# Properties

 Label 168.2.v.a Level $168$ Weight $2$ Character orbit 168.v Analytic conductor $1.341$ Analytic rank $0$ Dimension $56$ CM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,2,Mod(11,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 3, 4]))

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

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

chi := DirichletCharacter("168.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$168 = 2^{3} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 168.v (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: $$1.34148675396$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$56 q - 2 q^{3} - 2 q^{4} - 8 q^{6} - 2 q^{9}+O(q^{10})$$ 56 * q - 2 * q^3 - 2 * q^4 - 8 * q^6 - 2 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$56 q - 2 q^{3} - 2 q^{4} - 8 q^{6} - 2 q^{9} + 6 q^{10} + 10 q^{12} - 10 q^{16} - 10 q^{18} - 4 q^{19} - 20 q^{22} - 8 q^{24} - 16 q^{25} - 8 q^{27} - 22 q^{28} - 12 q^{30} - 14 q^{33} - 56 q^{34} + 4 q^{36} + 14 q^{40} - 8 q^{42} - 16 q^{43} - 12 q^{46} + 64 q^{48} - 16 q^{49} - 34 q^{51} - 8 q^{52} + 32 q^{54} + 4 q^{57} - 18 q^{58} + 40 q^{60} + 4 q^{64} - 20 q^{66} - 36 q^{67} - 42 q^{70} + 22 q^{72} + 4 q^{73} + 104 q^{76} + 28 q^{78} - 10 q^{81} + 52 q^{82} + 4 q^{84} + 46 q^{88} + 140 q^{90} + 72 q^{91} - 46 q^{96} - 32 q^{97} - 44 q^{99}+O(q^{100})$$ 56 * q - 2 * q^3 - 2 * q^4 - 8 * q^6 - 2 * q^9 + 6 * q^10 + 10 * q^12 - 10 * q^16 - 10 * q^18 - 4 * q^19 - 20 * q^22 - 8 * q^24 - 16 * q^25 - 8 * q^27 - 22 * q^28 - 12 * q^30 - 14 * q^33 - 56 * q^34 + 4 * q^36 + 14 * q^40 - 8 * q^42 - 16 * q^43 - 12 * q^46 + 64 * q^48 - 16 * q^49 - 34 * q^51 - 8 * q^52 + 32 * q^54 + 4 * q^57 - 18 * q^58 + 40 * q^60 + 4 * q^64 - 20 * q^66 - 36 * q^67 - 42 * q^70 + 22 * q^72 + 4 * q^73 + 104 * q^76 + 28 * q^78 - 10 * q^81 + 52 * q^82 + 4 * q^84 + 46 * q^88 + 140 * q^90 + 72 * q^91 - 46 * q^96 - 32 * q^97 - 44 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −1.40790 + 0.133456i −1.71959 0.207430i 1.96438 0.375786i 0.692094 + 1.19874i 2.44869 + 0.0625523i −2.08863 1.62408i −2.71550 + 0.791228i 2.91395 + 0.713387i −1.13438 1.59535i
11.2 −1.40124 + 0.191140i 1.66017 0.493786i 1.92693 0.535666i 1.09212 + 1.89161i −2.23191 + 1.00924i −0.451847 + 2.60688i −2.59770 + 1.11891i 2.51235 1.63954i −1.89188 2.44184i
11.3 −1.32999 + 0.480762i −0.316681 + 1.70285i 1.53774 1.27882i −1.86088 3.22314i −0.397486 2.41702i −2.46429 + 0.962962i −1.43036 + 2.44009i −2.79943 1.07852i 4.02451 + 3.39209i
11.4 −1.32990 0.481016i 1.28463 + 1.16178i 1.53725 + 1.27940i −0.646402 1.11960i −1.14959 2.16297i 2.42742 1.05244i −1.42896 2.44091i 0.300539 + 2.98491i 0.321101 + 1.79988i
11.5 −1.18883 0.765954i 0.728136 1.57157i 0.826627 + 1.82118i −1.00081 1.73346i −2.06938 + 1.31060i −2.50986 0.837013i 0.412221 2.79823i −1.93964 2.28863i −0.137955 + 2.82736i
11.6 −1.08135 + 0.911422i −0.316681 + 1.70285i 0.338619 1.97113i 1.86088 + 3.22314i −1.20958 2.13001i 2.46429 0.962962i 1.43036 + 2.44009i −2.79943 1.07852i −4.94989 1.78928i
11.7 −0.982844 1.01687i −0.987527 1.42295i −0.0680365 + 1.99884i 1.77783 + 3.07929i −0.476367 + 2.40272i 0.793456 + 2.52397i 2.09943 1.89537i −1.04958 + 2.81041i 1.38390 4.83428i
11.8 −0.935706 1.06040i −1.45135 + 0.945295i −0.248908 + 1.98445i −0.187787 0.325257i 2.36043 + 0.654497i 0.198565 2.63829i 2.33722 1.59292i 1.21283 2.74391i −0.169190 + 0.503475i
