# Properties

 Label 966.2.q.f Level $966$ Weight $2$ Character orbit 966.q Analytic conductor $7.714$ Analytic rank $0$ Dimension $30$ 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: $$30$$ Relative dimension: $$3$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30q + 3q^{2} - 3q^{3} - 3q^{4} - 10q^{5} + 3q^{6} + 3q^{7} + 3q^{8} - 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q + 3q^{2} - 3q^{3} - 3q^{4} - 10q^{5} + 3q^{6} + 3q^{7} + 3q^{8} - 3q^{9} - 12q^{10} - 9q^{11} - 3q^{12} + 12q^{13} - 3q^{14} + q^{15} - 3q^{16} + 9q^{17} + 3q^{18} - 18q^{19} + q^{20} + 3q^{21} + 20q^{22} + q^{23} - 30q^{24} + 5q^{25} + 32q^{26} - 3q^{27} + 3q^{28} - 23q^{29} - 12q^{30} - q^{31} + 3q^{32} + 2q^{33} - 9q^{34} - q^{35} - 3q^{36} - q^{37} + 7q^{38} + 12q^{39} - q^{40} + 7q^{41} - 3q^{42} - 10q^{43} - 9q^{44} - 10q^{45} - q^{46} + 68q^{47} - 3q^{48} - 3q^{49} + 50q^{50} + 9q^{51} + q^{52} + 42q^{53} + 3q^{54} - 66q^{55} - 3q^{56} + 4q^{57} + q^{58} + 25q^{59} - 10q^{60} + 10q^{61} - 10q^{62} + 3q^{63} - 3q^{64} - 54q^{65} - 2q^{66} - 6q^{67} - 2q^{68} - 21q^{69} - 10q^{70} - 13q^{71} + 3q^{72} + 33q^{73} + q^{74} - 50q^{75} + 26q^{76} - 2q^{77} - 12q^{78} - 8q^{79} + q^{80} - 3q^{81} + 4q^{82} - 2q^{83} + 3q^{84} - 77q^{85} - 45q^{86} - q^{87} - 13q^{88} + 64q^{89} - q^{90} - 12q^{91} + 12q^{92} - 56q^{93} - 24q^{94} + 59q^{95} + 3q^{96} + 2q^{97} + 3q^{98} + 2q^{99} + O(q^{100})$$

## 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 −1.82570 3.99772i 0.142315 0.989821i 0.959493 0.281733i −0.841254 0.540641i 0.415415 0.909632i −4.21685 1.23818i
85.2 0.654861 0.755750i 0.841254 0.540641i −0.142315 0.989821i −0.454603 0.995441i 0.142315 0.989821i 0.959493 0.281733i −0.841254 0.540641i 0.415415 0.909632i −1.05001 0.308309i
85.3 0.654861 0.755750i 0.841254 0.540641i −0.142315 0.989821i 0.722570 + 1.58221i 0.142315 0.989821i 0.959493 0.281733i −0.841254 0.540641i 0.415415 0.909632i 1.66893 + 0.490044i
127.1 −0.841254 + 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i −1.76124 + 0.517147i −0.415415 0.909632i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i 1.20206 1.38725i
127.2 −0.841254 + 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i −0.421404 + 0.123735i −0.415415 0.909632i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i 0.287611 0.331921i
127.3 −0.841254 + 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i 2.55755 0.750965i −0.415415 0.909632i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i −1.74555 + 2.01447i
169.1 0.142315 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −2.78120 3.20968i 0.959493 0.281733i −0.841254 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i −3.57281 + 2.29611i
169.2 0.142315 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −0.150642 0.173850i 0.959493 0.281733i −0.841254 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i −0.193519 + 0.124367i
169.3 0.142315 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i 1.62721 + 1.87790i 0.959493 0.281733i −0.841254 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i 2.09036 1.34340i
211.1 −0.415415 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i −2.44943 1.57415i 0.654861 + 0.755750i 0.142315 + 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i −0.414370 + 2.88200i
211.2 −0.415415 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i −2.10097 1.35021i 0.654861 + 0.755750i 0.142315 + 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i −0.355420 + 2.47200i
211.3 −0.415415 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i 2.05428 + 1.32020i 0.654861 + 0.755750i 0.142315 + 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i 0.347522 2.41707i
463.1 0.142315 + 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i −2.78120 + 3.20968i 0.959493 + 0.281733i −0.841254 + 0.540641i −0.415415 0.909632i −0.654861 0.755750i −3.57281 2.29611i
463.2 0.142315 + 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i −0.150642 + 0.173850i 0.959493 + 0.281733i −0.841254 + 0.540641i −0.415415 0.909632i −0.654861 0.755750i −0.193519 0.124367i
463.3 0.142315 + 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i 1.62721 1.87790i 0.959493 + 0.281733i −0.841254 + 0.540641i −0.415415 0.909632i −0.654861 0.755750i 2.09036 + 1.34340i
547.1 0.959493 + 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i −0.220101 + 1.53083i −0.841254 + 0.540641i −0.415415 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i −0.642471 + 1.40681i
547.2 0.959493 + 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i 0.0353399 0.245794i −0.841254 + 0.540641i −0.415415 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i 0.103157 0.225881i
547.3 0.959493 + 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i 0.168329 1.17076i −0.841254 + 0.540641i −0.415415 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i 0.491350 1.07591i
673.1 −0.415415 + 0.909632i −0.959493 0.281733i −0.654861 0.755750i −2.44943 + 1.57415i 0.654861 0.755750i 0.142315 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i −0.414370 2.88200i
673.2 −0.415415 + 0.909632i −0.959493 0.281733i −0.654861 0.755750i −2.10097 + 1.35021i 0.654861 0.755750i 0.142315 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i −0.355420 2.47200i
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 883.3 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.f 30
23.c even 11 1 inner 966.2.q.f 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.q.f 30 1.a even 1 1 trivial
966.2.q.f 30 23.c even 11 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{30} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(966, [\chi])$$.