# Properties

 Label 966.2.q.h 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.

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

## 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 −0.480114 1.05130i −0.142315 + 0.989821i −0.959493 + 0.281733i −0.841254 0.540641i 0.415415 0.909632i −1.10893 0.325611i
85.2 0.654861 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i 0.702155 + 1.53751i −0.142315 + 0.989821i −0.959493 + 0.281733i −0.841254 0.540641i 0.415415 0.909632i 1.62178 + 0.476199i
85.3 0.654861 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i 1.33569 + 2.92475i −0.142315 + 0.989821i −0.959493 + 0.281733i −0.841254 0.540641i 0.415415 0.909632i 3.08507 + 0.905858i
127.1 −0.841254 + 0.540641i 0.142315 0.989821i 0.415415 0.909632i −3.32453 + 0.976169i 0.415415 + 0.909632i −0.654861 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i 2.26901 2.61858i
127.2 −0.841254 + 0.540641i 0.142315 0.989821i 0.415415 0.909632i 0.584519 0.171630i 0.415415 + 0.909632i −0.654861 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i −0.398938 + 0.460399i
127.3 −0.841254 + 0.540641i 0.142315 0.989821i 0.415415 0.909632i 2.36510 0.694456i 0.415415 + 0.909632i −0.654861 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i −1.61420 + 1.86288i
169.1 0.142315 0.989821i −0.415415 0.909632i −0.959493 0.281733i −0.257442 0.297104i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i −0.330718 + 0.212539i
169.2 0.142315 0.989821i −0.415415 0.909632i −0.959493 0.281733i 0.0358687 + 0.0413947i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i 0.0460780 0.0296125i
169.3 0.142315 0.989821i −0.415415 0.909632i −0.959493 0.281733i 1.52621 + 1.76134i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i 1.96061 1.26001i
211.1 −0.415415 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i −1.29551 0.832573i −0.654861 0.755750i −0.142315 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i −0.219161 + 1.52430i
211.2 −0.415415 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i 0.520240 + 0.334338i −0.654861 0.755750i −0.142315 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i 0.0880091 0.612116i
211.3 −0.415415 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i 3.27138 + 2.10239i −0.654861 0.755750i −0.142315 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i 0.553420 3.84912i
463.1 0.142315 + 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.257442 + 0.297104i −0.959493 0.281733i 0.841254 0.540641i −0.415415 0.909632i −0.654861 0.755750i −0.330718 0.212539i
463.2 0.142315 + 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.0358687 0.0413947i −0.959493 0.281733i 0.841254 0.540641i −0.415415 0.909632i −0.654861 0.755750i 0.0460780 + 0.0296125i
463.3 0.142315 + 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.52621 1.76134i −0.959493 0.281733i 0.841254 0.540641i −0.415415 0.909632i −0.654861 0.755750i 1.96061 + 1.26001i
547.1 0.959493 + 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i −0.431201 + 2.99907i 0.841254 0.540641i 0.415415 + 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i −1.25867 + 2.75610i
547.2 0.959493 + 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i −0.0682669 + 0.474806i 0.841254 0.540641i 0.415415 + 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i −0.199270 + 0.436340i
547.3 0.959493 + 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i 0.515900 3.58816i 0.841254 0.540641i 0.415415 + 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i 1.50590 3.29747i
673.1 −0.415415 + 0.909632i 0.959493 + 0.281733i −0.654861 0.755750i −1.29551 + 0.832573i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i −0.219161 1.52430i
673.2 −0.415415 + 0.909632i 0.959493 + 0.281733i −0.654861 0.755750i 0.520240 0.334338i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i 0.0880091 + 0.612116i
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.h 30
23.c even 11 1 inner 966.2.q.h 30

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