# Properties

 Label 966.2.bf.a Level $966$ Weight $2$ Character orbit 966.bf Analytic conductor $7.714$ Analytic rank $0$ Dimension $1280$ 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.bf (of order $$66$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.71354883526$$ Analytic rank: $$0$$ Dimension: $$1280$$ Relative dimension: $$64$$ over $$\Q(\zeta_{66})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$1280q - 64q^{4} - 8q^{6} + 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$1280q - 64q^{4} - 8q^{6} + 4q^{9} + 64q^{16} + 44q^{18} - 132q^{21} - 4q^{24} + 72q^{25} + 108q^{27} + 44q^{30} + 8q^{36} + 44q^{37} - 4q^{39} - 88q^{43} - 12q^{46} + 60q^{49} - 48q^{54} + 96q^{55} + 44q^{58} - 176q^{61} - 110q^{63} + 128q^{64} + 56q^{69} - 120q^{70} + 44q^{72} + 40q^{73} - 268q^{75} + 16q^{78} + 88q^{79} + 8q^{81} - 56q^{82} - 22q^{84} + 64q^{85} - 42q^{87} - 32q^{93} + 8q^{94} + 4q^{96} + 132q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −0.458227 + 0.888835i −1.72757 + 0.124485i −0.580057 0.814576i 1.87856 + 1.79120i 0.680972 1.59257i 0.248917 + 2.63402i 0.989821 0.142315i 2.96901 0.430114i −2.45289 + 0.848953i
11.2 −0.458227 + 0.888835i −1.71624 + 0.233522i −0.580057 0.814576i −2.07174 1.97540i 0.578862 1.63246i −2.59178 + 0.531663i 0.989821 0.142315i 2.89093 0.801559i 2.70513 0.936254i
11.3 −0.458227 + 0.888835i −1.61819 + 0.617630i −0.580057 0.814576i 2.31266 + 2.20512i 0.192526 1.72132i −1.37219 2.26210i 0.989821 0.142315i 2.23707 1.99888i −3.01971 + 1.04513i
11.4 −0.458227 + 0.888835i −1.61477 0.626501i −0.580057 0.814576i −1.38105 1.31683i 1.29679 1.14819i 2.39472 + 1.12486i 0.989821 0.142315i 2.21499 + 2.02332i 1.80327 0.624119i
11.5 −0.458227 + 0.888835i −1.59740 0.669558i −0.580057 0.814576i −1.48546 1.41639i 1.32710 1.11302i 0.598705 2.57712i 0.989821 0.142315i 2.10338 + 2.13911i 1.93961 0.671307i
11.6 −0.458227 + 0.888835i −1.47486 + 0.908181i −0.580057 0.814576i 0.367129 + 0.350057i −0.131404 1.72706i 2.61845 0.379131i 0.989821 0.142315i 1.35042 2.67888i −0.479371 + 0.165912i
11.7 −0.458227 + 0.888835i −1.24755 1.20151i −0.580057 0.814576i 2.79695 + 2.66688i 1.63960 0.558302i 2.62506 0.330282i 0.989821 0.142315i 0.112753 + 2.99788i −3.65206 + 1.26399i
11.8 −0.458227 + 0.888835i −1.17110 1.27614i −0.580057 0.814576i 0.549032 + 0.523501i 1.67091 0.456154i −2.28022 1.34186i 0.989821 0.142315i −0.257056 + 2.98897i −0.716888 + 0.248117i
11.9 −0.458227 + 0.888835i −1.08373 1.35113i −0.580057 0.814576i 0.598114 + 0.570300i 1.69752 0.344132i −0.225504 + 2.63612i 0.989821 0.142315i −0.651080 + 2.92850i −0.780975 + 0.270298i
11.10 −0.458227 + 0.888835i −1.07755 + 1.35606i −0.580057 0.814576i −1.05877 1.00954i −0.711553 1.57914i −2.64409 + 0.0936177i 0.989821 0.142315i −0.677791 2.92243i 1.38247 0.478477i
