# Properties

 Label 138.2.f.a Level $138$ Weight $2$ Character orbit 138.f Analytic conductor $1.102$ Analytic rank $0$ Dimension $80$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,2,Mod(5,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 1]))

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

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

chi := DirichletCharacter("138.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 138.f (of order $$22$$, degree $$10$$, 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: $$1.10193554789$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$8$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$80 q + 4 q^{3} + 8 q^{4} + 4 q^{6}+O(q^{10})$$ 80 * q + 4 * q^3 + 8 * q^4 + 4 * q^6 $$\operatorname{Tr}(f)(q) =$$ $$80 q + 4 q^{3} + 8 q^{4} + 4 q^{6} - 4 q^{12} + 8 q^{13} - 22 q^{15} - 8 q^{16} - 28 q^{18} - 66 q^{21} - 4 q^{24} - 48 q^{25} - 38 q^{27} - 44 q^{30} - 16 q^{31} - 22 q^{33} - 44 q^{37} - 24 q^{39} - 44 q^{43} - 16 q^{46} + 4 q^{48} - 76 q^{49} - 8 q^{52} - 6 q^{54} + 64 q^{55} + 66 q^{57} + 36 q^{58} + 22 q^{60} + 88 q^{61} + 110 q^{63} + 8 q^{64} + 88 q^{66} + 44 q^{67} + 82 q^{69} + 112 q^{70} + 28 q^{72} + 52 q^{73} + 136 q^{75} + 82 q^{78} + 88 q^{79} + 36 q^{81} + 44 q^{82} + 22 q^{84} + 20 q^{85} - 10 q^{87} + 8 q^{93} - 56 q^{94} + 4 q^{96} - 132 q^{97} - 66 q^{99}+O(q^{100})$$ 80 * q + 4 * q^3 + 8 * q^4 + 4 * q^6 - 4 * q^12 + 8 * q^13 - 22 * q^15 - 8 * q^16 - 28 * q^18 - 66 * q^21 - 4 * q^24 - 48 * q^25 - 38 * q^27 - 44 * q^30 - 16 * q^31 - 22 * q^33 - 44 * q^37 - 24 * q^39 - 44 * q^43 - 16 * q^46 + 4 * q^48 - 76 * q^49 - 8 * q^52 - 6 * q^54 + 64 * q^55 + 66 * q^57 + 36 * q^58 + 22 * q^60 + 88 * q^61 + 110 * q^63 + 8 * q^64 + 88 * q^66 + 44 * q^67 + 82 * q^69 + 112 * q^70 + 28 * q^72 + 52 * q^73 + 136 * q^75 + 82 * q^78 + 88 * q^79 + 36 * q^81 + 44 * q^82 + 22 * q^84 + 20 * q^85 - 10 * q^87 + 8 * q^93 - 56 * q^94 + 4 * q^96 - 132 * q^97 - 66 * q^99

