# Properties

 Label 966.2.q.g 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30q + 3q^{2} + 3q^{3} - 3q^{4} - 10q^{5} - 3q^{6} + 3q^{7} + 3q^{8} - 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q + 3q^{2} + 3q^{3} - 3q^{4} - 10q^{5} - 3q^{6} + 3q^{7} + 3q^{8} - 3q^{9} + 10q^{10} + 9q^{11} + 3q^{12} - 12q^{13} - 3q^{14} - q^{15} - 3q^{16} + 5q^{17} + 3q^{18} + 18q^{19} + q^{20} - 3q^{21} + 2q^{22} + 21q^{23} + 30q^{24} + 13q^{25} - 10q^{26} + 3q^{27} + 3q^{28} + 17q^{29} - 10q^{30} + 12q^{31} + 3q^{32} + 13q^{33} + 17q^{34} - q^{35} - 3q^{36} + 16q^{37} + 15q^{38} + 12q^{39} - 12q^{40} + 10q^{41} + 3q^{42} - 35q^{43} + 9q^{44} + 12q^{45} + q^{46} - 8q^{47} + 3q^{48} - 3q^{49} - 2q^{50} + 6q^{51} - q^{52} + 42q^{53} - 3q^{54} + 49q^{55} - 3q^{56} + 15q^{57} + 5q^{58} - 6q^{59} + 10q^{60} - 18q^{61} - 34q^{62} + 3q^{63} - 3q^{64} + 34q^{65} - 2q^{66} + 72q^{67} - 6q^{68} - 10q^{69} + 12q^{70} + 17q^{71} + 3q^{72} + 9q^{73} - 16q^{74} - 2q^{75} + 18q^{76} + 2q^{77} + 10q^{78} - 56q^{79} + q^{80} - 3q^{81} + 12q^{82} + 52q^{83} - 3q^{84} - 53q^{85} - 31q^{86} + 5q^{87} + 13q^{88} - 104q^{89} - q^{90} + 34q^{91} - 12q^{92} + 32q^{93} - 14q^{94} - 92q^{95} - 3q^{96} - 82q^{97} + 3q^{98} - 13q^{99} + O(q^{100})$$

## 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 −0.635376 1.39128i −0.142315 + 0.989821i 0.959493 0.281733i −0.841254 0.540641i 0.415415 0.909632i −1.46754 0.430909i
85.2 0.654861 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i 0.477791 + 1.04622i −0.142315 + 0.989821i 0.959493 0.281733i −0.841254 0.540641i 0.415415 0.909632i 1.10356 + 0.324036i
85.3 0.654861 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i 0.906796 + 1.98561i −0.142315 + 0.989821i 0.959493 0.281733i −0.841254 0.540641i 0.415415 0.909632i 2.09445 + 0.614985i
127.1 −0.841254 + 0.540641i 0.142315 0.989821i 0.415415 0.909632i −4.09857 + 1.20345i 0.415415 + 0.909632i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i 2.79730 3.22826i
127.2 −0.841254 + 0.540641i 0.142315 0.989821i 0.415415 0.909632i −1.03525 + 0.303978i 0.415415 + 0.909632i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i 0.706568 0.815423i
127.3 −0.841254 + 0.540641i 0.142315 0.989821i 0.415415 0.909632i 0.279230 0.0819895i 0.415415 + 0.909632i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i −0.190577 + 0.219937i
169.1 0.142315 0.989821i −0.415415 0.909632i −0.959493 0.281733i −1.94500 2.24464i −0.959493 + 0.281733i −0.841254 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i −2.49860 + 1.60575i
169.2 0.142315 0.989821i −0.415415 0.909632i −0.959493 0.281733i 1.14899 + 1.32600i −0.959493 + 0.281733i −0.841254 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i 1.47603 0.948584i
169.3 0.142315 0.989821i −0.415415 0.909632i −0.959493 0.281733i 1.64194 + 1.89490i −0.959493 + 0.281733i −0.841254 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i 2.10928 1.35555i
211.1 −0.415415 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i −2.54794 1.63746i −0.654861 0.755750i 0.142315 + 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i −0.431035 + 2.99791i
211.2 −0.415415 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i −0.642037 0.412612i −0.654861 0.755750i 0.142315 + 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i −0.108613 + 0.755423i
211.3 −0.415415 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i 1.75345 + 1.12688i −0.654861 0.755750i 0.142315 + 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i 0.296631 2.06312i
463.1 0.142315 + 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i −1.94500 + 2.24464i −0.959493 0.281733i −0.841254 + 0.540641i −0.415415 0.909632i −0.654861 0.755750i −2.49860 1.60575i
463.2 0.142315 + 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.14899 1.32600i −0.959493 0.281733i −0.841254 + 0.540641i −0.415415 0.909632i −0.654861 0.755750i 1.47603 + 0.948584i
463.3 0.142315 + 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.64194 1.89490i −0.959493 0.281733i −0.841254 + 0.540641i −0.415415 0.909632i −0.654861 0.755750i 2.10928 + 1.35555i
547.1 0.959493 + 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i −0.471267 + 3.27773i 0.841254 0.540641i −0.415415 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i −1.37562 + 3.01219i
547.2 0.959493 + 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i −0.121006 + 0.841612i 0.841254 0.540641i −0.415415 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i −0.353213 + 0.773429i
547.3 0.959493 + 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i 0.288244 2.00478i 0.841254 0.540641i −0.415415 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i 0.841381 1.84237i
673.1 −0.415415 + 0.909632i 0.959493 + 0.281733i −0.654861 0.755750i −2.54794 + 1.63746i −0.654861 + 0.755750i 0.142315 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i −0.431035 2.99791i
673.2 −0.415415 + 0.909632i 0.959493 + 0.281733i −0.654861 0.755750i −0.642037 + 0.412612i −0.654861 + 0.755750i 0.142315 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i −0.108613 0.755423i
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.g 30
23.c even 11 1 inner 966.2.q.g 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.q.g 30 1.a even 1 1 trivial
966.2.q.g 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 $$S_{2}^{\mathrm{new}}(966, [\chi])$$.