# Properties

 Label 567.2.ba.a Level $567$ Weight $2$ Character orbit 567.ba Analytic conductor $4.528$ Analytic rank $0$ Dimension $132$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [567,2,Mod(143,567)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(567, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([7, 3]))

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

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

chi := DirichletCharacter("567.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$567 = 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 567.ba (of order $$18$$, degree $$6$$, not 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: $$4.52751779461$$ Analytic rank: $$0$$ Dimension: $$132$$ Relative dimension: $$22$$ over $$\Q(\zeta_{18})$$ Twist minimal: no (minimal twist has level 189) Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8}+O(q^{10})$$ 132 * q + 3 * q^2 - 3 * q^4 + 9 * q^5 - 6 * q^7 + 18 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} + 9 q^{11} - 3 q^{14} + 3 q^{16} + 18 q^{17} - 18 q^{20} - 12 q^{22} + 6 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31} - 3 q^{32} - 18 q^{34} - 18 q^{35} + 3 q^{37} + 99 q^{38} - 54 q^{40} - 12 q^{43} + 9 q^{44} + 3 q^{46} - 45 q^{47} - 24 q^{49} + 9 q^{50} - 9 q^{52} + 45 q^{53} - 3 q^{56} - 3 q^{58} - 36 q^{59} - 9 q^{61} + 99 q^{62} + 18 q^{64} - 69 q^{65} - 3 q^{67} - 36 q^{68} + 66 q^{70} - 18 q^{71} - 9 q^{73} - 75 q^{74} + 36 q^{76} - 15 q^{77} - 21 q^{79} - 72 q^{80} - 18 q^{82} + 90 q^{83} + 9 q^{85} + 105 q^{86} - 63 q^{88} + 18 q^{89} + 6 q^{91} - 150 q^{92} - 9 q^{94} - 45 q^{95} - 27 q^{98}+O(q^{100})$$ 132 * q + 3 * q^2 - 3 * q^4 + 9 * q^5 - 6 * q^7 + 18 * q^8 + 9 * q^11 - 3 * q^14 + 3 * q^16 + 18 * q^17 - 18 * q^20 - 12 * q^22 + 6 * q^23 - 3 * q^25 - 12 * q^28 - 6 * q^29 - 9 * q^31 - 3 * q^32 - 18 * q^34 - 18 * q^35 + 3 * q^37 + 99 * q^38 - 54 * q^40 - 12 * q^43 + 9 * q^44 + 3 * q^46 - 45 * q^47 - 24 * q^49 + 9 * q^50 - 9 * q^52 + 45 * q^53 - 3 * q^56 - 3 * q^58 - 36 * q^59 - 9 * q^61 + 99 * q^62 + 18 * q^64 - 69 * q^65 - 3 * q^67 - 36 * q^68 + 66 * q^70 - 18 * q^71 - 9 * q^73 - 75 * q^74 + 36 * q^76 - 15 * q^77 - 21 * q^79 - 72 * q^80 - 18 * q^82 + 90 * q^83 + 9 * q^85 + 105 * q^86 - 63 * q^88 + 18 * q^89 + 6 * q^91 - 150 * q^92 - 9 * q^94 - 45 * q^95 - 27 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1 −1.75048 + 2.08614i 0 −0.940509 5.33389i −2.08375 + 1.74847i 0 −0.122613 + 2.64291i 8.05677 + 4.65158i 0 7.40767i
143.2 −1.61592 + 1.92578i 0 −0.750127 4.25418i 2.10349 1.76504i 0 −0.122508 2.64291i 5.05050 + 2.91591i 0 6.90302i
143.3 −1.47978 + 1.76353i 0 −0.573000 3.24965i 1.41208 1.18487i 0 −2.40040 + 1.11270i 2.59137 + 1.49613i 0 4.24358i
