# 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 $4$

# Related objects

## 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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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