# Properties

 Label 966.2.f.b Level $966$ Weight $2$ Character orbit 966.f Analytic conductor $7.714$ Analytic rank $0$ Dimension $24$ 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(461,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

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

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

chi := DirichletCharacter("966.461");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.f (of order $$2$$, degree $$1$$, 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: $$24$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 24 q^{4} - 4 q^{7} + 8 q^{9}+O(q^{10})$$ 24 * q - 24 * q^4 - 4 * q^7 + 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 24 q^{4} - 4 q^{7} + 8 q^{9} + 24 q^{16} + 16 q^{18} - 28 q^{21} + 8 q^{22} - 24 q^{25} + 4 q^{28} + 24 q^{30} - 8 q^{36} + 40 q^{37} + 72 q^{39} + 64 q^{43} + 24 q^{46} - 24 q^{51} + 16 q^{58} + 12 q^{63} - 24 q^{64} - 64 q^{67} + 16 q^{70} - 16 q^{72} - 32 q^{78} + 88 q^{79} + 48 q^{81} + 28 q^{84} + 64 q^{85} - 8 q^{88} - 56 q^{91} + 8 q^{93} - 8 q^{99}+O(q^{100})$$ 24 * q - 24 * q^4 - 4 * q^7 + 8 * q^9 + 24 * q^16 + 16 * q^18 - 28 * q^21 + 8 * q^22 - 24 * q^25 + 4 * q^28 + 24 * q^30 - 8 * q^36 + 40 * q^37 + 72 * q^39 + 64 * q^43 + 24 * q^46 - 24 * q^51 + 16 * q^58 + 12 * q^63 - 24 * q^64 - 64 * q^67 + 16 * q^70 - 16 * q^72 - 32 * q^78 + 88 * q^79 + 48 * q^81 + 28 * q^84 + 64 * q^85 - 8 * q^88 - 56 * q^91 + 8 * q^93 - 8 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
461.1 1.00000i −1.72384 0.168468i −1.00000 1.25574 −0.168468 + 1.72384i −1.98144 1.75325i 1.00000i 2.94324 + 0.580822i 1.25574i
461.2 1.00000i −1.66919 0.462387i −1.00000 −1.95027 −0.462387 + 1.66919i −0.602835 + 2.57616i 1.00000i 2.57240 + 1.54363i 1.95027i
461.3 1.00000i −1.62776 + 0.591954i −1.00000 2.12506 0.591954 + 1.62776i 2.32692 + 1.25915i 1.00000i 2.29918 1.92711i 2.12506i
461.4 1.00000i −0.950180 1.44816i −1.00000 0.575987 −1.44816 + 0.950180i 2.10044 1.60877i 1.00000i −1.19432 + 2.75202i 0.575987i
461.5 1.00000i −0.740879 1.56560i −1.00000 −3.67536 −1.56560 + 0.740879i −0.229372 2.63579i 1.00000i −1.90220 + 2.31984i 3.67536i
461.6 1.00000i −0.375300 + 1.69090i −1.00000 −0.513446 1.69090 + 0.375300i −2.61371 + 0.410499i 1.00000i −2.71830 1.26919i 0.513446i
461.7 1.00000i 0.375300 1.69090i −1.00000 0.513446 −1.69090 0.375300i −2.61371 0.410499i 1.00000i −2.71830 1.26919i 0.513446i
461.8 1.00000i 0.740879 + 1.56560i −1.00000 3.67536 1.56560 0.740879i −0.229372 + 2.63579i 1.00000i −1.90220 + 2.31984i 3.67536i
461.9 1.00000i 0.950180 + 1.44816i −1.00000 −0.575987 1.44816 0.950180i 2.10044 + 1.60877i 1.00000i −1.19432 + 2.75202i 0.575987i
461.10 1.00000i 1.62776 0.591954i −1.00000 −2.12506 −0.591954 1.62776i 2.32692 1.25915i 1.00000i 2.29918 1.92711i 2.12506i
461.11 1.00000i 1.66919 + 0.462387i −1.00000 1.95027 0.462387 1.66919i −0.602835 2.57616i 1.00000i 2.57240 + 1.54363i 1.95027i
461.12 1.00000i 1.72384 + 0.168468i −1.00000 −1.25574 0.168468 1.72384i −1.98144 + 1.75325i 1.00000i 2.94324 + 0.580822i 1.25574i
461.13 1.00000i −1.72384 + 0.168468i −1.00000 1.25574 −0.168468 1.72384i −1.98144 + 1.75325i 1.00000i 2.94324 0.580822i 1.25574i
461.14 1.00000i −1.66919 + 0.462387i −1.00000 −1.95027 −0.462387 1.66919i −0.602835 2.57616i 1.00000i 2.57240 1.54363i 1.95027i
461.15 1.00000i −1.62776 0.591954i −1.00000 2.12506 0.591954 1.62776i 2.32692 1.25915i 1.00000i 2.29918 + 1.92711i 2.12506i
461.16 1.00000i −0.950180 + 1.44816i −1.00000 0.575987 −1.44816 0.950180i 2.10044 + 1.60877i 1.00000i −1.19432 2.75202i 0.575987i
461.17 1.00000i −0.740879 + 1.56560i −1.00000 −3.67536 −1.56560 0.740879i −0.229372 + 2.63579i 1.00000i −1.90220 2.31984i 3.67536i
461.18 1.00000i −0.375300 1.69090i −1.00000 −0.513446 1.69090 0.375300i −2.61371 0.410499i 1.00000i −2.71830 + 1.26919i 0.513446i
461.19 1.00000i 0.375300 + 1.69090i −1.00000 0.513446 −1.69090 + 0.375300i −2.61371 + 0.410499i 1.00000i −2.71830 + 1.26919i 0.513446i
461.20 1.00000i 0.740879 1.56560i −1.00000 3.67536 1.56560 + 0.740879i −0.229372 2.63579i 1.00000i −1.90220 2.31984i 3.67536i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 461.24 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.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.f.b 24
3.b odd 2 1 inner 966.2.f.b 24
7.b odd 2 1 inner 966.2.f.b 24
21.c even 2 1 inner 966.2.f.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.f.b 24 1.a even 1 1 trivial
966.2.f.b 24 3.b odd 2 1 inner
966.2.f.b 24 7.b odd 2 1 inner
966.2.f.b 24 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{12} - 24T_{5}^{10} + 178T_{5}^{8} - 536T_{5}^{6} + 640T_{5}^{4} - 256T_{5}^{2} + 32$$ acting on $$S_{2}^{\mathrm{new}}(966, [\chi])$$.