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Properties

Label 207.2.i.e
Level $207$
Weight $2$
Character orbit 207.i
Analytic conductor $1.653$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [207,2,Mod(55,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.55");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 207.i (of order \(11\), 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: \(1.65290332184\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(4\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{4} + 8 q^{7} - 56 q^{16} - 14 q^{19} - 28 q^{22} - 48 q^{25} - 64 q^{28} - 22 q^{31} - 10 q^{34} + 52 q^{37} + 6 q^{40} + 68 q^{43} + 84 q^{46} + 4 q^{49} + 110 q^{52} + 50 q^{55} + 18 q^{58} + 36 q^{61} + 116 q^{64} - 18 q^{67} + 96 q^{70} + 14 q^{73} + 34 q^{76} - 36 q^{79} - 72 q^{82} - 238 q^{85} - 160 q^{88} - 176 q^{91} - 206 q^{94} - 92 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1 −1.95757 + 0.574795i 0 1.81919 − 1.16913i 0.403163 + 2.80406i 0 0.0160895 − 0.0352312i −0.217083 + 0.250528i 0 −2.40098 − 5.25741i
55.2 −0.815320 + 0.239400i 0 −1.07507 + 0.690907i −0.413675 − 2.87717i 0 −0.728680 + 1.59559i 1.82405 − 2.10506i 0 1.02607 + 2.24678i
55.3 0.815320 − 0.239400i 0 −1.07507 + 0.690907i 0.413675 + 2.87717i 0 −0.728680 + 1.59559i −1.82405 + 2.10506i 0 1.02607 + 2.24678i
55.4 1.95757 − 0.574795i 0 1.81919 − 1.16913i −0.403163 − 2.80406i 0 0.0160895 − 0.0352312i 0.217083 − 0.250528i 0 −2.40098 − 5.25741i
64.1 −1.95757 − 0.574795i 0 1.81919 + 1.16913i 0.403163 − 2.80406i 0 0.0160895 + 0.0352312i −0.217083 − 0.250528i 0 −2.40098 + 5.25741i
64.2 −0.815320 − 0.239400i 0 −1.07507 − 0.690907i −0.413675 + 2.87717i 0 −0.728680 − 1.59559i 1.82405 + 2.10506i 0 1.02607 − 2.24678i
64.3 0.815320 + 0.239400i 0 −1.07507 − 0.690907i 0.413675 − 2.87717i 0 −0.728680 − 1.59559i −1.82405 − 2.10506i 0 1.02607 − 2.24678i
64.4 1.95757 + 0.574795i 0 1.81919 + 1.16913i −0.403163 + 2.80406i 0 0.0160895 + 0.0352312i 0.217083 + 0.250528i 0 −2.40098 + 5.25741i
73.1 −1.06109 − 2.32346i 0 −2.96283 + 3.41929i 1.79412 + 1.15301i 0 0.697257 + 4.84953i 6.18676 + 1.81660i 0 0.775255 − 5.39202i
73.2 −0.498977 − 1.09261i 0 0.364907 − 0.421126i −1.07050 − 0.687969i 0 −0.173182 − 1.20450i −2.94720 − 0.865377i 0 −0.217525 + 1.51292i
73.3 0.498977 + 1.09261i 0 0.364907 − 0.421126i 1.07050 + 0.687969i 0 −0.173182 − 1.20450i 2.94720 + 0.865377i 0 −0.217525 + 1.51292i
73.4 1.06109 + 2.32346i 0 −2.96283 + 3.41929i −1.79412 − 1.15301i 0 0.697257 + 4.84953i −6.18676 − 1.81660i 0 0.775255 − 5.39202i
82.1 −1.73622 − 2.00371i 0 −0.715745 + 4.97812i −0.667943 + 1.46259i 0 −0.586514 − 0.172216i 6.75657 − 4.34219i 0 4.09030 − 1.20102i
82.2 −0.858969 − 0.991303i 0 0.0397758 − 0.276647i 1.24955 − 2.73614i 0 3.30268 + 0.969753i −2.51532 + 1.61650i 0 −3.78567 + 1.11157i
82.3 0.858969 + 0.991303i 0 0.0397758 − 0.276647i −1.24955 + 2.73614i 0 3.30268 + 0.969753i 2.51532 − 1.61650i 0 −3.78567 + 1.11157i
82.4 1.73622 + 2.00371i 0 −0.715745 + 4.97812i 0.667943 − 1.46259i 0 −0.586514 − 0.172216i −6.75657 + 4.34219i 0 4.09030 − 1.20102i
100.1 −0.226138 + 1.57283i 0 −0.503658 − 0.147887i −0.696899 − 0.804264i 0 2.72743 + 1.75281i −0.973691 + 2.13209i 0 1.42256 − 0.914226i
100.2 −0.0988909 + 0.687801i 0 1.45569 + 0.427431i 2.69548 + 3.11074i 0 −3.30813 − 2.12601i −1.01526 + 2.22312i 0 −2.40613 + 1.54633i
100.3 0.0988909 − 0.687801i 0 1.45569 + 0.427431i −2.69548 − 3.11074i 0 −3.30813 − 2.12601i 1.01526 − 2.22312i 0 −2.40613 + 1.54633i
100.4 0.226138 − 1.57283i 0 −0.503658 − 0.147887i 0.696899 + 0.804264i 0 2.72743 + 1.75281i 0.973691 − 2.13209i 0 1.42256 − 0.914226i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.c even 11 1 inner
69.h odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.2.i.e ✓ 40
3.b odd 2 1 inner 207.2.i.e ✓ 40
23.c even 11 1 inner 207.2.i.e ✓ 40
23.c even 11 1 4761.2.a.bw 20
23.d odd 22 1 4761.2.a.bx 20
69.g even 22 1 4761.2.a.bx 20
69.h odd 22 1 inner 207.2.i.e ✓ 40
69.h odd 22 1 4761.2.a.bw 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.2.i.e ✓ 40 1.a even 1 1 trivial
207.2.i.e ✓ 40 3.b odd 2 1 inner
207.2.i.e ✓ 40 23.c even 11 1 inner
207.2.i.e ✓ 40 69.h odd 22 1 inner
4761.2.a.bw 20 23.c even 11 1
4761.2.a.bw 20 69.h odd 22 1
4761.2.a.bx 20 23.d odd 22 1
4761.2.a.bx 20 69.g even 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 6 T_{2}^{38} + 79 T_{2}^{36} + 248 T_{2}^{34} + 2069 T_{2}^{32} + 2948 T_{2}^{30} + \cdots + 279841 \) acting on \(S_{2}^{\mathrm{new}}(207, [\chi])\). Copy content Toggle raw display