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Properties

Label 980.2.x.j
Level $980$
Weight $2$
Character orbit 980.x
Analytic conductor $7.825$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,2,Mod(67,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 3, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.x (of order \(12\), degree \(4\), not 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: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 4 q^{2} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 4 q^{2} + 16 q^{8} - 8 q^{16} + 40 q^{18} - 72 q^{22} - 32 q^{25} + 36 q^{30} - 16 q^{32} - 176 q^{36} + 48 q^{37} + 56 q^{50} - 16 q^{53} - 32 q^{57} - 36 q^{58} + 80 q^{60} - 64 q^{65} - 56 q^{72} + 56 q^{78} - 56 q^{86} - 88 q^{88} + 272 q^{92} - 32 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 −1.33028 + 0.479962i −1.11218 + 0.298008i 1.53927 − 1.27696i 1.76393 + 1.37424i 1.33648 − 0.930237i 0 −1.43477 + 2.43751i −1.44994 + 0.837123i −3.00610 − 0.981506i
67.2 −1.33028 + 0.479962i 1.11218 − 0.298008i 1.53927 − 1.27696i −1.76393 − 1.37424i −1.33648 + 0.930237i 0 −1.43477 + 2.43751i −1.44994 + 0.837123i 3.00610 + 0.981506i
67.3 −1.26313 − 0.636013i −1.41415 + 0.378919i 1.19098 + 1.60673i 0.468031 − 2.18654i 2.02724 + 0.420792i 0 −0.482452 − 2.78698i −0.741848 + 0.428306i −1.98185 + 2.46420i
67.4 −1.26313 − 0.636013i 1.41415 − 0.378919i 1.19098 + 1.60673i −0.468031 + 2.18654i −2.02724 − 0.420792i 0 −0.482452 − 2.78698i −0.741848 + 0.428306i 1.98185 − 2.46420i
67.5 −0.702038 + 1.22766i −2.07451 + 0.555863i −1.01428 − 1.72372i −2.09358 + 0.785436i 0.773975 − 2.93702i 0 2.82821 − 0.0350723i 1.39652 − 0.806284i 0.505530 − 3.12161i
67.6 −0.702038 + 1.22766i 2.07451 − 0.555863i −1.01428 − 1.72372i 2.09358 − 0.785436i −0.773975 + 2.93702i 0 2.82821 − 0.0350723i 1.39652 − 0.806284i −0.505530 + 3.12161i
67.7 −0.445955 − 1.34206i −3.04202 + 0.815107i −1.60225 + 1.19700i 1.46712 + 1.68747i 2.45053 + 3.71907i 0 2.32097 + 1.61650i 5.99142 − 3.45915i 1.61042 − 2.72150i
67.8 −0.445955 − 1.34206i 3.04202 − 0.815107i −1.60225 + 1.19700i −1.46712 − 1.68747i −2.45053 − 3.71907i 0 2.32097 + 1.61650i 5.99142 − 3.45915i −1.61042 + 2.72150i
67.9 −0.00584552 + 1.41420i −2.07451 + 0.555863i −1.99993 − 0.0165335i 2.09358 − 0.785436i −0.773975 − 2.93702i 0 0.0350723 − 2.82821i 1.39652 − 0.806284i 1.09853 + 2.96534i
67.10 −0.00584552 + 1.41420i 2.07451 − 0.555863i −1.99993 − 0.0165335i −2.09358 + 0.785436i 0.773975 + 2.93702i 0 0.0350723 − 2.82821i 1.39652 − 0.806284i −1.09853 − 2.96534i
67.11 0.912073 + 1.08080i −1.11218 + 0.298008i −0.336247 + 1.97153i −1.76393 − 1.37424i −1.33648 − 0.930237i 0 −2.43751 + 1.43477i −1.44994 + 0.837123i −0.123553 − 3.15986i
