# Properties

 Label 966.2.f.c Level $966$ Weight $2$ Character orbit 966.f Analytic conductor $7.714$ Analytic rank $0$ Dimension $28$ 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: $$28$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 28 q^{4} + 4 q^{7} - 4 q^{9}+O(q^{10})$$ 28 * q - 28 * q^4 + 4 * q^7 - 4 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 28 q^{4} + 4 q^{7} - 4 q^{9} + 16 q^{15} + 28 q^{16} - 16 q^{18} + 4 q^{21} + 80 q^{25} - 4 q^{28} + 12 q^{30} + 4 q^{36} + 20 q^{37} - 20 q^{39} + 28 q^{42} - 28 q^{43} - 28 q^{46} - 28 q^{49} + 16 q^{51} - 8 q^{57} - 36 q^{58} - 16 q^{60} + 36 q^{63} - 28 q^{64} - 8 q^{67} - 60 q^{70} + 16 q^{72} + 16 q^{78} - 76 q^{81} - 4 q^{84} - 24 q^{85} + 36 q^{91} + 48 q^{93} + 72 q^{99}+O(q^{100})$$ 28 * q - 28 * q^4 + 4 * q^7 - 4 * q^9 + 16 * q^15 + 28 * q^16 - 16 * q^18 + 4 * q^21 + 80 * q^25 - 4 * q^28 + 12 * q^30 + 4 * q^36 + 20 * q^37 - 20 * q^39 + 28 * q^42 - 28 * q^43 - 28 * q^46 - 28 * q^49 + 16 * q^51 - 8 * q^57 - 36 * q^58 - 16 * q^60 + 36 * q^63 - 28 * q^64 - 8 * q^67 - 60 * q^70 + 16 * q^72 + 16 * q^78 - 76 * q^81 - 4 * q^84 - 24 * q^85 + 36 * q^91 + 48 * q^93 + 72 * 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 −1.64221 0.550576i −1.00000 −1.99217 −0.550576 + 1.64221i 2.64482 0.0703415i 1.00000i 2.39373 + 1.80833i 1.99217i
461.2 1.00000i −1.55720 + 0.758370i −1.00000 −2.55377 0.758370 + 1.55720i 0.900399 2.48783i 1.00000i 1.84975 2.36187i 2.55377i
461.3 1.00000i −1.37202 + 1.05715i −1.00000 −2.38722 1.05715 + 1.37202i −2.59254 0.527932i 1.00000i 0.764872 2.90086i 2.38722i
461.4 1.00000i −1.18002 1.26789i −1.00000 3.87735 −1.26789 + 1.18002i 0.302862 2.62836i 1.00000i −0.215085 + 2.99228i 3.87735i
461.5 1.00000i −0.952183 + 1.44684i −1.00000 3.39470 1.44684 + 0.952183i 1.73823 1.99463i 1.00000i −1.18670 2.75531i 3.39470i
461.6 1.00000i −0.711304 + 1.57925i −1.00000 0.334212 1.57925 + 0.711304i −0.168570 + 2.64038i 1.00000i −1.98809 2.24666i 0.334212i
461.7 1.00000i −0.436762 1.67608i −1.00000 −3.48442 −1.67608 + 0.436762i −1.82520 + 1.91537i 1.00000i −2.61848 + 1.46409i 3.48442i
461.8 1.00000i 0.436762 + 1.67608i −1.00000 3.48442 1.67608 0.436762i −1.82520 1.91537i 1.00000i −2.61848 + 1.46409i 3.48442i
461.9 1.00000i 0.711304 1.57925i −1.00000 −0.334212 −1.57925 0.711304i −0.168570 2.64038i 1.00000i −1.98809 2.24666i 0.334212i
461.10 1.00000i 0.952183 1.44684i −1.00000 −3.39470 −1.44684 0.952183i 1.73823 + 1.99463i 1.00000i −1.18670 2.75531i 3.39470i
461.11 1.00000i 1.18002 + 1.26789i −1.00000 −3.87735 1.26789 1.18002i 0.302862 + 2.62836i 1.00000i −0.215085 + 2.99228i 3.87735i
461.12 1.00000i 1.37202 1.05715i −1.00000 2.38722 −1.05715 1.37202i −2.59254 + 0.527932i 1.00000i 0.764872 2.90086i 2.38722i
461.13 1.00000i 1.55720 0.758370i −1.00000 2.55377 −0.758370 1.55720i 0.900399 + 2.48783i 1.00000i 1.84975 2.36187i 2.55377i
461.14 1.00000i 1.64221 + 0.550576i −1.00000 1.99217 0.550576 1.64221i 2.64482 + 0.0703415i 1.00000i 2.39373 + 1.80833i 1.99217i
461.15 1.00000i −1.64221 + 0.550576i −1.00000 −1.99217 −0.550576 1.64221i 2.64482 + 0.0703415i 1.00000i 2.39373 1.80833i 1.99217i
461.16 1.00000i −1.55720 0.758370i −1.00000 −2.55377 0.758370 1.55720i 0.900399 + 2.48783i 1.00000i 1.84975 + 2.36187i 2.55377i
461.17 1.00000i −1.37202 1.05715i −1.00000 −2.38722 1.05715 1.37202i −2.59254 + 0.527932i 1.00000i 0.764872 + 2.90086i 2.38722i
461.18 1.00000i −1.18002 + 1.26789i −1.00000 3.87735 −1.26789 1.18002i 0.302862 + 2.62836i 1.00000i −0.215085 2.99228i 3.87735i
461.19 1.00000i −0.952183 1.44684i −1.00000 3.39470 1.44684 0.952183i 1.73823 + 1.99463i 1.00000i −1.18670 + 2.75531i 3.39470i
461.20 1.00000i −0.711304 1.57925i −1.00000 0.334212 1.57925 0.711304i −0.168570 2.64038i 1.00000i −1.98809 + 2.24666i 0.334212i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 461.28 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.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.c 28
3.b odd 2 1 inner 966.2.f.c 28
7.b odd 2 1 inner 966.2.f.c 28
21.c even 2 1 inner 966.2.f.c 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.f.c 28 1.a even 1 1 trivial
966.2.f.c 28 3.b odd 2 1 inner
966.2.f.c 28 7.b odd 2 1 inner
966.2.f.c 28 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{14} - 55T_{5}^{12} + 1214T_{5}^{10} - 13726T_{5}^{8} + 83744T_{5}^{6} - 262496T_{5}^{4} + 338560T_{5}^{2} - 34656$$ acting on $$S_{2}^{\mathrm{new}}(966, [\chi])$$.