# Properties

 Label 567.2.v.b Level $567$ Weight $2$ Character orbit 567.v Analytic conductor $4.528$ Analytic rank $0$ Dimension $54$ 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(64,567)]

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([2, 0]))

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

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

chi := DirichletCharacter("567.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$567 = 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 567.v (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: $$54$$ Relative dimension: $$9$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$54 q + 3 q^{5} + 27 q^{8}+O(q^{10})$$ 54 * q + 3 * q^5 + 27 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$54 q + 3 q^{5} + 27 q^{8} + 6 q^{11} - 9 q^{13} + 30 q^{17} + 12 q^{20} - 9 q^{22} + 12 q^{23} + 27 q^{25} - 18 q^{26} + 54 q^{28} - 6 q^{29} - 9 q^{31} + 9 q^{32} - 9 q^{34} + 12 q^{35} - 54 q^{38} - 45 q^{40} + 15 q^{41} - 9 q^{43} + 42 q^{44} + 45 q^{47} - 18 q^{50} - 63 q^{52} - 132 q^{53} + 9 q^{56} - 27 q^{58} + 36 q^{62} - 27 q^{64} - 66 q^{65} + 45 q^{67} - 87 q^{68} + 72 q^{71} + 72 q^{74} + 54 q^{76} - 3 q^{77} - 36 q^{79} - 42 q^{80} - 24 q^{83} + 18 q^{85} + 90 q^{86} + 54 q^{88} + 42 q^{89} - 87 q^{92} - 90 q^{94} - 12 q^{95} - 18 q^{97} + 9 q^{98}+O(q^{100})$$ 54 * q + 3 * q^5 + 27 * q^8 + 6 * q^11 - 9 * q^13 + 30 * q^17 + 12 * q^20 - 9 * q^22 + 12 * q^23 + 27 * q^25 - 18 * q^26 + 54 * q^28 - 6 * q^29 - 9 * q^31 + 9 * q^32 - 9 * q^34 + 12 * q^35 - 54 * q^38 - 45 * q^40 + 15 * q^41 - 9 * q^43 + 42 * q^44 + 45 * q^47 - 18 * q^50 - 63 * q^52 - 132 * q^53 + 9 * q^56 - 27 * q^58 + 36 * q^62 - 27 * q^64 - 66 * q^65 + 45 * q^67 - 87 * q^68 + 72 * q^71 + 72 * q^74 + 54 * q^76 - 3 * q^77 - 36 * q^79 - 42 * q^80 - 24 * q^83 + 18 * q^85 + 90 * q^86 + 54 * q^88 + 42 * q^89 - 87 * q^92 - 90 * q^94 - 12 * q^95 - 18 * q^97 + 9 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1 −2.16628 0.788461i 0 2.53900 + 2.13048i −0.276602 + 1.56869i 0 0.766044 0.642788i −1.51508 2.62420i 0 1.83605 3.18012i
64.2 −1.66659 0.606588i 0 0.877471 + 0.736286i 0.343651 1.94894i 0 0.766044 0.642788i 0.757784 + 1.31252i 0 −1.75493 + 3.03962i
64.3 −0.754940 0.274776i 0 −1.03766 0.870696i −0.713335 + 4.04552i 0 0.766044 0.642788i 1.34751 + 2.33396i 0 1.65014 2.85812i
64.4 −0.435562 0.158531i 0 −1.36751 1.14747i 0.0549550 0.311665i 0 0.766044 0.642788i 0.877238 + 1.51942i 0 −0.0733450 + 0.127037i
64.5 0.252618 + 0.0919454i 0 −1.47673 1.23912i −0.344603 + 1.95434i 0 0.766044 0.642788i −0.527947 0.914430i 0 −0.266746 + 0.462017i
