}^{20} - \)\(11\!\cdots\!96\)\( T_{5}^{18} + \)\(29\!\cdots\!96\)\( T_{5}^{16} - \)\(73\!\cdots\!08\)\( T_{5}^{14} + \)\(49\!\cdots\!36\)\( T_{5}^{12} - \)\(27\!\cdots\!32\)\( T_{5}^{10} + \)\(27\!\cdots\!20\)\( T_{5}^{8} + \)\(98\!\cdots\!60\)\( T_{5}^{6} + \)\(25\!\cdots\!04\)\( T_{5}^{4} - \)\(63\!\cdots\!32\)\( T_{5}^{2} + \)\(45\!\cdots\!76\)\( \)">\(T_{5}^{160} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).