# Properties

 Label 567.2.bd.a Level $567$ Weight $2$ Character orbit 567.bd 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(17,567)]

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([11, 3]))

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

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

chi := DirichletCharacter("567.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$567 = 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 567.bd (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 $$\operatorname{Tr}(f)(q) =$$ $$132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} - 9 q^{10} - 9 q^{11} + 42 q^{14} - 15 q^{16} + 9 q^{17} - 9 q^{19} + 18 q^{20} - 12 q^{22} - 30 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31} + 51 q^{32} + 18 q^{34} + 9 q^{35} - 6 q^{37} + 9 q^{38} - 9 q^{40} - 12 q^{43} - 6 q^{46} - 45 q^{47} + 30 q^{49} + 9 q^{50} - 9 q^{52} - 45 q^{53} + 51 q^{56} - 3 q^{58} + 9 q^{59} - 63 q^{61} - 99 q^{62} + 18 q^{64} + 102 q^{65} - 3 q^{67} - 144 q^{68} - 15 q^{70} - 18 q^{71} + 33 q^{74} - 36 q^{76} + 57 q^{77} - 21 q^{79} + 72 q^{80} - 18 q^{82} - 90 q^{83} + 9 q^{85} + 33 q^{86} + 45 q^{88} + 9 q^{89} - 21 q^{91} - 150 q^{92} - 9 q^{94} - 27 q^{95} + 180 q^{98}+O(q^{100})$$ 132 * q + 3 * q^2 - 3 * q^4 + 9 * q^5 - 6 * q^7 + 18 * q^8 - 9 * q^10 - 9 * q^11 + 42 * q^14 - 15 * q^16 + 9 * q^17 - 9 * q^19 + 18 * q^20 - 12 * q^22 - 30 * q^23 - 3 * q^25 - 12 * q^28 - 6 * q^29 - 9 * q^31 + 51 * q^32 + 18 * q^34 + 9 * q^35 - 6 * q^37 + 9 * q^38 - 9 * q^40 - 12 * q^43 - 6 * q^46 - 45 * q^47 + 30 * q^49 + 9 * q^50 - 9 * q^52 - 45 * q^53 + 51 * q^56 - 3 * q^58 + 9 * q^59 - 63 * q^61 - 99 * q^62 + 18 * q^64 + 102 * q^65 - 3 * q^67 - 144 * q^68 - 15 * q^70 - 18 * q^71 + 33 * q^74 - 36 * q^76 + 57 * q^77 - 21 * q^79 + 72 * q^80 - 18 * q^82 - 90 * q^83 + 9 * q^85 + 33 * q^86 + 45 * q^88 + 9 * q^89 - 21 * q^91 - 150 * q^92 - 9 * q^94 - 27 * q^95 + 180 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 −2.50283 + 0.441316i 0 4.19001 1.52504i 0.437341 0.159179i 0 −1.48170 + 2.19193i −5.41196 + 3.12460i 0 −1.02434 + 0.591403i
17.2 −2.45739 + 0.433304i 0 3.97163 1.44555i 0.335948 0.122275i 0 0.0666399 2.64491i −4.81149 + 2.77792i 0 −0.772573 + 0.446045i
17.3 −2.22112 + 0.391643i 0 2.90060 1.05573i 3.20807 1.16764i 0 −2.62457 0.334082i −2.12267 + 1.22552i 0 −6.66821 + 3.84989i
17.4 −1.87320 + 0.330296i 0 1.52040 0.553380i 0.848304 0.308757i 0 2.48993 + 0.894564i 0.629295 0.363324i 0 −1.48706 + 0.858555i
17.5 −1.66985 + 0.294440i 0 0.822331 0.299304i −1.39543 + 0.507895i 0 −0.356869 2.62157i 1.65184 0.953692i 0 2.18062 1.25898i
17.6 −1.65422 + 0.291684i 0 0.771989 0.280981i −3.52188 + 1.28186i 0 −2.17331 + 1.50888i 1.71431 0.989760i 0 5.45209 3.14776i
17.7 −1.10401 + 0.194666i 0 −0.698446 + 0.254214i 3.85196 1.40200i 0 2.61259 0.417595i 2.66330 1.53766i 0 −3.97967 + 2.29767i
17.8 −0.892534 + 0.157378i 0 −1.10754 + 0.403110i 1.14226 0.415747i 0 −1.98483 + 1.74942i 2.49483 1.44039i 0 −0.954073 + 0.550834i
17.9 −0.882178 + 0.155552i 0 −1.12534 + 0.409592i −0.123522 + 0.0449584i 0 1.69913 + 2.02805i 2.48059 1.43217i 0 0.101975 0.0588754i
17.10 −0.245063 + 0.0432112i 0 −1.82120 + 0.662861i −1.99870 + 0.727467i 0 0.302346 2.62842i 0.848674 0.489982i 0 0.458373 0.264642i
17.11 −0.0159182 + 0.00280680i 0 −1.87914 + 0.683951i −3.75147 + 1.36542i 0 0.157938 + 2.64103i 0.0559891 0.0323253i 0 0.0558840 0.0322646i
17.12 0.0147002 0.00259205i 0 −1.87918 + 0.683964i −1.54651 + 0.562885i 0 2.21011 1.45445i −0.0517058 + 0.0298524i 0 −0.0212751 + 0.0122832i
17.13 0.313923 0.0553531i 0 −1.78390 + 0.649287i 2.68563 0.977491i 0 −2.48320 0.913082i −1.07619 + 0.621337i 0 0.788975 0.455515i
17.14 0.877614 0.154747i 0 −1.13313 + 0.412424i 1.30288 0.474210i 0 1.81190 + 1.92796i −2.47415 + 1.42845i 0 1.07004 0.617790i
17.15 0.910598 0.160563i 0 −1.07598 + 0.391623i 0.473927 0.172495i 0 −2.46079 0.971863i −2.51844 + 1.45402i 0 0.403861 0.233169i
17.16 1.38665 0.244504i 0 −0.0163608 + 0.00595485i −1.98299 + 0.721749i 0 −2.18605 1.49037i −2.46004 + 1.42030i 0 −2.57325 + 1.48566i
17.17 1.57155 0.277106i 0 0.513586 0.186930i −1.78318 + 0.649025i 0 −1.28547 + 2.31248i −2.00866 + 1.15970i 0 −2.62250 + 1.51410i
17.18 1.63190 0.287749i 0 0.700923 0.255115i 3.60525 1.31220i 0 2.03118 1.69538i −1.79971 + 1.03907i 0 5.50583 3.17879i
17.19 2.06025 0.363278i 0 2.23328 0.812849i −0.338534 + 0.123216i 0 1.84499 1.89632i 0.682329 0.393943i 0 −0.652704 + 0.376839i
17.20 2.26715 0.399760i 0 3.10078 1.12859i 1.73217 0.630457i 0 0.678971 + 2.55715i 2.59137 1.49613i 0 3.67505 2.12179i
See next 80 embeddings (of 132 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.22 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.bd 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.bd.a 132
3.b odd 2 1 189.2.bd.a yes 132
7.d odd 6 1 567.2.ba.a 132
21.g even 6 1 189.2.ba.a 132
27.e even 9 1 189.2.ba.a 132
27.f odd 18 1 567.2.ba.a 132
189.z odd 18 1 189.2.bd.a yes 132
189.bd even 18 1 inner 567.2.bd.a 132

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

## Hecke kernels

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