# Properties

 Label 966.2.t.a Level $966$ Weight $2$ Character orbit 966.t Analytic conductor $7.714$ Analytic rank $0$ Dimension $640$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.t (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.71354883526$$ Analytic rank: $$0$$ Dimension: $$640$$ Relative dimension: $$64$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$640q + 64q^{4} + 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$640q + 64q^{4} + 8q^{9} + 8q^{15} - 64q^{16} + 20q^{18} + 54q^{21} - 8q^{22} - 56q^{25} - 44q^{30} - 8q^{36} + 20q^{37} - 16q^{39} - 4q^{42} - 28q^{43} + 24q^{46} - 68q^{49} - 40q^{51} + 32q^{57} - 80q^{58} - 8q^{60} + 86q^{63} + 64q^{64} + 24q^{67} + 72q^{70} - 64q^{72} - 8q^{78} - 144q^{79} + 20q^{81} + 34q^{84} + 32q^{85} + 8q^{88} - 40q^{91} + 16q^{93} + 68q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1 −0.281733 + 0.959493i −1.72699 + 0.132270i −0.841254 0.540641i −0.371345 + 2.58276i 0.359638 1.69430i −2.27409 1.35223i 0.755750 0.654861i 2.96501 0.456859i −2.37352 1.08395i
41.2 −0.281733 + 0.959493i −1.72562 + 0.149127i −0.841254 0.540641i 0.250877 1.74489i 0.343077 1.69773i −0.458506 2.60572i 0.755750 0.654861i 2.95552 0.514671i 1.60353 + 0.732307i
41.3 −0.281733 + 0.959493i −1.72464 + 0.159999i −0.841254 0.540641i 0.116658 0.811375i 0.332371 1.69986i −1.16903 + 2.37347i 0.755750 0.654861i 2.94880 0.551882i 0.745642 + 0.340523i
41.4 −0.281733 + 0.959493i −1.70014 0.330969i −0.841254 0.540641i 0.212651 1.47902i 0.796546 1.53802i 1.89318 + 1.84820i 0.755750 0.654861i 2.78092 + 1.12539i 1.35920 + 0.620726i
41.5 −0.281733 + 0.959493i −1.53070 0.810522i −0.841254 0.540641i 0.429144 2.98476i 1.20894 1.24035i 0.998575 2.45007i 0.755750 0.654861i 1.68611 + 2.48134i 2.74295 + 1.25266i
41.6 −0.281733 + 0.959493i −1.26614 1.18191i −0.841254 0.540641i −0.345722 + 2.40455i 1.49074 0.881867i 1.74088 1.99232i 0.755750 0.654861i 0.206198 + 2.99291i −2.20975 1.00916i
41.7 −0.281733 + 0.959493i −1.25848 + 1.19006i −0.841254 0.540641i −0.602743 + 4.19217i −0.787299 1.54278i −1.84332 + 1.89794i 0.755750 0.654861i 0.167522 2.99532i −3.85255 1.75940i
41.8 −0.281733 + 0.959493i −1.25685 1.19177i −0.841254 0.540641i −0.205721 + 1.43082i 1.49759 0.870178i −2.56843 + 0.634953i 0.755750 0.654861i 0.159350 + 2.99576i −1.31490 0.600496i
41.9 −0.281733 + 0.959493i −1.13682 + 1.30677i −0.841254 0.540641i 0.133168 0.926206i −0.933559 1.45893i 1.08253 + 2.41415i 0.755750 0.654861i −0.415299 2.97112i 0.851170 + 0.388716i
41.10 −0.281733 + 0.959493i −0.927012 + 1.46310i −0.841254 0.540641i −0.216850 + 1.50823i −1.14266 1.30166i −0.318183 2.62655i 0.755750 0.654861i −1.28130 2.71261i −1.38604 0.632983i
41.11 −0.281733 + 0.959493i −0.781340 + 1.54580i −0.841254 0.540641i −0.0180311 + 0.125409i −1.26306 1.18519i 2.64141 0.151432i 0.755750 0.654861i −1.77902 2.41559i −0.115249 0.0526324i
41.12 −0.281733 + 0.959493i −0.678660 1.59356i −0.841254 0.540641i 0.563772 3.92112i 1.72021 0.202213i −2.63555 + 0.232107i 0.755750 0.654861i −2.07884 + 2.16297i 3.60345 + 1.64564i
41.13 −0.281733 + 0.959493i −0.543127 1.64469i −0.841254 0.540641i 0.342096 2.37933i 1.73109 0.0577632i 1.62492 + 2.08797i 0.755750 0.654861i −2.41003 + 1.78655i 2.18657 + 0.998573i
41.14 −0.281733 + 0.959493i −0.459895 1.66988i −0.841254 0.540641i −0.0361830 + 0.251658i 1.73180 + 0.0291935i 2.47549 0.933780i 0.755750 0.654861i −2.57699 + 1.53594i −0.231270 0.105618i
41.15 −0.281733 + 0.959493i −0.134057 + 1.72686i −0.841254 0.540641i 0.595091 4.13895i −1.61914 0.615138i 0.717684 + 2.54655i 0.755750 0.654861i −2.96406 0.462995i 3.80364 + 1.73706i
41.16 −0.281733 + 0.959493i −0.0227709 1.73190i −0.841254 0.540641i −0.321156 + 2.23369i 1.66816 + 0.466084i −1.72364 2.00725i 0.755750 0.654861i −2.99896 + 0.0788740i −2.05273 0.937451i
41.17 −0.281733 + 0.959493i 0.0227709 + 1.73190i −0.841254 0.540641i 0.321156 2.23369i −1.66816 0.466084i −0.388235 2.61711i 0.755750 0.654861i −2.99896 + 0.0788740i 2.05273 + 0.937451i
41.18 −0.281733 + 0.959493i 0.134057 1.72686i −0.841254 0.540641i −0.595091 + 4.13895i 1.61914 + 0.615138i 1.45457 + 2.21003i 0.755750 0.654861i −2.96406 0.462995i −3.80364 1.73706i
41.19 −0.281733 + 0.959493i 0.459895 + 1.66988i −0.841254 0.540641i 0.0361830 0.251658i −1.73180 0.0291935i −2.32681 + 1.25936i 0.755750 0.654861i −2.57699 + 1.53594i 0.231270 + 0.105618i
41.20 −0.281733 + 0.959493i 0.543127 + 1.64469i −0.841254 0.540641i −0.342096 + 2.37933i −1.73109 + 0.0577632i 0.513888 + 2.59537i 0.755750 0.654861i −2.41003 + 1.78655i −2.18657 0.998573i
See next 80 embeddings (of 640 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 923.64 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner
23.c even 11 1 inner
69.h odd 22 1 inner
161.l odd 22 1 inner
483.v 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.t.a 640
3.b odd 2 1 inner 966.2.t.a 640
7.b odd 2 1 inner 966.2.t.a 640
21.c even 2 1 inner 966.2.t.a 640
23.c even 11 1 inner 966.2.t.a 640
69.h odd 22 1 inner 966.2.t.a 640
161.l odd 22 1 inner 966.2.t.a 640
483.v even 22 1 inner 966.2.t.a 640

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.t.a 640 1.a even 1 1 trivial
966.2.t.a 640 3.b odd 2 1 inner
966.2.t.a 640 7.b odd 2 1 inner
966.2.t.a 640 21.c even 2 1 inner
966.2.t.a 640 23.c even 11 1 inner
966.2.t.a 640 69.h odd 22 1 inner
966.2.t.a 640 161.l odd 22 1 inner
966.2.t.a 640 483.v even 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(966, [\chi])$$.