# Properties

 Label 966.2.k.b Level $966$ Weight $2$ Character orbit 966.k Analytic conductor $7.714$ Analytic rank $0$ Dimension $32$ CM no 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 dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, 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 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 16 q^{2} - 16 q^{4} - 32 q^{8} + 16 q^{9} - 16 q^{16} - 16 q^{18} + 4 q^{23} - 40 q^{25} - 12 q^{26} + 24 q^{29} - 24 q^{31} + 16 q^{32} + 12 q^{35} - 32 q^{36} + 8 q^{39} - 4 q^{46} + 12 q^{47} + 24 q^{49} - 80 q^{50} - 12 q^{52} + 12 q^{58} - 12 q^{59} + 32 q^{64} + 24 q^{70} + 16 q^{71} - 16 q^{72} - 84 q^{73} + 12 q^{75} - 40 q^{77} + 16 q^{78} - 16 q^{81} - 36 q^{82} - 112 q^{85} - 48 q^{87} - 8 q^{92} - 8 q^{93} + 12 q^{94} + 24 q^{98}+O(q^{100})$$ 32 * q + 16 * q^2 - 16 * q^4 - 32 * q^8 + 16 * q^9 - 16 * q^16 - 16 * q^18 + 4 * q^23 - 40 * q^25 - 12 * q^26 + 24 * q^29 - 24 * q^31 + 16 * q^32 + 12 * q^35 - 32 * q^36 + 8 * q^39 - 4 * q^46 + 12 * q^47 + 24 * q^49 - 80 * q^50 - 12 * q^52 + 12 * q^58 - 12 * q^59 + 32 * q^64 + 24 * q^70 + 16 * q^71 - 16 * q^72 - 84 * q^73 + 12 * q^75 - 40 * q^77 + 16 * q^78 - 16 * q^81 - 36 * q^82 - 112 * q^85 - 48 * q^87 - 8 * q^92 - 8 * q^93 + 12 * q^94 + 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 −2.02969 + 3.51552i 1.00000i 2.03303 + 1.69316i −1.00000 0.500000 0.866025i 2.02969 + 3.51552i
229.2 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i −1.92123 + 3.32767i 1.00000i 0.661545 2.56171i −1.00000 0.500000 0.866025i 1.92123 + 3.32767i
229.3 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i −0.320775 + 0.555599i 1.00000i −2.04200 + 1.68233i −1.00000 0.500000 0.866025i 0.320775 + 0.555599i
229.4 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i −0.139134 + 0.240988i 1.00000i 2.42763 + 1.05197i −1.00000 0.500000 0.866025i 0.139134 + 0.240988i
229.5 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i 0.139134 0.240988i 1.00000i −2.42763 1.05197i −1.00000 0.500000 0.866025i −0.139134 0.240988i
229.6 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i 0.320775 0.555599i 1.00000i 2.04200 1.68233i −1.00000 0.500000 0.866025i −0.320775 0.555599i
229.7 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i 1.92123 3.32767i 1.00000i −0.661545 + 2.56171i −1.00000 0.500000 0.866025i −1.92123 3.32767i
229.8 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i 2.02969 3.51552i 1.00000i −2.03303 1.69316i −1.00000 0.500000 0.866025i −2.02969 3.51552i
229.9 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i −1.70402 + 2.95145i 1.00000i −2.64298 0.121148i −1.00000 0.500000 0.866025i 1.70402 + 2.95145i
229.10 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i −1.36115 + 2.35758i 1.00000i 2.49823 0.871108i −1.00000 0.500000 0.866025i 1.36115 + 2.35758i
229.11 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i −1.28362 + 2.22329i 1.00000i −1.74878 1.98539i −1.00000 0.500000 0.866025i 1.28362 + 2.22329i
229.12 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i −0.814185 + 1.41021i 1.00000i −0.285113 + 2.63034i −1.00000 0.500000 0.866025i 0.814185 + 1.41021i
229.13 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i 0.814185 1.41021i 1.00000i 0.285113 2.63034i −1.00000 0.500000 0.866025i −0.814185 1.41021i
229.14 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i 1.28362 2.22329i 1.00000i 1.74878 + 1.98539i −1.00000 0.500000 0.866025i −1.28362 2.22329i
229.15 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i 1.36115 2.35758i 1.00000i −2.49823 + 0.871108i −1.00000 0.500000 0.866025i −1.36115 2.35758i
229.16 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i 1.70402 2.95145i 1.00000i 2.64298 + 0.121148i −1.00000 0.500000 0.866025i −1.70402 2.95145i
367.1 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −2.02969 3.51552i 1.00000i 2.03303 1.69316i −1.00000 0.500000 + 0.866025i 2.02969 3.51552i
367.2 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −1.92123 3.32767i 1.00000i 0.661545 + 2.56171i −1.00000 0.500000 + 0.866025i 1.92123 3.32767i
367.3 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −0.320775 0.555599i 1.00000i −2.04200 1.68233i −1.00000 0.500000 + 0.866025i 0.320775 0.555599i
367.4 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −0.139134 0.240988i 1.00000i 2.42763 1.05197i −1.00000 0.500000 + 0.866025i 0.139134 0.240988i
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.b 32
7.d odd 6 1 inner 966.2.k.b 32
23.b odd 2 1 inner 966.2.k.b 32
161.g even 6 1 inner 966.2.k.b 32

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{32} + 60 T_{5}^{30} + 2165 T_{5}^{28} + 51252 T_{5}^{26} + 900834 T_{5}^{24} + 11822508 T_{5}^{22} + \cdots + 136048896$$ acting on $$S_{2}^{\mathrm{new}}(966, [\chi])$$.