64.6 1.08295 + 0.394160i 0 −0.514680 0.431868i 0.132560 0.751786i 0 0.766044 0.642788i −1.53959 2.66665i 0 0.439879 0.761893i
64.7 1.80206 + 0.655894i 0 1.28512 + 1.07834i 0.489098 2.77381i 0 0.766044 0.642788i −0.309134 0.535436i 0 2.70071 4.67777i
64.8 2.17921 + 0.793168i 0 2.58775 + 2.17138i −0.443161 + 2.51329i 0 0.766044 0.642788i 1.59792 + 2.76768i 0 −2.95920 + 5.12549i
64.9 2.52562 + 0.919249i 0 4.00163 + 3.35776i −0.203200 + 1.15240i 0 0.766044 0.642788i 4.33224 + 7.50367i 0 −1.57255 + 2.72374i
127.1 −2.00215 1.68000i 0 0.838893 + 4.75760i 3.29720 + 1.20008i 0 0.173648 0.984808i 3.69957 6.40784i 0 −4.58534 7.94204i
127.2 −1.89179 1.58740i 0 0.711733 + 4.03644i −2.16637 0.788496i 0 0.173648 0.984808i 2.59144 4.48850i 0 2.84667 + 4.93057i
127.3 −0.969537 0.813538i 0 −0.0691389 0.392106i 0.855069 + 0.311220i 0 0.173648 0.984808i −1.51760 + 2.62856i 0 −0.575832 0.997370i
127.4 −0.786541 0.659987i 0 −0.164231 0.931402i −3.56827 1.29874i 0 0.173648 0.984808i −1.51229 + 2.61937i 0 1.94944 + 3.37653i
127.5 −0.731503 0.613803i 0 −0.188955 1.07162i 3.78665 + 1.37823i 0 0.173648 0.984808i −1.47445 + 2.55382i 0 −1.92398 3.33244i
127.6 −0.0808704 0.0678583i 0 −0.345361 1.95864i −0.0449304 0.0163533i 0 0.173648 0.984808i −0.210549 + 0.364682i 0 0.00252383 + 0.00437140i
127.7 1.09798 + 0.921318i 0 0.00944557 + 0.0535685i −1.35272 0.492351i 0 0.173648 0.984808i 1.39433 2.41506i 0 −1.03166 1.78688i
127.8 1.17963 + 0.989828i 0 0.0644735 + 0.365648i 1.99174 + 0.724932i 0 0.173648 0.984808i 1.25403 2.17204i 0 1.63195 + 2.82663i
127.9 1.88664 + 1.58308i 0 0.705975 + 4.00378i 1.28676 + 0.468343i 0 0.173648 0.984808i −2.54355 + 4.40555i 0 1.68623 + 2.92064i
253.1 −0.483276 2.74079i 0 −5.39901 + 1.96508i 1.86046 + 1.56111i 0 −0.939693 0.342020i 5.21200 + 9.02746i 0 3.37956 5.85358i
253.2 −0.399059 2.26318i 0 −3.08333 + 1.12224i −2.84760 2.38942i 0 −0.939693 0.342020i 1.47217 + 2.54988i 0 −4.27132 + 7.39815i
See all 54 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 64.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e 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.v.b 54
3.b odd 2 1 189.2.v.a 54
27.e even 9 1 inner 567.2.v.b 54
27.e even 9 1 5103.2.a.f 27
27.f odd 18 1 189.2.v.a 54
27.f odd 18 1 5103.2.a.i 27

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.v.a 54 3.b odd 2 1
189.2.v.a 54 27.f odd 18 1
567.2.v.b 54 1.a even 1 1 trivial
567.2.v.b 54 27.e even 9 1 inner
5103.2.a.f 27 27.e even 9 1
5103.2.a.i 27 27.f odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{54} - 21 T_{2}^{51} + 27 T_{2}^{49} + 612 T_{2}^{48} - 117 T_{2}^{47} - 648 T_{2}^{46} + \cdots + 729$$ acting on $$S_{2}^{\mathrm{new}}(567, [\chi])$$.