# Properties

 Label 567.2.u.a Level $567$ Weight $2$ Character orbit 567.u Analytic conductor $4.528$ Analytic rank $0$ Dimension $132$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [567,2,Mod(100,567)]

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([16, 6]))

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

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

chi := DirichletCharacter("567.100");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$567 = 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 567.u (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 algebraic $$q$$-expansion of this newform has not been computed, 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 $$\operatorname{Tr}(f)(q) =$$ $$132 q + 3 q^{2} - 3 q^{4} + 3 q^{5} - 6 q^{7} + 6 q^{8} + 3 q^{10} + 15 q^{11} - 12 q^{13} + 30 q^{14} + 9 q^{16} - 27 q^{17} + 3 q^{19} + 18 q^{20} - 12 q^{22} + 36 q^{23} - 3 q^{25} - 30 q^{26} - 12 q^{28} + 30 q^{29} - 3 q^{31} + 75 q^{32} - 18 q^{34} - 15 q^{35} - 6 q^{37} - 69 q^{38} + 51 q^{40} - 12 q^{43} + 6 q^{44} - 6 q^{46} + 21 q^{47} - 42 q^{49} + 39 q^{50} + 9 q^{52} - 9 q^{53} - 24 q^{55} - 111 q^{56} - 3 q^{58} - 27 q^{59} - 21 q^{61} - 75 q^{62} - 30 q^{64} + 90 q^{65} - 3 q^{67} + 30 q^{68} + 39 q^{70} + 18 q^{71} - 42 q^{73} - 51 q^{74} - 24 q^{76} - 15 q^{77} + 15 q^{79} - 102 q^{80} - 6 q^{82} + 42 q^{83} - 63 q^{85} + 93 q^{86} - 51 q^{88} - 75 q^{89} - 21 q^{91} + 66 q^{92} + 33 q^{94} - 15 q^{95} - 12 q^{97} + 36 q^{98}+O(q^{100})$$ 132 * q + 3 * q^2 - 3 * q^4 + 3 * q^5 - 6 * q^7 + 6 * q^8 + 3 * q^10 + 15 * q^11 - 12 * q^13 + 30 * q^14 + 9 * q^16 - 27 * q^17 + 3 * q^19 + 18 * q^20 - 12 * q^22 + 36 * q^23 - 3 * q^25 - 30 * q^26 - 12 * q^28 + 30 * q^29 - 3 * q^31 + 75 * q^32 - 18 * q^34 - 15 * q^35 - 6 * q^37 - 69 * q^38 + 51 * q^40 - 12 * q^43 + 6 * q^44 - 6 * q^46 + 21 * q^47 - 42 * q^49 + 39 * q^50 + 9 * q^52 - 9 * q^53 - 24 * q^55 - 111 * q^56 - 3 * q^58 - 27 * q^59 - 21 * q^61 - 75 * q^62 - 30 * q^64 + 90 * q^65 - 3 * q^67 + 30 * q^68 + 39 * q^70 + 18 * q^71 - 42 * q^73 - 51 * q^74 - 24 * q^76 - 15 * q^77 + 15 * q^79 - 102 * q^80 - 6 * q^82 + 42 * q^83 - 63 * q^85 + 93 * q^86 - 51 * q^88 - 75 * q^89 - 21 * q^91 + 66 * q^92 + 33 * q^94 - 15 * q^95 - 12 * q^97 + 36 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
100.1 −0.456041 + 2.58633i 0 −4.60177 1.67491i −2.97048 1.08117i 0 −2.55032 + 0.704184i 3.80423 6.58912i 0 4.15092 7.18961i
100.2 −0.427861 + 2.42652i 0 −3.82554 1.39238i 2.97164 + 1.08159i 0 0.768747 2.53161i 2.55149 4.41931i 0 −3.89594 + 6.74796i
100.3 −0.373934 + 2.12069i 0 −2.47810 0.901956i 1.18205 + 0.430229i 0 −2.62839 + 0.302569i 0.686013 1.18821i 0 −1.35439 + 2.34587i
