# Properties

 Label 966.2.k.a Level $966$ Weight $2$ Character orbit 966.k Analytic conductor $7.714$ Analytic rank $0$ Dimension $32$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(229,966)]

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([0, 5, 3]))

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

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

chi := DirichletCharacter("966.229");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.k (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: $$32$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 16 q^{2} - 16 q^{4} + 32 q^{8} + 16 q^{9}+O(q^{10})$$ 32 * q - 16 * q^2 - 16 * q^4 + 32 * q^8 + 16 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 16 q^{2} - 16 q^{4} + 32 q^{8} + 16 q^{9} - 16 q^{16} + 16 q^{18} + 8 q^{23} - 24 q^{25} - 12 q^{26} - 8 q^{29} + 48 q^{31} - 16 q^{32} - 20 q^{35} - 32 q^{36} - 8 q^{39} + 8 q^{46} + 12 q^{47} + 24 q^{49} + 48 q^{50} + 12 q^{52} + 4 q^{58} - 12 q^{59} + 32 q^{64} + 64 q^{70} + 48 q^{71} + 16 q^{72} + 12 q^{73} + 36 q^{75} + 64 q^{77} + 16 q^{78} - 16 q^{81} - 12 q^{82} + 32 q^{85} - 24 q^{87} - 16 q^{92} - 12 q^{94} + 40 q^{95} - 24 q^{98}+O(q^{100})$$ 32 * q - 16 * q^2 - 16 * q^4 + 32 * q^8 + 16 * q^9 - 16 * q^16 + 16 * q^18 + 8 * q^23 - 24 * q^25 - 12 * q^26 - 8 * q^29 + 48 * q^31 - 16 * q^32 - 20 * q^35 - 32 * q^36 - 8 * q^39 + 8 * q^46 + 12 * q^47 + 24 * q^49 + 48 * q^50 + 12 * q^52 + 4 * q^58 - 12 * q^59 + 32 * q^64 + 64 * q^70 + 48 * q^71 + 16 * q^72 + 12 * q^73 + 36 * q^75 + 64 * q^77 + 16 * q^78 - 16 * q^81 - 12 * q^82 + 32 * q^85 - 24 * q^87 - 16 * q^92 - 12 * q^94 + 40 * q^95 - 24 * q^98

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
229.1 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i −1.81715 + 3.14740i 1.00000i −1.58446 + 2.11884i 1.00000 0.500000 0.866025i −1.81715 3.14740i
229.2 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i −1.52184 + 2.63590i 1.00000i −1.49399 2.18358i 1.00000 0.500000 0.866025i −1.52184 2.63590i
229.3 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i −1.44868 + 2.50918i 1.00000i 2.53486 0.757941i 1.00000 0.500000 0.866025i −1.44868 2.50918i
229.4 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i −0.286936 + 0.496987i 1.00000i 1.61243 + 2.09764i 1.00000 0.500000 0.866025i −0.286936 0.496987i
229.5 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i 0.286936 0.496987i 1.00000i −1.61243 2.09764i 1.00000 0.500000 0.866025i 0.286936 + 0.496987i
229.6 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i 1.44868 2.50918i 1.00000i −2.53486 + 0.757941i 1.00000 0.500000 0.866025i 1.44868 + 2.50918i
229.7 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i 1.52184 2.63590i 1.00000i 1.49399 + 2.18358i 1.00000 0.500000 0.866025i 1.52184 + 2.63590i
229.8 −0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i 1.81715 3.14740i 1.00000i 1.58446 2.11884i 1.00000 0.500000 0.866025i 1.81715 + 3.14740i
229.9 −0.500000 + 0.866025i 0.866025 0.500000i −0.500000 0.866025i −2.07087 + 3.58686i 1.00000i −1.78719 + 1.95088i 1.00000 0.500000 0.866025i −2.07087 3.58686i
229.10 −0.500000 + 0.866025i 0.866025 0.500000i −0.500000 0.866025i −0.856094 + 1.48280i 1.00000i 0.871223 + 2.49819i 1.00000 0.500000 0.866025i −0.856094 1.48280i
229.11 −0.500000 + 0.866025i 0.866025 0.500000i −0.500000 0.866025i −0.347206 + 0.601379i 1.00000i −2.64100 + 0.158433i 1.00000 0.500000 0.866025i −0.347206 0.601379i
229.12 −0.500000 + 0.866025i 0.866025 0.500000i −0.500000 0.866025i −0.242904 + 0.420722i 1.00000i −2.51079 0.834227i 1.00000 0.500000 0.866025i −0.242904 0.420722i
229.13 −0.500000 + 0.866025i 0.866025 0.500000i −0.500000 0.866025i 0.242904 0.420722i 1.00000i 2.51079 + 0.834227i 1.00000 0.500000 0.866025i 0.242904 + 0.420722i
229.14 −0.500000 + 0.866025i 0.866025 0.500000i −0.500000 0.866025i 0.347206 0.601379i 1.00000i 2.64100 0.158433i 1.00000 0.500000 0.866025i 0.347206 + 0.601379i
229.15 −0.500000 + 0.866025i 0.866025 0.500000i −0.500000 0.866025i 0.856094 1.48280i 1.00000i −0.871223 2.49819i 1.00000 0.500000 0.866025i 0.856094 + 1.48280i
229.16 −0.500000 + 0.866025i 0.866025 0.500000i −0.500000 0.866025i 2.07087 3.58686i 1.00000i 1.78719 1.95088i 1.00000 0.500000 0.866025i 2.07087 + 3.58686i
367.1 −0.500000 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −1.81715 3.14740i 1.00000i −1.58446 2.11884i 1.00000 0.500000 + 0.866025i −1.81715 + 3.14740i
367.2 −0.500000 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −1.52184 2.63590i 1.00000i −1.49399 + 2.18358i 1.00000 0.500000 + 0.866025i −1.52184 + 2.63590i
367.3 −0.500000 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −1.44868 2.50918i 1.00000i 2.53486 + 0.757941i 1.00000 0.500000 + 0.866025i −1.44868 + 2.50918i
367.4 −0.500000 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −0.286936 0.496987i 1.00000i 1.61243 2.09764i 1.00000 0.500000 + 0.866025i −0.286936 + 0.496987i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 229.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
23.b odd 2 1 inner
161.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.k.a 32
7.d odd 6 1 inner 966.2.k.a 32
23.b odd 2 1 inner 966.2.k.a 32
161.g even 6 1 inner 966.2.k.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.k.a 32 1.a even 1 1 trivial
966.2.k.a 32 7.d odd 6 1 inner
966.2.k.a 32 23.b odd 2 1 inner
966.2.k.a 32 161.g even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{32} + 52 T_{5}^{30} + 1669 T_{5}^{28} + 34076 T_{5}^{26} + 512018 T_{5}^{24} + 5540548 T_{5}^{22} + \cdots + 3748096$$ acting on $$S_{2}^{\mathrm{new}}(966, [\chi])$$.