Properties

Label 637.2.bd.c
Level $637$
Weight $2$
Character orbit 637.bd
Analytic conductor $5.086$
Analytic rank $0$
Dimension $112$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(97,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.bd (of order \(12\), degree \(4\), 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: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(28\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 56 q^{9} + 16 q^{11} - 48 q^{15} + 56 q^{16} + 32 q^{18} - 96 q^{30} - 192 q^{36} + 32 q^{39} - 128 q^{44} - 32 q^{46} - 56 q^{50} - 32 q^{53} + 96 q^{57} - 48 q^{58} - 224 q^{60} + 64 q^{65} - 32 q^{71} + 80 q^{72} + 104 q^{74} - 64 q^{78} + 128 q^{79} - 8 q^{81} + 120 q^{85} - 64 q^{86} + 288 q^{88} + 320 q^{92} - 64 q^{93} + 96 q^{95} + 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 −0.698623 2.60730i −1.59953 0.923490i −4.57787 + 2.64304i 0.611546 + 0.611546i −1.29034 + 4.81563i 0 6.27204 + 6.27204i 0.205668 + 0.356227i 1.16724 2.02172i
97.2 −0.698623 2.60730i 1.59953 + 0.923490i −4.57787 + 2.64304i −0.611546 0.611546i 1.29034 4.81563i 0 6.27204 + 6.27204i 0.205668 + 0.356227i −1.16724 + 2.02172i
97.3 −0.593883 2.21640i −0.730443 0.421721i −2.82768 + 1.63256i −0.109871 0.109871i −0.500906 + 1.86941i 0 2.05269 + 2.05269i −1.14430 1.98199i −0.178268 + 0.308770i
97.4 −0.593883 2.21640i 0.730443 + 0.421721i −2.82768 + 1.63256i 0.109871 + 0.109871i 0.500906 1.86941i 0 2.05269 + 2.05269i −1.14430 1.98199i 0.178268 0.308770i
97.5 −0.527888 1.97010i −2.81776 1.62683i −1.87060 + 1.07999i −2.21395 2.21395i −1.71757 + 6.41007i 0 0.230723 + 0.230723i 3.79318 + 6.56999i −3.19299 + 5.53043i
97.6 −0.527888 1.97010i 2.81776 + 1.62683i −1.87060 + 1.07999i 2.21395 + 2.21395i 1.71757 6.41007i 0 0.230723 + 0.230723i 3.79318 + 6.56999i 3.19299 5.53043i
97.7 −0.382384 1.42708i −2.31749 1.33800i −0.158275 + 0.0913803i 2.34842 + 2.34842i −1.02326 + 3.81886i 0 −1.89845 1.89845i 2.08049 + 3.60352i 2.45338 4.24938i
97.8 −0.382384 1.42708i 2.31749 + 1.33800i −0.158275 + 0.0913803i −2.34842 2.34842i 1.02326 3.81886i 0 −1.89845 1.89845i 2.08049 + 3.60352i −2.45338 + 4.24938i
97.9 −0.304268 1.13554i −0.530560 0.306319i 0.535168 0.308979i 0.927001 + 0.927001i −0.186406 + 0.695677i 0 −2.17625 2.17625i −1.31234 2.27304i 0.770594 1.33471i
97.10 −0.304268 1.13554i 0.530560 + 0.306319i 0.535168 0.308979i −0.927001 0.927001i 0.186406 0.695677i 0 −2.17625 2.17625i −1.31234 2.27304i −0.770594 + 1.33471i
97.11 −0.201540 0.752157i −2.47059 1.42640i 1.20693 0.696820i 0.942403 + 0.942403i −0.574953 + 2.14575i 0 −1.86860 1.86860i 2.56922 + 4.45003i 0.518904 0.898767i
97.12 −0.201540 0.752157i 2.47059 + 1.42640i 1.20693 0.696820i −0.942403 0.942403i 0.574953 2.14575i 0 −1.86860 1.86860i 2.56922 + 4.45003i −0.518904 + 0.898767i
