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Properties

Label 639.2.j.d
Level $639$
Weight $2$
Character orbit 639.j
Analytic conductor $5.102$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [639,2,Mod(37,639)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(639, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("639.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 639 = 3^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 639.j (of order \(7\), degree \(6\), 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: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{7})\)
Twist minimal: no (minimal twist has level 213)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 2 q^{2} - 18 q^{4} - 2 q^{5} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 2 q^{2} - 18 q^{4} - 2 q^{5} + 12 q^{8} + 19 q^{10} - 2 q^{11} + 46 q^{14} - 8 q^{16} + 20 q^{17} + 24 q^{20} + 24 q^{22} + 10 q^{23} + 50 q^{25} - 16 q^{26} + 6 q^{28} + 5 q^{29} + 23 q^{31} - 9 q^{32} - 12 q^{34} - 12 q^{35} - 36 q^{37} + 2 q^{38} - 66 q^{40} - 11 q^{41} + 27 q^{43} + 86 q^{44} + 72 q^{46} - 61 q^{47} + 16 q^{49} + 26 q^{50} + 53 q^{52} + 34 q^{53} - 62 q^{55} - 71 q^{56} - 21 q^{58} + 10 q^{59} + 19 q^{61} - 43 q^{62} - 24 q^{64} + 2 q^{65} + 24 q^{67} - 43 q^{68} - 20 q^{70} - 40 q^{71} + 78 q^{73} - 46 q^{74} + 6 q^{76} - 56 q^{77} - 82 q^{79} - 168 q^{80} + 3 q^{82} + 12 q^{83} - 118 q^{85} + 9 q^{86} + 16 q^{88} - 46 q^{89} - 54 q^{91} - 26 q^{92} + 6 q^{94} + 105 q^{95} + 31 q^{97} + 16 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −1.54982 + 1.94341i 0 −0.929870 − 4.07403i −1.04138 0 −0.718319 − 0.900743i 4.87954 + 2.34986i 0 1.61395 − 2.02383i
37.2 −0.863971 + 1.08339i 0 0.0177635 + 0.0778269i −0.526726 0 0.516999 + 0.648296i −2.59661 − 1.25046i 0 0.455076 − 0.570648i
37.3 −0.0398697 + 0.0499950i 0 0.444132 + 1.94587i −3.84276 0 −2.17217 − 2.72381i −0.230218 − 0.110867i 0 0.153210 − 0.192119i
37.4 0.208977 − 0.262049i 0 0.420044 + 1.84033i 2.86351 0 0.716473 + 0.898429i 1.17400 + 0.565368i 0 0.598408 − 0.750380i
37.5 0.894657 − 1.12186i 0 −0.0131267 − 0.0575118i −1.89146 0 1.98843 + 2.49341i 2.50937 + 1.20845i 0 −1.69220 + 2.12196i
37.6 1.62751 − 2.04083i 0 −1.07116 − 4.69306i 3.99377 0 −0.331414 − 0.415580i −6.61743 − 3.18679i 0 6.49989 − 8.15061i
91.1 −0.604226 − 2.64729i 0 −4.84110 + 2.33135i −3.96130 0 −0.306748 + 1.34395i 5.71087 + 7.16120i 0 2.39352 + 10.4867i
91.2 −0.498434 − 2.18378i 0 −2.71853 + 1.30917i 4.23158 0 −1.05163 + 4.60749i 1.42080 + 1.78162i 0 −2.10916 − 9.24084i
91.3 −0.320773 − 1.40540i 0 −0.0703122 + 0.0338606i −0.128425 0 0.484223 − 2.12152i −1.72743 − 2.16613i 0 0.0411951 + 0.180488i
91.4 0.231990 + 1.01641i 0 0.822662 − 0.396173i −1.99212 0 −0.316299 + 1.38580i 1.89357 + 2.37446i 0 −0.462152 − 2.02482i
91.5 0.239019 + 1.04721i 0 0.762418 − 0.367161i 2.33653 0 0.294907 − 1.29207i 1.90616 + 2.39025i 0 0.558476 + 2.44684i
91.6 0.551455 + 2.41608i 0 −3.73142 + 1.79696i −2.28821 0 0.895546 − 3.92364i −3.30903 − 4.14939i 0 −1.26184 − 5.52850i
172.1 −1.78589 − 0.860038i 0 1.20275 + 1.50820i −2.20136 0 0.0958702 − 0.0461686i 0.0312890 + 0.137086i 0 3.93138 + 1.89325i
172.2 −1.01823 − 0.490355i 0 −0.450631 − 0.565073i 1.10889 0 3.88291 − 1.86991i 0.684726 + 2.99998i 0 −1.12910 − 0.543747i
172.3 −0.623135 − 0.300086i 0 −0.948734 − 1.18967i 2.61453 0 −2.58710 + 1.24588i 0.541988 + 2.37460i 0 −1.62921 − 0.784585i
172.4 1.04914 + 0.505239i 0 −0.401552 − 0.503530i −0.111351 0 −4.35452 + 2.09703i −0.685113 − 3.00168i 0 −0.116822 − 0.0562587i
172.5 1.31498 + 0.633261i 0 0.0811733 + 0.101788i −2.30988 0 2.67882 − 1.29005i −0.607264 − 2.66060i 0 −3.03745 − 1.46276i
172.6 2.18663 + 1.05302i 0 2.42549 + 3.04147i 2.14616 0 0.284028 − 0.136781i 1.02080 + 4.47242i 0 4.69284 + 2.25995i
190.1 −1.54982 − 1.94341i 0 −0.929870 + 4.07403i −1.04138 0 −0.718319 + 0.900743i 4.87954 − 2.34986i 0 1.61395 + 2.02383i
190.2 −0.863971 − 1.08339i 0 0.0177635 − 0.0778269i −0.526726 0 0.516999 − 0.648296i −2.59661 + 1.25046i 0 0.455076 + 0.570648i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 639.2.j.d 36
3.b odd 2 1 213.2.f.b ✓ 36
71.d even 7 1 inner 639.2.j.d 36
213.k odd 14 1 213.2.f.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
213.2.f.b ✓ 36 3.b odd 2 1
213.2.f.b ✓ 36 213.k odd 14 1
639.2.j.d 36 1.a even 1 1 trivial
639.2.j.d 36 71.d even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 2 T_{2}^{35} + 17 T_{2}^{34} - 38 T_{2}^{33} + 166 T_{2}^{32} - 319 T_{2}^{31} + \cdots + 1849 \) acting on \(S_{2}^{\mathrm{new}}(639, [\chi])\). Copy content Toggle raw display