# Properties

 Label 966.2.bd.a Level $966$ Weight $2$ Character orbit 966.bd 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.bd (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.

 $$\operatorname{Tr}(f)(q) =$$ $$1280q - 64q^{4} + 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$1280q - 64q^{4} + 4q^{9} + 16q^{15} + 64q^{16} - 44q^{18} + 120q^{21} - 16q^{22} - 12q^{24} + 56q^{25} + 32q^{30} - 24q^{33} + 8q^{36} - 44q^{37} - 20q^{39} + 4q^{42} + 136q^{43} + 12q^{45} + 12q^{46} + 92q^{49} + 4q^{51} - 36q^{54} - 56q^{57} - 28q^{58} + 8q^{60} + 72q^{61} - 134q^{63} + 128q^{64} + 24q^{67} - 72q^{70} - 44q^{72} - 72q^{73} + 48q^{75} - 16q^{78} - 72q^{79} + 40q^{81} + 48q^{82} - 10q^{84} - 32q^{85} + 222q^{87} - 8q^{88} - 8q^{91} - 16q^{93} + 72q^{94} - 12q^{96} - 68q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1 −0.945001 0.327068i −1.72725 0.128835i 0.786053 + 0.618159i −2.25085 + 0.214930i 1.59012 + 0.686678i −2.10733 1.59974i −0.540641 0.841254i 2.96680 + 0.445061i 2.19735 + 0.533071i
59.2 −0.945001 0.327068i −1.72300 + 0.176876i 0.786053 + 0.618159i −3.77287 + 0.360266i 1.68608 + 0.396389i 1.25466 + 2.32934i −0.540641 0.841254i 2.93743 0.609512i 3.68320 + 0.893535i
59.3 −0.945001 0.327068i −1.71282 0.257380i 0.786053 + 0.618159i −0.220141 + 0.0210209i 1.53444 + 0.803434i 2.60042 + 0.487642i −0.540641 0.841254i 2.86751 + 0.881693i 0.214909 + 0.0521363i
59.4 −0.945001 0.327068i −1.66777 + 0.467492i 0.786053 + 0.618159i 3.60829 0.344550i 1.72894 + 0.103693i −2.58647 0.556950i −0.540641 0.841254i 2.56290 1.55934i −3.52253 0.854556i
59.5 −0.945001 0.327068i −1.49813 0.869251i 0.786053 + 0.618159i 2.42488 0.231548i 1.13143 + 1.31143i −1.25611 + 2.32856i −0.540641 0.841254i 1.48881 + 2.60451i −2.36724 0.574287i
59.6 −0.945001 0.327068i −1.40527 + 1.01253i 0.786053 + 0.618159i 1.29830 0.123973i 1.65915 0.497225i −0.324812 2.62574i −0.540641 0.841254i 0.949560 2.84576i −1.26744 0.307478i
59.7 −0.945001 0.327068i −1.38916 + 1.03452i 0.786053 + 0.618159i −0.847161 + 0.0808940i 1.65112 0.523271i −1.22496 + 2.34510i −0.540641 0.841254i 0.859542 2.87423i 0.827025 + 0.200634i
59.8 −0.945001 0.327068i −1.30303 1.14110i 0.786053 + 0.618159i 0.986781 0.0942262i 0.858148 + 1.50452i −0.514065 + 2.59533i −0.540641 0.841254i 0.395779 + 2.97378i −0.963327 0.233701i
59.9 −0.945001 0.327068i −1.02432 1.39670i 0.786053 + 0.618159i −3.14301 + 0.300121i 0.511165 + 1.65491i 1.51112 2.17176i −0.540641 0.841254i −0.901547 + 2.86133i 3.06831 + 0.744364i
59.10 −0.945001 0.327068i −0.968062 1.43626i 0.786053 + 0.618159i −0.185246 + 0.0176888i 0.445064 + 1.67389i −0.860764 2.50182i −0.540641 0.841254i −1.12571 + 2.78079i 0.180843 + 0.0438720i
