# Properties

 Label 966.2.l.d Level $966$ Weight $2$ Character orbit 966.l Analytic conductor $7.714$ Analytic rank $0$ Dimension $56$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(47,966)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("966.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.l (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$7.71354883526$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$28$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$56 q + 28 q^{4} + 8 q^{7} + 10 q^{9}+O(q^{10})$$ 56 * q + 28 * q^4 + 8 * q^7 + 10 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$56 q + 28 q^{4} + 8 q^{7} + 10 q^{9} + 12 q^{10} + 8 q^{15} - 28 q^{16} - 8 q^{18} - 24 q^{19} - 4 q^{21} - 12 q^{22} + 6 q^{24} - 8 q^{25} + 10 q^{28} + 6 q^{30} - 6 q^{31} + 30 q^{33} + 20 q^{36} + 4 q^{37} - 10 q^{39} + 12 q^{40} + 8 q^{42} - 104 q^{43} - 6 q^{45} + 28 q^{46} + 40 q^{49} - 22 q^{51} - 6 q^{52} + 18 q^{54} + 68 q^{57} - 30 q^{58} + 4 q^{60} - 84 q^{61} - 6 q^{63} - 56 q^{64} - 30 q^{66} + 2 q^{67} + 30 q^{70} + 8 q^{72} + 54 q^{73} - 24 q^{75} + 68 q^{78} - 6 q^{79} + 22 q^{81} + 4 q^{84} - 108 q^{85} - 126 q^{87} - 6 q^{88} - 42 q^{91} + 36 q^{93} - 18 q^{94} + 6 q^{96} + 48 q^{99}+O(q^{100})$$ 56 * q + 28 * q^4 + 8 * q^7 + 10 * q^9 + 12 * q^10 + 8 * q^15 - 28 * q^16 - 8 * q^18 - 24 * q^19 - 4 * q^21 - 12 * q^22 + 6 * q^24 - 8 * q^25 + 10 * q^28 + 6 * q^30 - 6 * q^31 + 30 * q^33 + 20 * q^36 + 4 * q^37 - 10 * q^39 + 12 * q^40 + 8 * q^42 - 104 * q^43 - 6 * q^45 + 28 * q^46 + 40 * q^49 - 22 * q^51 - 6 * q^52 + 18 * q^54 + 68 * q^57 - 30 * q^58 + 4 * q^60 - 84 * q^61 - 6 * q^63 - 56 * q^64 - 30 * q^66 + 2 * q^67 + 30 * q^70 + 8 * q^72 + 54 * q^73 - 24 * q^75 + 68 * q^78 - 6 * q^79 + 22 * q^81 + 4 * q^84 - 108 * q^85 - 126 * q^87 - 6 * q^88 - 42 * q^91 + 36 * q^93 - 18 * q^94 + 6 * q^96 + 48 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1 −0.866025 0.500000i −1.72705 + 0.131543i 0.500000 + 0.866025i 1.22807 2.12708i 1.56144 + 0.749605i −0.955407 2.46722i 1.00000i 2.96539 0.454363i −2.12708 + 1.22807i
47.2 −0.866025 0.500000i −1.71118 0.268099i 0.500000 + 0.866025i −0.519082 + 0.899076i 1.34787 + 1.08777i 2.02474 + 1.70307i 1.00000i 2.85625 + 0.917529i 0.899076 0.519082i
47.3 −0.866025 0.500000i −1.67384 + 0.445275i 0.500000 + 0.866025i −1.04467 + 1.80941i 1.67222 + 0.451299i −2.62205 0.353363i 1.00000i 2.60346 1.49064i 1.80941 1.04467i
47.4 −0.866025 0.500000i −1.20460 + 1.24456i 0.500000 + 0.866025i 0.662008 1.14663i 1.66550 0.475517i 2.07946 + 1.63580i 1.00000i −0.0978548 2.99840i −1.14663 + 0.662008i
47.5 −0.866025 0.500000i −0.896677 + 1.48188i 0.500000 + 0.866025i 1.18968 2.06059i 1.51749 0.835007i 1.24146 2.33640i 1.00000i −1.39194 2.65754i −2.06059 + 1.18968i
47.6 −0.866025 0.500000i −0.778308 1.54733i 0.500000 + 0.866025i −1.34998 + 2.33823i −0.0996314 + 1.72918i −2.25095 + 1.39041i 1.00000i −1.78847 + 2.40860i 2.33823 1.34998i
