# Properties

 Label 966.2.l.c Level $966$ Weight $2$ Character orbit 966.l Analytic conductor $7.714$ Analytic rank $0$ Dimension $56$ 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(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 dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$56 q - 6 q^{3} + 28 q^{4} + 20 q^{7} - 2 q^{9}+O(q^{10})$$ 56 * q - 6 * q^3 + 28 * q^4 + 20 * q^7 - 2 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$56 q - 6 q^{3} + 28 q^{4} + 20 q^{7} - 2 q^{9} - 6 q^{10} - 6 q^{12} - 36 q^{15} - 28 q^{16} + 8 q^{18} - 36 q^{19} + 34 q^{21} + 16 q^{22} - 46 q^{25} + 4 q^{28} - 6 q^{30} - 6 q^{31} + 30 q^{33} - 4 q^{36} - 8 q^{37} + 42 q^{39} - 6 q^{40} + 18 q^{42} + 64 q^{43} - 6 q^{45} - 28 q^{46} - 40 q^{49} - 24 q^{51} - 36 q^{52} - 60 q^{57} - 16 q^{58} - 18 q^{60} + 12 q^{61} + 36 q^{63} - 56 q^{64} + 48 q^{66} - 16 q^{67} - 34 q^{70} - 8 q^{72} - 102 q^{73} + 18 q^{75} - 16 q^{78} + 24 q^{79} + 54 q^{81} + 12 q^{82} - 4 q^{84} + 128 q^{85} + 102 q^{87} + 8 q^{88} + 68 q^{91} - 32 q^{93} + 6 q^{94} - 112 q^{99}+O(q^{100})$$ 56 * q - 6 * q^3 + 28 * q^4 + 20 * q^7 - 2 * q^9 - 6 * q^10 - 6 * q^12 - 36 * q^15 - 28 * q^16 + 8 * q^18 - 36 * q^19 + 34 * q^21 + 16 * q^22 - 46 * q^25 + 4 * q^28 - 6 * q^30 - 6 * q^31 + 30 * q^33 - 4 * q^36 - 8 * q^37 + 42 * q^39 - 6 * q^40 + 18 * q^42 + 64 * q^43 - 6 * q^45 - 28 * q^46 - 40 * q^49 - 24 * q^51 - 36 * q^52 - 60 * q^57 - 16 * q^58 - 18 * q^60 + 12 * q^61 + 36 * q^63 - 56 * q^64 + 48 * q^66 - 16 * q^67 - 34 * q^70 - 8 * q^72 - 102 * q^73 + 18 * q^75 - 16 * q^78 + 24 * q^79 + 54 * q^81 + 12 * q^82 - 4 * q^84 + 128 * q^85 + 102 * q^87 + 8 * q^88 + 68 * q^91 - 32 * q^93 + 6 * q^94 - 112 * 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 −1.72955 0.0930981i 0.500000 + 0.866025i 1.85941 3.22059i 1.45128 + 0.945399i −1.67325 + 2.04945i 1.00000i 2.98267 + 0.322035i −3.22059 + 1.85941i
47.2 −0.866025 0.500000i −1.63883 0.560573i 0.500000 + 0.866025i −0.519026 + 0.898980i 1.13898 + 1.30488i 1.91946 1.82090i 1.00000i 2.37152 + 1.83737i 0.898980 0.519026i
47.3 −0.866025 0.500000i −1.43165 + 0.974877i 0.500000 + 0.866025i −0.723421 + 1.25300i 1.72728 0.128444i −1.91613 1.82440i 1.00000i 1.09923 2.79136i 1.25300 0.723421i
47.4 −0.866025 0.500000i −1.24955 + 1.19943i 0.500000 + 0.866025i −1.89177 + 3.27664i 1.68186 0.413957i 2.63339 0.255430i 1.00000i 0.122757 2.99749i 3.27664 1.89177i
47.5 −0.866025 0.500000i −1.13768 1.30602i 0.500000 + 0.866025i 0.890465 1.54233i 0.332245 + 1.69989i −2.00001 1.73204i 1.00000i −0.411389 + 2.97166i −1.54233 + 0.890465i
47.6 −0.866025 0.500000i −0.319643 1.70230i 0.500000 + 0.866025i 0.224543 0.388920i −0.574332 + 1.63406i 2.31951 1.27274i 1.00000i −2.79566 + 1.08826i −0.388920 + 0.224543i
