# Properties

 Label 567.2.w.a Level $567$ Weight $2$ Character orbit 567.w 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(37,567)]

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([14, 6]))

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

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

chi := DirichletCharacter("567.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$567 = 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 567.w (of order $$9$$, 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 $$\Q(\zeta_{9})$$ Twist minimal: no (minimal twist has level 189) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$132 q + 3 q^{2} - 3 q^{4} + 3 q^{5} - 6 q^{7} + 6 q^{8}+O(q^{10})$$ 132 * q + 3 * q^2 - 3 * q^4 + 3 * q^5 - 6 * q^7 + 6 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$132 q + 3 q^{2} - 3 q^{4} + 3 q^{5} - 6 q^{7} + 6 q^{8} - 6 q^{10} - 3 q^{11} - 12 q^{13} - 15 q^{14} - 9 q^{16} + 54 q^{17} - 6 q^{19} + 18 q^{20} - 12 q^{22} - 3 q^{25} - 30 q^{26} - 12 q^{28} + 30 q^{29} - 3 q^{31} - 51 q^{32} - 18 q^{34} + 12 q^{35} + 3 q^{37} + 57 q^{38} - 66 q^{40} - 12 q^{43} - 3 q^{44} + 3 q^{46} + 21 q^{47} + 12 q^{49} + 39 q^{50} + 9 q^{52} - 9 q^{53} - 24 q^{55} - 57 q^{56} - 3 q^{58} + 18 q^{59} + 33 q^{61} - 75 q^{62} - 30 q^{64} - 81 q^{65} - 3 q^{67} - 6 q^{68} - 42 q^{70} + 18 q^{71} + 21 q^{73} + 93 q^{74} - 24 q^{76} - 87 q^{77} + 15 q^{79} - 102 q^{80} - 6 q^{82} + 42 q^{83} - 63 q^{85} - 159 q^{86} + 57 q^{88} + 150 q^{89} + 6 q^{91} + 66 q^{92} + 33 q^{94} + 147 q^{95} - 12 q^{97} - 99 q^{98}+O(q^{100})$$ 132 * q + 3 * q^2 - 3 * q^4 + 3 * q^5 - 6 * q^7 + 6 * q^8 - 6 * q^10 - 3 * q^11 - 12 * q^13 - 15 * q^14 - 9 * q^16 + 54 * q^17 - 6 * q^19 + 18 * q^20 - 12 * q^22 - 3 * q^25 - 30 * q^26 - 12 * q^28 + 30 * q^29 - 3 * q^31 - 51 * q^32 - 18 * q^34 + 12 * q^35 + 3 * q^37 + 57 * q^38 - 66 * q^40 - 12 * q^43 - 3 * q^44 + 3 * q^46 + 21 * q^47 + 12 * q^49 + 39 * q^50 + 9 * q^52 - 9 * q^53 - 24 * q^55 - 57 * q^56 - 3 * q^58 + 18 * q^59 + 33 * q^61 - 75 * q^62 - 30 * q^64 - 81 * q^65 - 3 * q^67 - 6 * q^68 - 42 * q^70 + 18 * q^71 + 21 * q^73 + 93 * q^74 - 24 * q^76 - 87 * q^77 + 15 * q^79 - 102 * q^80 - 6 * q^82 + 42 * q^83 - 63 * q^85 - 159 * q^86 + 57 * q^88 + 150 * q^89 + 6 * q^91 + 66 * q^92 + 33 * q^94 + 147 * q^95 - 12 * q^97 - 99 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 −0.475852 2.69869i 0 −5.17712 + 1.88432i −0.0114982 + 0.0652096i 0 2.53330 + 0.763145i 4.80842 + 8.32842i 0 0.181452
37.2 −0.442261 2.50819i 0 −4.21603 + 1.53451i 0.331479 1.87991i 0 0.0136229 2.64572i 3.16655 + 5.48462i 0 −4.86178
37.3 −0.379821 2.15407i 0 −2.61637 + 0.952279i 0.347563 1.97113i 0 0.0877467 + 2.64430i 0.857725 + 1.48562i 0 −4.37795