143.4 −1.34473 + 1.60259i 0 −0.412694 2.34051i −0.275976 + 0.231571i 0 2.18789 + 1.48767i 0.682329 + 0.393943i 0 0.753678i
143.5 −1.06515 + 1.26940i 0 −0.129525 0.734574i 2.93903 2.46614i 0 2.02233 + 1.70592i −1.79971 1.03907i 0 6.35759i
143.6 −1.02575 + 1.22245i 0 −0.0949069 0.538244i −1.45366 + 1.21977i 0 −2.50057 0.864386i −2.00866 1.15970i 0 3.02821i
143.7 −0.905074 + 1.07862i 0 0.00302336 + 0.0171463i −1.61655 + 1.35644i 0 1.08812 2.41164i −2.46004 1.42030i 0 2.97133i
143.8 −0.594351 + 0.708320i 0 0.198832 + 1.12763i 0.386349 0.324185i 0 0.529787 2.59217i −2.51844 1.45402i 0 0.466338i
143.9 −0.572822 + 0.682663i 0 0.209393 + 1.18753i 1.06212 0.891222i 0 −1.58403 + 2.11916i −2.47415 1.42845i 0 1.23558i
143.10 −0.204899 + 0.244189i 0 0.329652 + 1.86955i 2.18935 1.83708i 0 0.468007 2.60403i −1.07619 0.621337i 0 0.911030i
143.11 −0.00959490 + 0.0114348i 0 0.347258 + 1.96940i −1.26073 + 1.05788i 0 1.81613 + 1.92397i −0.0517058 0.0298524i 0 0.0245663i
143.12 0.0103898 0.0123821i 0 0.347251 + 1.96936i −3.05823 + 2.56616i 0 −2.57348 + 0.614149i 0.0559891 + 0.0323253i 0 0.0645293i
143.13 0.159953 0.190625i 0 0.336544 + 1.90863i −1.62936 + 1.36719i 0 2.64099 0.158668i 0.848674 + 0.489982i 0 0.529283i
143.14 0.575801 0.686212i 0 0.207955 + 1.17937i −0.100696 + 0.0844942i 0 −1.70219 + 2.02548i 2.48059 + 1.43217i 0 0.117751i
143.15 0.582560 0.694268i 0 0.204665 + 1.16071i 0.931176 0.781349i 0 −2.06750 1.65089i 2.49483 + 1.44039i 0 1.10167i
143.16 0.720590 0.858766i 0 0.129068 + 0.731979i 3.14015 2.63490i 0 0.864922 + 2.50038i 2.66330 + 1.53766i 0 4.59533i
143.17 1.07972 1.28676i 0 −0.142658 0.809053i −2.87107 + 2.40911i 0 −1.86335 1.87828i 1.71431 + 0.989760i 0 6.29553i
143.18 1.08992 1.29892i 0 −0.151961 0.861812i −1.13756 + 0.954530i 0 2.51978 0.806679i 1.65184 + 0.953692i 0 2.51796i
143.19 1.22264 1.45709i 0 −0.280959 1.59339i 0.691544 0.580274i 0 −0.448602 + 2.60744i 0.629295 + 0.363324i 0 1.71711i
143.20 1.44973 1.72772i 0 −0.536009 3.03986i 2.61524 2.19445i 0 −0.126746 2.64271i −2.12267 1.22552i 0 7.69979i
See next 80 embeddings (of 132 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 143.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.ba even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.ba.a 132
3.b odd 2 1 189.2.ba.a 132
7.d odd 6 1 567.2.bd.a 132
21.g even 6 1 189.2.bd.a yes 132
27.e even 9 1 189.2.bd.a yes 132
27.f odd 18 1 567.2.bd.a 132
189.x odd 18 1 189.2.ba.a 132
189.ba even 18 1 inner 567.2.ba.a 132

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.ba.a 132 3.b odd 2 1
189.2.ba.a 132 189.x odd 18 1
189.2.bd.a yes 132 21.g even 6 1
189.2.bd.a yes 132 27.e even 9 1
567.2.ba.a 132 1.a even 1 1 trivial
567.2.ba.a 132 189.ba even 18 1 inner
567.2.bd.a 132 7.d odd 6 1
567.2.bd.a 132 27.f odd 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(567, [\chi])$$.