11.9 −0.866151 + 1.11794i 1.66017 0.493786i −0.499565 1.93660i −1.09212 1.89161i −0.885940 + 2.28366i 0.451847 2.60688i 2.59770 + 1.11891i 2.51235 1.63954i 3.06064 + 0.417497i
11.10 −0.819527 + 1.15255i −1.71959 0.207430i −0.656749 1.88910i −0.692094 1.19874i 1.64832 1.81192i 2.08863 + 1.62408i 2.71550 + 0.791228i 2.91395 + 0.713387i 1.94880 + 0.184728i
11.11 −0.450483 1.34055i −0.0929747 + 1.72955i −1.59413 + 1.20779i −0.187787 0.325257i 2.36043 0.654497i −0.198565 + 2.63829i 2.33722 + 1.59292i −2.98271 0.321609i −0.351427 + 0.398260i
11.12 −0.389211 1.35960i 1.72608 + 0.143748i −1.69703 + 1.05834i 1.77783 + 3.07929i −0.476367 2.40272i −0.793456 2.52397i 2.09943 + 1.89537i 2.95867 + 0.496241i 3.49466 3.61563i
11.13 −0.248376 + 1.39223i 1.28463 + 1.16178i −1.87662 0.691593i 0.646402 + 1.11960i −1.93654 + 1.49994i −2.42742 + 1.05244i 1.42896 2.44091i 0.300539 + 2.98491i −1.71929 + 0.621859i
11.14 −0.0689217 1.41253i 0.996948 1.41637i −1.99050 + 0.194708i −1.00081 1.73346i −2.06938 1.31060i 2.50986 + 0.837013i 0.412221 + 2.79823i −1.01219 2.82409i −2.37959 + 1.53315i
11.15 0.0689217 + 1.41253i 0.728136 1.57157i −1.99050 + 0.194708i 1.00081 + 1.73346i 2.27007 + 0.920201i 2.50986 + 0.837013i −0.412221 2.79823i −1.93964 2.28863i −2.37959 + 1.53315i
11.16 0.248376 1.39223i −1.64844 0.531631i −1.87662 0.691593i −0.646402 1.11960i −1.14959 + 2.16297i −2.42742 + 1.05244i −1.42896 + 2.44091i 2.43474 + 1.75273i −1.71929 + 0.621859i
11.17 0.389211 + 1.35960i −0.987527 1.42295i −1.69703 + 1.05834i −1.77783 3.07929i 1.55029 1.89647i −0.793456 2.52397i −2.09943 1.89537i −1.04958 + 2.81041i 3.49466 3.61563i
11.18 0.450483 + 1.34055i −1.45135 + 0.945295i −1.59413 + 1.20779i 0.187787 + 0.325257i −1.92102 1.51976i −0.198565 + 2.63829i −2.33722 1.59292i 1.21283 2.74391i −0.351427 + 0.398260i
11.19 0.819527 1.15255i 1.03943 + 1.38549i −0.656749 1.88910i 0.692094 + 1.19874i 2.44869 0.0625523i 2.08863 + 1.62408i −2.71550 0.791228i −0.839162 + 2.88024i 1.94880 + 0.184728i
11.20 0.866151 1.11794i −0.402456 1.68465i −0.499565 1.93660i 1.09212 + 1.89161i −2.23191 1.00924i 0.451847 2.60688i −2.59770 1.11891i −2.67606 + 1.35599i 3.06064 + 0.417497i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.28 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.c even 3 1 inner
8.d odd 2 1 inner
21.h odd 6 1 inner
24.f even 2 1 inner
56.k odd 6 1 inner
168.v even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.2.v.a 56
3.b odd 2 1 inner 168.2.v.a 56
4.b odd 2 1 672.2.bd.a 56
7.c even 3 1 inner 168.2.v.a 56
8.b even 2 1 672.2.bd.a 56
8.d odd 2 1 inner 168.2.v.a 56
12.b even 2 1 672.2.bd.a 56
21.h odd 6 1 inner 168.2.v.a 56
24.f even 2 1 inner 168.2.v.a 56
24.h odd 2 1 672.2.bd.a 56
28.g odd 6 1 672.2.bd.a 56
56.k odd 6 1 inner 168.2.v.a 56
56.p even 6 1 672.2.bd.a 56
84.n even 6 1 672.2.bd.a 56
168.s odd 6 1 672.2.bd.a 56
168.v even 6 1 inner 168.2.v.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.v.a 56 1.a even 1 1 trivial
168.2.v.a 56 3.b odd 2 1 inner
168.2.v.a 56 7.c even 3 1 inner
168.2.v.a 56 8.d odd 2 1 inner
168.2.v.a 56 21.h odd 6 1 inner
168.2.v.a 56 24.f even 2 1 inner
168.2.v.a 56 56.k odd 6 1 inner
168.2.v.a 56 168.v even 6 1 inner
672.2.bd.a 56 4.b odd 2 1
672.2.bd.a 56 8.b even 2 1
672.2.bd.a 56 12.b even 2 1
672.2.bd.a 56 24.h odd 2 1
672.2.bd.a 56 28.g odd 6 1
672.2.bd.a 56 56.p even 6 1
672.2.bd.a 56 84.n even 6 1
672.2.bd.a 56 168.s odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(168, [\chi])$$.