97.13 −0.0641123 0.239270i −1.55865 0.899886i 1.67891 0.969320i −2.60215 2.60215i −0.115387 + 0.430632i 0 −0.689884 0.689884i 0.119590 + 0.207136i −0.455788 + 0.789448i
97.14 −0.0641123 0.239270i 1.55865 + 0.899886i 1.67891 0.969320i 2.60215 + 2.60215i 0.115387 0.430632i 0 −0.689884 0.689884i 0.119590 + 0.207136i 0.455788 0.789448i
97.15 0.0305678 + 0.114081i −0.864337 0.499025i 1.71997 0.993026i 0.483986 + 0.483986i 0.0305082 0.113858i 0 0.332887 + 0.332887i −1.00195 1.73543i −0.0404190 + 0.0700078i
97.16 0.0305678 + 0.114081i 0.864337 + 0.499025i 1.71997 0.993026i −0.483986 0.483986i −0.0305082 + 0.113858i 0 0.332887 + 0.332887i −1.00195 1.73543i 0.0404190 0.0700078i
97.17 0.244980 + 0.914278i −0.510427 0.294695i 0.956161 0.552040i −1.45945 1.45945i 0.144389 0.538867i 0 2.07756 + 2.07756i −1.32631 2.29724i 0.976808 1.69188i
97.18 0.244980 + 0.914278i 0.510427 + 0.294695i 0.956161 0.552040i 1.45945 + 1.45945i −0.144389 + 0.538867i 0 2.07756 + 2.07756i −1.32631 2.29724i −0.976808 + 1.69188i
97.19 0.280827 + 1.04806i −2.30381 1.33011i 0.712486 0.411354i 2.92094 + 2.92094i 0.747060 2.78806i 0 2.16567 + 2.16567i 2.03837 + 3.53056i −2.24104 + 3.88160i
97.20 0.280827 + 1.04806i 2.30381 + 1.33011i 0.712486 0.411354i −2.92094 2.92094i −0.747060 + 2.78806i 0 2.16567 + 2.16567i 2.03837 + 3.53056i 2.24104 3.88160i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.f odd 12 1 inner
91.bc even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.bd.c 112
7.b odd 2 1 inner 637.2.bd.c 112
7.c even 3 1 637.2.x.c 112
7.c even 3 1 637.2.bb.c 112
7.d odd 6 1 637.2.x.c 112
7.d odd 6 1 637.2.bb.c 112
13.f odd 12 1 inner 637.2.bd.c 112
91.w even 12 1 637.2.bb.c 112
91.x odd 12 1 637.2.x.c 112
91.ba even 12 1 637.2.x.c 112
91.bc even 12 1 inner 637.2.bd.c 112
91.bd odd 12 1 637.2.bb.c 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.x.c 112 7.c even 3 1
637.2.x.c 112 7.d odd 6 1
637.2.x.c 112 91.x odd 12 1
637.2.x.c 112 91.ba even 12 1
637.2.bb.c 112 7.c even 3 1
637.2.bb.c 112 7.d odd 6 1
637.2.bb.c 112 91.w even 12 1
637.2.bb.c 112 91.bd odd 12 1
637.2.bd.c 112 1.a even 1 1 trivial
637.2.bd.c 112 7.b odd 2 1 inner
637.2.bd.c 112 13.f odd 12 1 inner
637.2.bd.c 112 91.bc even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):

\( T_{2}^{56} - 105 T_{2}^{52} + 8 T_{2}^{49} + 7501 T_{2}^{48} - 216 T_{2}^{47} - 420 T_{2}^{46} - 1704 T_{2}^{45} - 279952 T_{2}^{44} + 15616 T_{2}^{43} - 3148 T_{2}^{42} + 2328 T_{2}^{41} + 7470684 T_{2}^{40} - 137736 T_{2}^{39} + \cdots + 58081 \) Copy content Toggle raw display
\( T_{3}^{112} - 112 T_{3}^{110} + 6652 T_{3}^{108} - 272672 T_{3}^{106} + 8572342 T_{3}^{104} - 218614832 T_{3}^{102} + 4681717696 T_{3}^{100} - 86173122064 T_{3}^{98} + 1386034819121 T_{3}^{96} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display