# 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 $4$

# 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.

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

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