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −0.281733 + 0.959493i −1.72467 + 0.159749i −0.841254 0.540641i 0.591943 4.11706i 0.332618 1.69981i 0.656690 0.299900i 0.755750 0.654861i 2.94896 0.551026i 3.78352 + 1.72787i
5.2 −0.281733 + 0.959493i −1.37386 1.05476i −0.841254 0.540641i −0.493501 + 3.43237i 1.39909 1.02105i −1.02427 + 0.467769i 0.755750 0.654861i 0.774979 + 2.89817i −3.15430 1.44052i
5.3 −0.281733 + 0.959493i 0.583374 1.63085i −0.841254 0.540641i 0.140990 0.980604i 1.40043 + 1.01921i 1.30276 0.594948i 0.755750 0.654861i −2.31935 1.90279i 0.901161 + 0.411546i
5.4 −0.281733 + 0.959493i 1.60896 + 0.641290i −0.841254 0.540641i −0.239432 + 1.66529i −1.06861 + 1.36311i −0.935173 + 0.427079i 0.755750 0.654861i 2.17749 + 2.06362i −1.53038 0.698899i
5.5 0.281733 0.959493i −0.863742 1.50132i −0.841254 0.540641i 0.239432 1.66529i −1.68385 + 0.405784i −0.935173 + 0.427079i −0.755750 + 0.654861i −1.50790 + 2.59350i −1.53038 0.698899i
5.6 0.281733 0.959493i 0.0873233 + 1.72985i −0.841254 0.540641i −0.591943 + 4.11706i 1.68438 + 0.403568i 0.656690 0.299900i −0.755750 + 0.654861i −2.98475 + 0.302112i 3.78352 + 1.72787i
5.7 0.281733 0.959493i 1.23954 + 1.20977i −0.841254 0.540641i 0.493501 3.43237i 1.50998 0.848500i −1.02427 + 0.467769i −0.755750 + 0.654861i 0.0729229 + 2.99911i −3.15430 1.44052i
5.8 0.281733 0.959493i 1.53123 0.809531i −0.841254 0.540641i −0.140990 + 0.980604i −0.345342 1.69727i 1.30276 0.594948i −0.755750 + 0.654861i 1.68932 2.47915i 0.901161 + 0.411546i
11.1 −0.540641 0.841254i −1.59667 + 0.671295i −0.415415 + 0.909632i 1.80115 0.528864i 1.42796 + 0.980277i 1.49075 1.29175i 0.989821 0.142315i 2.09873 2.14368i −1.41868 1.22930i
11.2 −0.540641 0.841254i 0.0700914 + 1.73063i −0.415415 + 0.909632i −3.21268 + 0.943327i 1.41801 0.994615i −1.80758 + 1.56628i 0.989821 0.142315i −2.99017 + 0.242605i 2.53048 + 2.19268i
11.3 −0.540641 0.841254i 1.29667 1.14833i −0.415415 + 0.909632i −1.43749 + 0.422085i −1.66706 0.469995i 3.34037 2.89445i 0.989821 0.142315i 0.362698 2.97799i 1.13225 + 0.981098i
11.4 −0.540641 0.841254i 1.70890 + 0.282234i −0.415415 + 0.909632i 2.84902 0.836549i −0.686471 1.59021i −3.02354 + 2.61991i 0.989821 0.142315i 2.84069 + 0.964621i −2.24405 1.94448i
11.5 0.540641 + 0.841254i −1.71919 0.210651i −0.415415 + 0.909632i −2.84902 + 0.836549i −0.752255 1.56016i −3.02354 + 2.61991i −0.989821 + 0.142315i 2.91125 + 0.724301i −2.24405 1.94448i
11.6 0.540641 + 0.841254i −0.920624 1.46712i −0.415415 + 0.909632i 1.43749 0.422085i 0.736496 1.56766i 3.34037 2.89445i −0.989821 + 0.142315i −1.30490 + 2.70134i 1.13225 + 0.981098i
11.7 0.540641 + 0.841254i −0.554828 + 1.64078i −0.415415 + 0.909632i 3.21268 0.943327i −1.68028 + 0.420323i −1.80758 + 1.56628i −0.989821 + 0.142315i −2.38433 1.82070i 2.53048 + 2.19268i
11.8 0.540641 + 0.841254i 1.34287 + 1.09394i −0.415415 + 0.909632i −1.80115 + 0.528864i −0.194269 + 1.72112i 1.49075 1.29175i −0.989821 + 0.142315i 0.606601 + 2.93803i −1.41868 1.22930i
17.1 −0.909632 0.415415i −1.72827 0.114314i 0.654861 + 0.755750i −1.19773 + 0.769732i 1.52461 + 0.821935i 3.68797 + 0.530250i −0.281733 0.959493i 2.97386 + 0.395131i 1.40925 0.202619i
17.2 −0.909632 0.415415i −0.580581 + 1.63185i 0.654861 + 0.755750i −1.63636 + 1.05163i 1.20601 1.24320i −3.52778 0.507219i −0.281733 0.959493i −2.32585 1.89484i 1.92535 0.276823i
17.3 −0.909632 0.415415i 0.322552 1.70175i 0.654861 + 0.755750i 1.12015 0.719874i −1.00034 + 1.41398i 1.49094 + 0.214365i −0.281733 0.959493i −2.79192 1.09781i −1.31797 + 0.189495i
17.4 −0.909632 0.415415i 1.47913 + 0.901205i 0.654861 + 0.755750i 1.71394 1.10148i −0.971091 1.43422i −1.65113 0.237396i −0.281733 0.959493i 1.37566 + 2.66600i −2.01663 + 0.289948i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.d odd 22 1 inner
69.g even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.2.f.a 80
3.b odd 2 1 inner 138.2.f.a 80
23.d odd 22 1 inner 138.2.f.a 80
69.g even 22 1 inner 138.2.f.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.f.a 80 1.a even 1 1 trivial
138.2.f.a 80 3.b odd 2 1 inner
138.2.f.a 80 23.d odd 22 1 inner
138.2.f.a 80 69.g even 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(138, [\chi])$$.