47.7 −0.866025 0.500000i −0.165511 + 1.72412i 0.500000 + 0.866025i −0.592581 + 1.02638i 1.00540 1.41038i −2.13419 + 1.56372i 1.00000i −2.94521 0.570723i 1.02638 0.592581i
47.8 −0.866025 0.500000i 0.110462 + 1.72852i 0.500000 + 0.866025i −0.0289992 + 0.0502281i 0.768599 1.55218i 1.41346 2.23654i 1.00000i −2.97560 + 0.381874i 0.0502281 0.0289992i
47.9 −0.866025 0.500000i 0.185627 + 1.72208i 0.500000 + 0.866025i 1.97297 3.41729i 0.700280 1.58417i 1.45851 + 2.20743i 1.00000i −2.93109 + 0.639328i −3.41729 + 1.97297i
47.10 −0.866025 0.500000i 0.454222 1.67143i 0.500000 + 0.866025i 2.02138 3.50113i −1.22908 + 1.22039i 2.64571 + 0.0151948i 1.00000i −2.58737 1.51840i −3.50113 + 2.02138i
47.11 −0.866025 0.500000i 0.633338 1.61211i 0.500000 + 0.866025i −1.42388 + 2.46624i −1.35454 + 1.07946i 0.837175 + 2.50981i 1.00000i −2.19777 2.04201i 2.46624 1.42388i
47.12 −0.866025 0.500000i 1.55959 0.753450i 0.500000 + 0.866025i 0.416548 0.721483i −1.72737 0.127287i −0.928188 + 2.47759i 1.00000i 1.86463 2.35014i −0.721483 + 0.416548i
47.13 −0.866025 0.500000i 1.56059 + 0.751362i 0.500000 + 0.866025i −1.92458 + 3.33347i −0.975834 1.43100i 0.526310 2.59287i 1.00000i 1.87091 + 2.34514i 3.33347 1.92458i
47.14 −0.866025 0.500000i 1.66857 + 0.464617i 0.500000 + 0.866025i 0.584963 1.01319i −1.21272 1.23666i −0.101753 + 2.64379i 1.00000i 2.56826 + 1.55049i −1.01319 + 0.584963i
47.15 0.866025 + 0.500000i −1.66351 + 0.482430i 0.500000 + 0.866025i 1.89177 3.27664i −1.68186 0.413957i 2.63339 0.255430i 1.00000i 2.53452 1.60505i 3.27664 1.89177i
47.16 0.866025 + 0.500000i −1.57589 0.718726i 0.500000 + 0.866025i 0.592581 1.02638i −1.00540 1.41038i −2.13419 + 1.56372i 1.00000i 1.96687 + 2.26527i 1.02638 0.592581i
47.17 0.866025 + 0.500000i −1.56009 + 0.752405i 0.500000 + 0.866025i 0.723421 1.25300i −1.72728 0.128444i −1.91613 1.82440i 1.00000i 1.86777 2.34764i 1.25300 0.723421i
47.18 0.866025 + 0.500000i −1.44172 0.959926i 0.500000 + 0.866025i 0.0289992 0.0502281i −0.768599 1.55218i 1.41346 2.23654i 1.00000i 1.15709 + 2.76788i 0.0502281 0.0289992i
47.19 0.866025 + 0.500000i −1.39855 1.02180i 0.500000 + 0.866025i −1.97297 + 3.41729i −0.700280 1.58417i 1.45851 + 2.20743i 1.00000i 0.911868 + 2.85806i −3.41729 + 1.97297i
47.20 0.866025 + 0.500000i −0.784148 + 1.54438i 0.500000 + 0.866025i −1.85941 + 3.22059i −1.45128 + 0.945399i −1.67325 + 2.04945i 1.00000i −1.77022 2.42205i −3.22059 + 1.85941i
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.c 56
3.b odd 2 1 inner 966.2.l.c 56
7.d odd 6 1 inner 966.2.l.c 56
21.g even 6 1 inner 966.2.l.c 56

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{56} + 93 T_{5}^{54} + 4930 T_{5}^{52} + 178669 T_{5}^{50} + 4895734 T_{5}^{48} + \cdots + 1536953616$$ acting on $$S_{2}^{\mathrm{new}}(966, [\chi])$$.