59.11 −0.945001 0.327068i −0.755746 + 1.55848i 0.786053 + 0.618159i −0.102884 + 0.00982421i 1.22391 1.22558i 2.61358 0.411367i −0.540641 0.841254i −1.85770 2.35562i 0.100438 + 0.0243661i
59.12 −0.945001 0.327068i −0.689077 1.58908i 0.786053 + 0.618159i 3.18017 0.303670i 0.131442 + 1.72706i −2.23440 1.41685i −0.540641 0.841254i −2.05034 + 2.19000i −3.10459 0.753165i
59.13 −0.945001 0.327068i −0.321085 + 1.70203i 0.786053 + 0.618159i −2.90323 + 0.277225i 0.860105 1.50340i −2.41573 + 1.07900i −0.540641 0.841254i −2.79381 1.09299i 2.83423 + 0.687576i
59.14 −0.945001 0.327068i −0.133141 + 1.72693i 0.786053 + 0.618159i 3.12559 0.298458i 0.690641 1.58840i 0.569043 + 2.58383i −0.540641 0.841254i −2.96455 0.459850i −3.05130 0.740238i
59.15 −0.945001 0.327068i −0.0481888 1.73138i 0.786053 + 0.618159i −1.37494 + 0.131291i −0.520741 + 1.65192i 2.07811 + 1.63751i −0.540641 0.841254i −2.99536 + 0.166866i 1.34226 + 0.325630i
59.16 −0.945001 0.327068i −0.0112042 1.73201i 0.786053 + 0.618159i 3.78210 0.361147i −0.555899 + 1.64042i 2.61932 0.373018i −0.540641 0.841254i −2.99975 + 0.0388116i −3.69221 0.895721i
59.17 −0.945001 0.327068i −0.00143408 + 1.73205i 0.786053 + 0.618159i −2.94984 + 0.281676i 0.567853 1.63632i −1.51302 2.17043i −0.540641 0.841254i −3.00000 0.00496781i 2.87973 + 0.698615i
59.18 −0.945001 0.327068i 0.129478 + 1.72720i 0.786053 + 0.618159i 2.15460 0.205739i 0.442557 1.67456i 0.734488 2.54176i −0.540641 0.841254i −2.96647 + 0.447269i −2.10339 0.510276i
59.19 −0.945001 0.327068i 0.238443 1.71556i 0.786053 + 0.618159i −0.133592 + 0.0127565i −0.786433 + 1.54322i −2.64105 0.157712i −0.540641 0.841254i −2.88629 0.818126i 0.130417 + 0.0316389i
59.20 −0.945001 0.327068i 0.974657 + 1.43180i 0.786053 + 0.618159i −3.35223 + 0.320099i −0.452757 1.67183i 2.60360 + 0.470376i −0.540641 0.841254i −1.10009 + 2.79102i 3.27255 + 0.793912i
See next 80 embeddings (of 1280 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 929.64 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.d odd 6 1 inner
21.g even 6 1 inner
23.c even 11 1 inner
69.h odd 22 1 inner
161.n odd 66 1 inner
483.bb 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.bd.a 1280
3.b odd 2 1 inner 966.2.bd.a 1280
7.d odd 6 1 inner 966.2.bd.a 1280
21.g even 6 1 inner 966.2.bd.a 1280
23.c even 11 1 inner 966.2.bd.a 1280
69.h odd 22 1 inner 966.2.bd.a 1280
161.n odd 66 1 inner 966.2.bd.a 1280
483.bb even 66 1 inner 966.2.bd.a 1280

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.bd.a 1280 1.a even 1 1 trivial
966.2.bd.a 1280 3.b odd 2 1 inner
966.2.bd.a 1280 7.d odd 6 1 inner
966.2.bd.a 1280 21.g even 6 1 inner
966.2.bd.a 1280 23.c even 11 1 inner
966.2.bd.a 1280 69.h odd 22 1 inner
966.2.bd.a 1280 161.n odd 66 1 inner
966.2.bd.a 1280 483.bb even 66 1 inner

## Hecke kernels

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