100.4 −0.371090 + 2.10456i 0 −2.41207 0.877922i −2.18602 0.795646i 0 1.27150 + 2.32019i 0.605711 1.04912i 0 2.48569 4.30535i
100.5 −0.340870 + 1.93317i 0 −1.74157 0.633879i −1.35943 0.494793i 0 −0.147009 2.64166i −0.143948 + 0.249325i 0 1.41991 2.45935i
100.6 −0.246187 + 1.39619i 0 −0.00936568 0.00340883i −3.04103 1.10685i 0 1.78778 1.95034i −1.41067 + 2.44335i 0 2.29403 3.97338i
100.7 −0.241593 + 1.37014i 0 0.0604707 + 0.0220095i 1.40760 + 0.512324i 0 1.17416 + 2.37094i −1.43604 + 2.48730i 0 −1.04202 + 1.80483i
100.8 −0.101938 + 0.578121i 0 1.55555 + 0.566175i 3.06658 + 1.11614i 0 −1.48906 2.18694i −1.07293 + 1.85836i 0 −0.957867 + 1.65908i
100.9 −0.0954712 + 0.541444i 0 1.59534 + 0.580656i 0.969531 + 0.352880i 0 2.17686 + 1.50376i −1.01650 + 1.76063i 0 −0.283627 + 0.491257i
100.10 −0.0570934 + 0.323793i 0 1.77780 + 0.647067i −1.85642 0.675680i 0 −2.33644 + 1.24140i −0.639805 + 1.10817i 0 0.324770 0.562517i
100.11 −0.0290601 + 0.164808i 0 1.85307 + 0.674462i 0.999212 + 0.363683i 0 1.97858 1.75648i −0.332357 + 0.575660i 0 −0.0889751 + 0.154109i
100.12 0.0305699 0.173370i 0 1.85026 + 0.673440i −2.52551 0.919210i 0 −1.78991 + 1.94839i 0.349362 0.605113i 0 −0.236568 + 0.409748i
100.13 0.134854 0.764795i 0 1.31266 + 0.477769i −1.10529 0.402293i 0 −0.975813 2.45923i 1.31901 2.28459i 0 −0.456725 + 0.791071i
100.14 0.165988 0.941364i 0 1.02077 + 0.371530i 2.75749 + 1.00364i 0 0.0940141 + 2.64408i 1.47507 2.55489i 0 1.40250 2.42921i
100.15 0.203963 1.15673i 0 0.582964 + 0.212181i 3.41115 + 1.24156i 0 −1.99948 1.73265i 1.53891 2.66548i 0 2.13189 3.69255i
100.16 0.266701 1.51254i 0 −0.337258 0.122752i −3.71318 1.35149i 0 2.56740 0.639091i 1.26026 2.18283i 0 −3.03449 + 5.25589i
100.17 0.279530 1.58529i 0 −0.555633 0.202234i −0.230811 0.0840085i 0 −1.31413 + 2.29631i 1.13383 1.96386i 0 −0.197697 + 0.342421i
100.18 0.310937 1.76341i 0 −1.13356 0.412582i 0.00172074 0.000626297i 0 2.59651 0.508047i 0.710598 1.23079i 0 0.00163946 0.00283963i
100.19 0.324131 1.83824i 0 −1.39467 0.507617i −1.37479 0.500384i 0 −2.42382 1.06071i 0.481419 0.833843i 0 −1.36544 + 2.36501i
100.20 0.422628 2.39684i 0 −3.68685 1.34190i 2.38847 + 0.869333i 0 2.35689 + 1.20212i −2.34068 + 4.05418i 0 3.09309 5.35738i
See next 80 embeddings (of 132 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 100.22 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.u 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.u.a 132
3.b odd 2 1 189.2.u.a 132
7.c even 3 1 567.2.w.a 132
21.h odd 6 1 189.2.w.a yes 132
27.e even 9 1 567.2.w.a 132
27.f odd 18 1 189.2.w.a yes 132
189.u even 9 1 inner 567.2.u.a 132
189.bc odd 18 1 189.2.u.a 132

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

## Hecke kernels

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