# Properties

 Label 690.2.w.b Level $690$ Weight $2$ Character orbit 690.w Analytic conductor $5.510$ Analytic rank $0$ Dimension $240$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,2,Mod(7,690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, base_ring=CyclotomicField(44))

chi = DirichletCharacter(H, H._module([0, 11, 38]))

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

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

chi := DirichletCharacter("690.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 690.w (of order $$44$$, degree $$20$$, 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: $$5.50967773947$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$12$$ over $$\Q(\zeta_{44})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$240 q + 24 q^{6}+O(q^{10})$$ 240 * q + 24 * q^6 $$\operatorname{Tr}(f)(q) =$$ $$240 q + 24 q^{6} - 44 q^{10} + 24 q^{16} - 44 q^{21} + 96 q^{23} + 16 q^{25} + 16 q^{26} + 44 q^{28} - 16 q^{31} + 44 q^{33} + 16 q^{35} - 24 q^{36} + 44 q^{37} - 88 q^{43} + 8 q^{46} + 96 q^{47} - 24 q^{50} - 24 q^{55} + 44 q^{57} - 16 q^{58} + 88 q^{61} + 56 q^{62} - 88 q^{65} + 132 q^{67} - 56 q^{70} + 16 q^{71} + 48 q^{73} + 24 q^{81} - 24 q^{82} + 44 q^{85} - 16 q^{87} + 44 q^{88} - 124 q^{92} + 32 q^{93} + 20 q^{95} - 24 q^{96} - 56 q^{98}+O(q^{100})$$ 240 * q + 24 * q^6 - 44 * q^10 + 24 * q^16 - 44 * q^21 + 96 * q^23 + 16 * q^25 + 16 * q^26 + 44 * q^28 - 16 * q^31 + 44 * q^33 + 16 * q^35 - 24 * q^36 + 44 * q^37 - 88 * q^43 + 8 * q^46 + 96 * q^47 - 24 * q^50 - 24 * q^55 + 44 * q^57 - 16 * q^58 + 88 * q^61 + 56 * q^62 - 88 * q^65 + 132 * q^67 - 56 * q^70 + 16 * q^71 + 48 * q^73 + 24 * q^81 - 24 * q^82 + 44 * q^85 - 16 * q^87 + 44 * q^88 - 124 * q^92 + 32 * q^93 + 20 * q^95 - 24 * q^96 - 56 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i −1.95274 + 1.08941i 0.142315 0.989821i 1.32241 2.42181i −0.977147 + 0.212565i −0.909632 0.415415i 1.87005 1.22594i
7.2 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i −1.89440 1.18796i 0.142315 0.989821i 0.244741 0.448210i −0.977147 + 0.212565i −0.909632 0.415415i 1.97432 + 1.04979i
7.3 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i −0.619349 2.14858i 0.142315 0.989821i −1.99507 + 3.65370i −0.977147 + 0.212565i −0.909632 0.415415i 0.771049 + 2.09892i
7.4 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i 1.62532 + 1.53569i 0.142315 0.989821i 1.83156 3.35425i −0.977147 + 0.212565i −0.909632 0.415415i −1.73074 1.41582i
7.5 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i 1.87213 1.22275i 0.142315 0.989821i −0.241465 + 0.442209i −0.977147 + 0.212565i −0.909632 0.415415i −1.78013 + 1.35319i
7.6 −0.997452 + 0.0713392i −0.212565 + 0.977147i 0.989821 0.142315i 1.98312 + 1.03307i 0.142315 0.989821i −0.653533 + 1.19686i −0.977147 + 0.212565i −0.909632 0.415415i −2.05177 0.888964i
7.7 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i −2.18918 + 0.455502i 0.142315 0.989821i 0.172948 0.316731i 0.977147 0.212565i −0.909632 0.415415i −2.15111 + 0.610516i
7.8 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i −1.40926 1.73608i 0.142315 0.989821i 1.26240 2.31192i 0.977147 0.212565i −0.909632 0.415415i −1.52952 1.63113i
7.9 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i 0.0285953 + 2.23589i 0.142315 0.989821i −1.81898 + 3.33121i 0.977147 0.212565i −0.909632 0.415415i 0.188029 + 2.22815i
7.10 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i 0.855176 + 2.06608i 0.142315 0.989821i 1.43032 2.61943i 0.977147 0.212565i −0.909632 0.415415i 1.00039 + 1.99981i
7.11 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i 1.40256 1.74150i 0.142315 0.989821i 1.87882 3.44079i 0.977147 0.212565i −0.909632 0.415415i 1.27475 1.83712i
7.12 0.997452 0.0713392i 0.212565 0.977147i 0.989821 0.142315i 1.93237 1.12513i 0.142315 0.989821i −0.968092 + 1.77293i 0.977147 0.212565i −0.909632 0.415415i 1.84719 1.26012i
37.1 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i −2.21203 0.326962i −0.841254 0.540641i 0.672927 + 1.80419i −0.0713392 0.997452i 0.989821 + 0.142315i 1.78476 + 1.34708i
37.2 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i −1.79792 1.32947i −0.841254 0.540641i −1.02771 2.75539i −0.0713392 0.997452i 0.989821 + 0.142315i 0.940853 + 2.02850i
37.3 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i 0.217321 2.22548i −0.841254 0.540641i 0.154967 + 0.415484i −0.0713392 0.997452i 0.989821 + 0.142315i −1.25730 + 1.84911i
37.4 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i 1.06201 + 1.96778i −0.841254 0.540641i −0.702461 1.88337i −0.0713392 0.997452i 0.989821 + 0.142315i 0.0109547 2.23604i
37.5 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i 2.06085 + 0.867694i −0.841254 0.540641i −1.60657 4.30737i −0.0713392 0.997452i 0.989821 + 0.142315i −1.39292 1.74922i
37.6 −0.877679 0.479249i 0.997452 + 0.0713392i 0.540641 + 0.841254i 2.18586 0.471182i −0.841254 0.540641i 1.33355 + 3.57540i −0.0713392 0.997452i 0.989821 + 0.142315i −2.14430 0.634025i
37.7 0.877679 + 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i −2.09572 0.779713i −0.841254 0.540641i −0.448960 1.20371i 0.0713392 + 0.997452i 0.989821 + 0.142315i −1.46569 1.68871i
37.8 0.877679 + 0.479249i −0.997452 0.0713392i 0.540641 + 0.841254i −1.84527 1.26292i −0.841254 0.540641i 1.33335 + 3.57484i 0.0713392 + 0.997452i 0.989821 + 0.142315i −1.01431 1.99278i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 613.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.d odd 22 1 inner
115.l even 44 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.w.b 240
5.c odd 4 1 inner 690.2.w.b 240
23.d odd 22 1 inner 690.2.w.b 240
115.l even 44 1 inner 690.2.w.b 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.w.b 240 1.a even 1 1 trivial
690.2.w.b 240 5.c odd 4 1 inner
690.2.w.b 240 23.d odd 22 1 inner
690.2.w.b 240 115.l even 44 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{240} + 88 T_{7}^{237} - 1408 T_{7}^{236} + 924 T_{7}^{235} + 3872 T_{7}^{234} - 121308 T_{7}^{233} + 966606 T_{7}^{232} - 1414952 T_{7}^{231} - 4796440 T_{7}^{230} + 77828432 T_{7}^{229} + \cdots + 55\!\cdots\!96$$ acting on $$S_{2}^{\mathrm{new}}(690, [\chi])$$.