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Properties

Label 639.2.i.b
Level $639$
Weight $2$
Character orbit 639.i
Analytic conductor $5.102$
Analytic rank $0$
Dimension $112$
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] = [639,2,Mod(212,639)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(639, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("639.212");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 639 = 3^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 639.i (of order \(6\), degree \(2\), 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: \(5.10244068916\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(56\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 6 q^{2} - 2 q^{3} + 38 q^{4} - 14 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 6 q^{2} - 2 q^{3} + 38 q^{4} - 14 q^{6} + 6 q^{9} - 16 q^{10} - 32 q^{12} - 28 q^{15} - 6 q^{16} - 40 q^{18} - 8 q^{19} - 78 q^{20} - 74 q^{24} - 16 q^{25} - 2 q^{27} - 6 q^{29} + 28 q^{30} + 54 q^{32} + 38 q^{36} + 4 q^{37} - 18 q^{38} - 8 q^{40} - 2 q^{43} - 14 q^{45} - 84 q^{48} + 154 q^{49} - 12 q^{50} - 30 q^{54} - 46 q^{57} - 28 q^{58} + 34 q^{60} + 148 q^{64} + 54 q^{72} - 8 q^{73} + 270 q^{74} - 62 q^{75} - 18 q^{76} + 114 q^{77} + 10 q^{79} - 26 q^{81} + 78 q^{83} + 102 q^{86} - 56 q^{87} + 208 q^{90} - 60 q^{91} - 132 q^{95} + 122 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
212.1 −2.18467 + 1.26132i 0.593196 + 1.62730i 2.18186 − 3.77910i −0.236451 − 0.136515i −3.34849 − 2.80692i −1.43165 + 0.826561i 5.96284i −2.29624 + 1.93062i 0.688757
212.2 −2.18467 + 1.26132i 0.593196 + 1.62730i 2.18186 − 3.77910i −0.236451 − 0.136515i −3.34849 − 2.80692i 1.43165 − 0.826561i 5.96284i −2.29624 + 1.93062i 0.688757
212.3 −2.15611 + 1.24483i 1.19523 − 1.25357i 2.09920 − 3.63592i 1.60079 + 0.924218i −1.01656 + 4.19068i −3.34137 + 1.92914i 5.47326i −0.142858 − 2.99660i −4.60197
212.4 −2.15611 + 1.24483i 1.19523 − 1.25357i 2.09920 − 3.63592i 1.60079 + 0.924218i −1.01656 + 4.19068i 3.34137 − 1.92914i 5.47326i −0.142858 − 2.99660i −4.60197
212.5 −2.12312 + 1.22578i 1.72247 + 0.181887i 2.00509 − 3.47291i −2.43689 − 1.40694i −3.87997 + 1.72521i −3.38907 + 1.95668i 4.92808i 2.93383 + 0.626591i 6.89840
212.6 −2.12312 + 1.22578i 1.72247 + 0.181887i 2.00509 − 3.47291i −2.43689 − 1.40694i −3.87997 + 1.72521i 3.38907 − 1.95668i 4.92808i 2.93383 + 0.626591i 6.89840
212.7 −2.00391 + 1.15696i −1.72962 + 0.0917325i 1.67711 − 2.90484i −2.39692 − 1.38386i 3.35988 − 2.18492i −2.67431 + 1.54401i 3.13356i 2.98317 − 0.317325i 6.40430
212.8 −2.00391 + 1.15696i −1.72962 + 0.0917325i 1.67711 − 2.90484i −2.39692 − 1.38386i 3.35988 − 2.18492i 2.67431 − 1.54401i 3.13356i 2.98317 − 0.317325i 6.40430
212.9 −1.87929 + 1.08501i −1.28600 − 1.16026i 1.35448 − 2.34602i 2.25481 + 1.30182i 3.67565 + 0.785147i −2.60786 + 1.50565i 1.53844i 0.307586 + 2.98419i −5.64992
212.10 −1.87929 + 1.08501i −1.28600 − 1.16026i 1.35448 − 2.34602i 2.25481 + 1.30182i 3.67565 + 0.785147i 2.60786 − 1.50565i 1.53844i 0.307586 + 2.98419i −5.64992
212.11 −1.73251 + 1.00026i −1.13721 + 1.30643i 1.00105 − 1.73387i 0.375224 + 0.216636i 0.663449 − 3.40091i −2.40175 + 1.38665i 0.00421387i −0.413513 − 2.97136i −0.866771
212.12 −1.73251 + 1.00026i −1.13721 + 1.30643i 1.00105 − 1.73387i 0.375224 + 0.216636i 0.663449 − 3.40091i 2.40175 − 1.38665i 0.00421387i −0.413513 − 2.97136i −0.866771
212.13 −1.20994 + 0.698561i −1.20289 − 1.24622i −0.0240261 + 0.0416144i −2.18157 − 1.25953i 2.32599 + 0.667560i −3.48167 + 2.01014i − 2.86138i −0.106114 + 2.99812i 3.51944
212.14 −1.20994 + 0.698561i −1.20289 − 1.24622i −0.0240261 + 0.0416144i −2.18157 − 1.25953i 2.32599 + 0.667560i 3.48167 − 2.01014i − 2.86138i −0.106114 + 2.99812i 3.51944
212.15 −1.16050 + 0.670016i 1.38146 + 1.04478i −0.102158 + 0.176943i −0.362195 − 0.209113i −2.30321 − 0.286876i −3.08744 + 1.78253i − 2.95385i 0.816849 + 2.88665i 0.560437
212.16 −1.16050 + 0.670016i 1.38146 + 1.04478i −0.102158 + 0.176943i −0.362195 − 0.209113i −2.30321 − 0.286876i 3.08744 − 1.78253i − 2.95385i 0.816849 + 2.88665i 0.560437
212.17 −1.03583 + 0.598035i 0.223409 − 1.71758i −0.284709 + 0.493130i 2.18036 + 1.25883i 0.795761 + 1.91272i −2.23587 + 1.29088i − 3.07320i −2.90018 − 0.767447i −3.01130
212.18 −1.03583 + 0.598035i 0.223409 − 1.71758i −0.284709 + 0.493130i 2.18036 + 1.25883i 0.795761 + 1.91272i 2.23587 − 1.29088i − 3.07320i −2.90018 − 0.767447i −3.01130
212.19 −0.901087 + 0.520243i −1.73092 + 0.0624675i −0.458695 + 0.794483i 1.22373 + 0.706522i 1.52721 − 0.956789i −0.741807 + 0.428283i − 3.03550i 2.99220 − 0.216253i −1.47025
212.20 −0.901087 + 0.520243i −1.73092 + 0.0624675i −0.458695 + 0.794483i 1.22373 + 0.706522i 1.52721 − 0.956789i 0.741807 − 0.428283i − 3.03550i 2.99220 − 0.216253i −1.47025
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 212.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
71.b odd 2 1 inner
639.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 639.2.i.b ✓ 112
9.d odd 6 1 inner 639.2.i.b ✓ 112
71.b odd 2 1 inner 639.2.i.b ✓ 112
639.i even 6 1 inner 639.2.i.b ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
639.2.i.b ✓ 112 1.a even 1 1 trivial
639.2.i.b ✓ 112 9.d odd 6 1 inner
639.2.i.b ✓ 112 71.b odd 2 1 inner
639.2.i.b ✓ 112 639.i even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} + 3 T_{2}^{55} - 33 T_{2}^{54} - 108 T_{2}^{53} + 623 T_{2}^{52} + 2157 T_{2}^{51} - 8089 T_{2}^{50} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(639, [\chi])\). Copy content Toggle raw display