# Properties

 Label 966.2.q.j Level $966$ Weight $2$ Character orbit 966.q Analytic conductor $7.714$ Analytic rank $0$ Dimension $40$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.q (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.71354883526$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$4$$ over $$\Q(\zeta_{11})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 4q^{2} + 4q^{3} - 4q^{4} - 4q^{5} + 4q^{6} + 4q^{7} - 4q^{8} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 4q^{2} + 4q^{3} - 4q^{4} - 4q^{5} + 4q^{6} + 4q^{7} - 4q^{8} - 4q^{9} - 4q^{10} - 11q^{11} + 4q^{12} - 20q^{13} + 4q^{14} - 7q^{15} - 4q^{16} + 5q^{17} - 4q^{18} + 44q^{19} + 7q^{20} - 4q^{21} + 22q^{22} - 6q^{23} - 40q^{24} + 2q^{25} + 24q^{26} + 4q^{27} + 4q^{28} + 35q^{29} + 4q^{30} - 2q^{31} - 4q^{32} - 11q^{33} + 5q^{34} - 7q^{35} - 4q^{36} + 20q^{37} - 11q^{38} + 20q^{39} - 4q^{40} + 11q^{41} - 4q^{42} - 42q^{43} - 11q^{44} - 4q^{45} - 6q^{46} + 40q^{47} + 4q^{48} - 4q^{49} - 53q^{50} + 17q^{51} - 9q^{52} - 32q^{53} + 4q^{54} + 29q^{55} + 4q^{56} - 11q^{57} - 31q^{58} - 25q^{59} + 4q^{60} - 30q^{61} - 24q^{62} + 4q^{63} - 4q^{64} + 80q^{65} - 88q^{67} - 28q^{68} + 6q^{69} + 4q^{70} - 45q^{71} - 4q^{72} - 39q^{73} + 20q^{74} + 53q^{75} - 22q^{76} - 2q^{78} + 26q^{79} + 7q^{80} - 4q^{81} + 22q^{83} - 4q^{84} + 77q^{85} - 31q^{86} + 9q^{87} - 11q^{88} + 20q^{89} + 7q^{90} - 24q^{91} - 17q^{92} - 42q^{93} - 26q^{94} + 84q^{95} + 4q^{96} + 74q^{97} - 4q^{98} + 11q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
85.1 −0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i −1.36169 2.98169i 0.142315 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i 3.14512 + 0.923492i
85.2 −0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i −0.568699 1.24528i 0.142315 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i 1.31354 + 0.385689i
85.3 −0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i 0.479771 + 1.05055i 0.142315 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i −1.10814 0.325379i
85.4 −0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i 1.60548 + 3.51551i 0.142315 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i −3.70821 1.08883i
127.1 0.841254 0.540641i 0.142315 0.989821i 0.415415 0.909632i −3.54832 + 1.04188i −0.415415 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i −0.959493 0.281733i −2.42175 + 2.79485i
127.2 0.841254 0.540641i 0.142315 0.989821i 0.415415 0.909632i −0.571809 + 0.167898i −0.415415 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i −0.959493 0.281733i −0.390264 + 0.450389i
127.3 0.841254 0.540641i 0.142315 0.989821i 0.415415 0.909632i −0.309565 + 0.0908964i −0.415415 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i −0.959493 0.281733i −0.211280 + 0.243830i
127.4 0.841254 0.540641i 0.142315 0.989821i 0.415415 0.909632i 3.08844 0.906847i −0.415415 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i −0.959493 0.281733i 2.10788 2.43262i
169.1 −0.142315 + 0.989821i −0.415415 0.909632i −0.959493 0.281733i −2.74645 3.16958i 0.959493 0.281733i −0.841254 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i 3.52818 2.26742i
169.2 −0.142315 + 0.989821i −0.415415 0.909632i −0.959493 0.281733i −0.463708 0.535148i 0.959493 0.281733i −0.841254 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i 0.595693 0.382829i
169.3 −0.142315 + 0.989821i −0.415415 0.909632i −0.959493 0.281733i 0.0335087 + 0.0386712i 0.959493 0.281733i −0.841254 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i −0.0430463 + 0.0276642i
169.4 −0.142315 + 0.989821i −0.415415 0.909632i −0.959493 0.281733i 2.81897 + 3.25326i 0.959493 0.281733i −0.841254 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i −3.62133 + 2.32729i
211.1 0.415415 + 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i −3.48521 2.23981i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i 0.841254 0.540641i 0.589593 4.10071i
211.2 0.415415 + 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i −2.25359 1.44829i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i 0.841254 0.540641i 0.381240 2.65158i
211.3 0.415415 + 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i 1.73540 + 1.11528i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i 0.841254 0.540641i −0.293578 + 2.04188i
211.4 0.415415 + 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i 3.08798 + 1.98452i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i 0.841254 0.540641i −0.522394 + 3.63333i
463.1 −0.142315 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i −2.74645 + 3.16958i 0.959493 + 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i 3.52818 + 2.26742i
463.2 −0.142315 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.463708 + 0.535148i 0.959493 + 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i 0.595693 + 0.382829i
463.3 −0.142315 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.0335087 0.0386712i 0.959493 + 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i −0.0430463 0.0276642i
463.4 −0.142315 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 2.81897 3.25326i 0.959493 + 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i −3.62133 2.32729i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 883.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.q.j 40
23.c even 11 1 inner 966.2.q.j 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.q.j 40 1.a even 1 1 trivial
966.2.q.j 40 23.c even 11 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$18\!\cdots\!18$$$$T_{5}^{18} +$$$$21\!\cdots\!84$$$$T_{5}^{17} +$$$$66\!\cdots\!51$$$$T_{5}^{16} +$$$$80\!\cdots\!67$$$$T_{5}^{15} +$$$$29\!\cdots\!83$$$$T_{5}^{14} +$$$$56\!\cdots\!56$$$$T_{5}^{13} +$$$$12\!\cdots\!69$$$$T_{5}^{12} +$$$$19\!\cdots\!34$$$$T_{5}^{11} +$$$$26\!\cdots\!45$$$$T_{5}^{10} +$$$$31\!\cdots\!92$$$$T_{5}^{9} +$$$$32\!\cdots\!36$$$$T_{5}^{8} +$$$$28\!\cdots\!32$$$$T_{5}^{7} +$$$$20\!\cdots\!32$$$$T_{5}^{6} +$$$$11\!\cdots\!04$$$$T_{5}^{5} +$$$$38\!\cdots\!40$$$$T_{5}^{4} +$$$$72\!\cdots\!48$$$$T_{5}^{3} + 376549054208 T_{5}^{2} - 35888090112 T_{5} + 2431673344$$">$$T_{5}^{40} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(966, [\chi])$$.