11.11 −0.458227 + 0.888835i −0.777131 + 1.54792i −0.580057 0.814576i −2.85699 2.72413i −1.01975 1.40004i 2.56739 + 0.639162i 0.989821 0.142315i −1.79214 2.40588i 3.73045 1.29112i
11.12 −0.458227 + 0.888835i −0.484961 + 1.66277i −0.580057 0.814576i 0.377293 + 0.359748i −1.25571 1.19298i −1.57508 2.12582i 0.989821 0.142315i −2.52963 1.61276i −0.492643 + 0.170505i
11.13 −0.458227 + 0.888835i −0.418014 1.68085i −0.580057 0.814576i −2.21751 2.11439i 1.68555 + 0.398666i 0.589589 + 2.57922i 0.989821 0.142315i −2.65053 + 1.40524i 2.89546 1.00213i
11.14 −0.458227 + 0.888835i −0.349901 + 1.69634i −0.580057 0.814576i 1.57677 + 1.50345i −1.34743 1.08831i 1.82632 1.91430i 0.989821 0.142315i −2.75514 1.18710i −2.05883 + 0.712569i
11.15 −0.458227 + 0.888835i −0.119815 1.72790i −0.580057 0.814576i 0.424017 + 0.404299i 1.59072 + 0.685274i −2.01888 + 1.71001i 0.989821 0.142315i −2.97129 + 0.414058i −0.553651 + 0.191621i
11.16 −0.458227 + 0.888835i −0.118210 1.72801i −0.580057 0.814576i 0.829499 + 0.790926i 1.59009 + 0.686752i 2.21783 1.44264i 0.989821 0.142315i −2.97205 + 0.408537i −1.08310 + 0.374865i
11.17 −0.458227 + 0.888835i 0.111990 + 1.72843i −0.580057 0.814576i −0.770390 0.734565i −1.58760 0.692470i 0.520510 + 2.59405i 0.989821 0.142315i −2.97492 + 0.387134i 1.00592 0.348153i
11.18 −0.458227 + 0.888835i 0.328863 1.70054i −0.580057 0.814576i −2.51640 2.39938i 1.36081 + 1.07154i 2.33268 1.24844i 0.989821 0.142315i −2.78370 1.11849i 3.28573 1.13720i
11.19 −0.458227 + 0.888835i 0.375913 1.69077i −0.580057 0.814576i 3.10185 + 2.95761i 1.33056 + 1.10888i −1.66460 2.05648i 0.989821 0.142315i −2.71738 1.27116i −4.05017 + 1.40178i
11.20 −0.458227 + 0.888835i 0.470566 + 1.66690i −0.580057 0.814576i 2.48669 + 2.37105i −1.69723 0.345564i 1.41557 + 2.23521i 0.989821 0.142315i −2.55714 + 1.56878i −3.24694 + 1.12378i
See next 80 embeddings (of 1280 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 935.64 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.c even 3 1 inner
21.h odd 6 1 inner
23.d odd 22 1 inner
69.g even 22 1 inner
161.p odd 66 1 inner
483.bc even 66 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.bf.a 1280
3.b odd 2 1 inner 966.2.bf.a 1280
7.c even 3 1 inner 966.2.bf.a 1280
21.h odd 6 1 inner 966.2.bf.a 1280
23.d odd 22 1 inner 966.2.bf.a 1280
69.g even 22 1 inner 966.2.bf.a 1280
161.p odd 66 1 inner 966.2.bf.a 1280
483.bc even 66 1 inner 966.2.bf.a 1280

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.bf.a 1280 1.a even 1 1 trivial
966.2.bf.a 1280 3.b odd 2 1 inner
966.2.bf.a 1280 7.c even 3 1 inner
966.2.bf.a 1280 21.h odd 6 1 inner
966.2.bf.a 1280 23.d odd 22 1 inner
966.2.bf.a 1280 69.g even 22 1 inner
966.2.bf.a 1280 161.p odd 66 1 inner
966.2.bf.a 1280 483.bc even 66 1 inner

## Hecke kernels

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