47.7 −0.866025 0.500000i 0.191760 1.72140i 0.500000 + 0.866025i −0.750837 + 1.30049i −1.02677 + 1.39490i −0.436788 2.60945i 1.00000i −2.92646 0.660191i 1.30049 0.750837i
47.8 −0.866025 0.500000i 0.219734 + 1.71806i 0.500000 + 0.866025i −1.55436 + 2.69223i 0.668732 1.59775i 1.70233 + 2.02535i 1.00000i −2.90343 + 0.755032i 2.69223 1.55436i
47.9 −0.866025 0.500000i 0.845576 1.51162i 0.500000 + 0.866025i 1.37030 2.37343i −1.48810 + 0.886316i −2.52742 + 0.782396i 1.00000i −1.57000 2.55638i −2.37343 + 1.37030i
47.10 −0.866025 0.500000i 1.43371 + 0.971841i 0.500000 + 0.866025i −0.427501 + 0.740454i −0.755709 1.55849i −2.36450 1.18708i 1.00000i 1.11105 + 2.78668i 0.740454 0.427501i
47.11 −0.866025 0.500000i 1.45188 + 0.944479i 0.500000 + 0.866025i −1.52157 + 2.63544i −0.785127 1.54388i −0.624219 + 2.57106i 1.00000i 1.21592 + 2.74254i 2.63544 1.52157i
47.12 −0.866025 0.500000i 1.45696 0.936628i 0.500000 + 0.866025i −1.06670 + 1.84758i −1.73008 + 0.0826645i 2.47771 0.927868i 1.00000i 1.24546 2.72926i 1.84758 1.06670i
47.13 −0.866025 0.500000i 1.54271 + 0.787427i 0.500000 + 0.866025i 1.84945 3.20334i −0.942314 1.45329i 1.61347 2.09684i 1.00000i 1.75992 + 2.42955i −3.20334 + 1.84945i
47.14 −0.866025 0.500000i 1.71535 0.239977i 0.500000 + 0.866025i 0.203133 0.351837i −1.60552 0.649847i 2.64215 + 0.138089i 1.00000i 2.88482 0.823286i −0.351837 + 0.203133i
47.15 0.866025 + 0.500000i −1.73168 + 0.0356051i 0.500000 + 0.866025i −1.18968 + 2.06059i −1.51749 0.835007i 1.24146 2.33640i 1.00000i 2.99746 0.123314i −2.06059 + 1.18968i
47.16 0.866025 + 0.500000i −1.68012 + 0.420939i 0.500000 + 0.866025i −0.662008 + 1.14663i −1.66550 0.475517i 2.07946 + 1.63580i 1.00000i 2.64562 1.41446i −1.14663 + 0.662008i
47.17 0.866025 + 0.500000i −1.37801 1.04932i 0.500000 + 0.866025i 1.55436 2.69223i −0.668732 1.59775i 1.70233 + 2.02535i 1.00000i 0.797840 + 2.89196i 2.69223 1.55436i
47.18 0.866025 + 0.500000i −1.22254 + 1.22695i 0.500000 + 0.866025i 1.04467 1.80941i −1.67222 + 0.451299i −2.62205 0.353363i 1.00000i −0.0108008 2.99998i 1.80941 1.04467i
47.19 0.866025 + 0.500000i −0.977444 + 1.42990i 0.500000 + 0.866025i −1.22807 + 2.12708i −1.56144 + 0.749605i −0.955407 2.46722i 1.00000i −1.08921 2.79529i −2.12708 + 1.22807i
47.20 0.866025 + 0.500000i −0.623408 + 1.61597i 0.500000 + 0.866025i 0.519082 0.899076i −1.34787 + 1.08777i 2.02474 + 1.70307i 1.00000i −2.22273 2.01482i 0.899076 0.519082i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 47.28 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.l.d 56
3.b odd 2 1 inner 966.2.l.d 56
7.d odd 6 1 inner 966.2.l.d 56
21.g even 6 1 inner 966.2.l.d 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.l.d 56 1.a even 1 1 trivial
966.2.l.d 56 3.b odd 2 1 inner
966.2.l.d 56 7.d odd 6 1 inner
966.2.l.d 56 21.g even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{56} + 74 T_{5}^{54} + 3035 T_{5}^{52} + 85738 T_{5}^{50} + 1842285 T_{5}^{48} + \cdots + 546946904514816$$ acting on $$S_{2}^{\mathrm{new}}(966, [\chi])$$.