37.4 −0.351787 1.99508i 0 −1.97722 + 0.719650i −0.155052 + 0.879341i 0 −1.72949 + 2.00222i 0.105462 + 0.182666i 0 1.80890
37.5 −0.337831 1.91594i 0 −1.67729 + 0.610485i −0.582582 + 3.30399i 0 0.666452 2.56044i −0.209199 0.362343i 0 6.52704
37.6 −0.306313 1.73719i 0 −1.04461 + 0.380207i 0.723302 4.10205i 0 −1.55501 2.14055i −0.783518 1.35709i 0 −7.34759
37.7 −0.278319 1.57843i 0 −0.534587 + 0.194574i −0.620906 + 3.52133i 0 −2.64260 + 0.129178i −1.14687 1.98644i 0 5.73098
37.8 −0.162199 0.919879i 0 1.05952 0.385633i 0.120368 0.682641i 0 1.39323 2.24920i −1.46066 2.52993i 0 −0.647471
37.9 −0.161602 0.916488i 0 1.06555 0.387829i −0.260888 + 1.47957i 0 1.80008 + 1.93899i −1.45826 2.52578i 0 1.39817
37.10 −0.0910389 0.516307i 0 1.62110 0.590032i −0.0290567 + 0.164789i 0 −0.352779 + 2.62213i −0.976493 1.69134i 0 0.0877269
37.11 −0.0119703 0.0678870i 0 1.87492 0.682415i −0.115724 + 0.656302i 0 −1.45670 2.20863i −0.137705 0.238512i 0 0.0459397
37.12 0.000974390 0.00552604i 0 1.87936 0.684030i 0.601605 3.41187i 0 2.41288 1.08535i 0.0112225 + 0.0194379i 0 0.0194403
37.13 0.0572325 + 0.324581i 0 1.77731 0.646887i 0.607976 3.44800i 0 −2.50814 + 0.842165i 0.641276 + 1.11072i 0 1.15395
37.14 0.162149 + 0.919594i 0 1.06003 0.385818i −0.575026 + 3.26114i 0 1.06445 + 2.42218i 1.46046 + 2.52959i 0 −3.09216
37.15 0.177652 + 1.00751i 0 0.895865 0.326068i −0.356801 + 2.02352i 0 −2.48566 + 0.906357i 1.51072 + 2.61665i 0 −2.10210
37.16 0.199381 + 1.13075i 0 0.640547 0.233140i 0.0295570 0.167626i 0 −1.82867 1.91206i 1.53953 + 2.66654i 0 0.195436
37.17 0.202689 + 1.14951i 0 0.599102 0.218055i 0.490607 2.78237i 0 2.49252 + 0.887321i 1.53933 + 2.66619i 0 3.29779
37.18 0.303930 + 1.72367i 0 −0.999292 + 0.363713i −0.0231991 + 0.131569i 0 1.74352 1.99001i 0.819627 + 1.41964i 0 −0.233833
37.19 0.345866 + 1.96151i 0 −1.84850 + 0.672798i −0.424474 + 2.40731i 0 2.63001 + 0.288173i 0.0327363 + 0.0567010i 0 −4.86876
37.20 0.402178 + 2.28086i 0 −3.16120 + 1.15058i −0.125418 + 0.711279i 0 −2.42560 + 1.05663i −1.57964 2.73601i 0 −1.67277
See next 80 embeddings (of 132 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 37.22 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.w even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.w.a 132
3.b odd 2 1 189.2.w.a yes 132
7.c even 3 1 567.2.u.a 132
21.h odd 6 1 189.2.u.a 132
27.e even 9 1 567.2.u.a 132
27.f odd 18 1 189.2.u.a 132
189.w even 9 1 inner 567.2.w.a 132
189.bf odd 18 1 189.2.w.a yes 132

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.u.a 132 21.h odd 6 1
189.2.u.a 132 27.f odd 18 1
189.2.w.a yes 132 3.b odd 2 1
189.2.w.a yes 132 189.bf odd 18 1
567.2.u.a 132 7.c even 3 1
567.2.u.a 132 27.e even 9 1
567.2.w.a 132 1.a even 1 1 trivial
567.2.w.a 132 189.w even 9 1 inner

## Hecke kernels

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