67.12 0.912073 + 1.08080i 1.11218 − 0.298008i −0.336247 + 1.97153i 1.76393 + 1.37424i 1.33648 + 0.930237i 0 −2.43751 + 1.43477i −1.44994 + 0.837123i 0.123553 + 3.15986i
67.13 1.05724 − 0.939280i −3.04202 + 0.815107i 0.235506 − 1.98609i −1.46712 − 1.68747i −2.45053 + 3.71907i 0 −1.61650 − 2.32097i 5.99142 − 3.45915i −3.13610 − 0.406030i
67.14 1.05724 − 0.939280i 3.04202 − 0.815107i 0.235506 − 1.98609i 1.46712 + 1.68747i 2.45053 − 3.71907i 0 −1.61650 − 2.32097i 5.99142 − 3.45915i 3.13610 + 0.406030i
67.15 1.41191 + 0.0807598i −1.41415 + 0.378919i 1.98696 + 0.228051i −0.468031 + 2.18654i −2.02724 + 0.420792i 0 2.78698 + 0.482452i −0.741848 + 0.428306i −0.837400 + 3.04939i
67.16 1.41191 + 0.0807598i 1.41415 − 0.378919i 1.98696 + 0.228051i 0.468031 − 2.18654i 2.02724 − 0.420792i 0 2.78698 + 0.482452i −0.741848 + 0.428306i 0.837400 − 3.04939i
263.1 −1.34206 + 0.445955i −0.815107 − 3.04202i 1.60225 − 1.19700i 0.727838 + 2.11430i 2.45053 + 3.71907i 0 −1.61650 + 2.32097i −5.99142 + 3.45915i −1.91968 − 2.51293i
263.2 −1.34206 + 0.445955i 0.815107 + 3.04202i 1.60225 − 1.19700i −0.727838 − 2.11430i −2.45053 − 3.71907i 0 −1.61650 + 2.32097i −5.99142 + 3.45915i 1.91968 + 2.51293i
263.3 −0.939280 − 1.05724i −0.815107 − 3.04202i −0.235506 + 1.98609i −0.727838 − 2.11430i −2.45053 + 3.71907i 0 2.32097 − 1.61650i −5.99142 + 3.45915i −1.55167 + 2.75542i
263.4 −0.939280 − 1.05724i 0.815107 + 3.04202i −0.235506 + 1.98609i 0.727838 + 2.11430i 2.45053 − 3.71907i 0 2.32097 − 1.61650i −5.99142 + 3.45915i 1.55167 − 2.75542i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
20.e even 4 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
35.f even 4 1 inner
35.k even 12 1 inner
35.l odd 12 1 inner
140.j odd 4 1 inner
140.w even 12 1 inner
140.x odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.x.j 64
4.b odd 2 1 inner 980.2.x.j 64
5.c odd 4 1 inner 980.2.x.j 64
7.b odd 2 1 inner 980.2.x.j 64
7.c even 3 1 980.2.k.i ✓ 32
7.c even 3 1 inner 980.2.x.j 64
7.d odd 6 1 980.2.k.i ✓ 32
7.d odd 6 1 inner 980.2.x.j 64
20.e even 4 1 inner 980.2.x.j 64
28.d even 2 1 inner 980.2.x.j 64
28.f even 6 1 980.2.k.i ✓ 32
28.f even 6 1 inner 980.2.x.j 64
28.g odd 6 1 980.2.k.i ✓ 32
28.g odd 6 1 inner 980.2.x.j 64
35.f even 4 1 inner 980.2.x.j 64
35.k even 12 1 980.2.k.i ✓ 32
35.k even 12 1 inner 980.2.x.j 64
35.l odd 12 1 980.2.k.i ✓ 32
35.l odd 12 1 inner 980.2.x.j 64
140.j odd 4 1 inner 980.2.x.j 64
140.w even 12 1 980.2.k.i ✓ 32
140.w even 12 1 inner 980.2.x.j 64
140.x odd 12 1 980.2.k.i ✓ 32
140.x odd 12 1 inner 980.2.x.j 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.k.i ✓ 32 7.c even 3 1
980.2.k.i ✓ 32 7.d odd 6 1
980.2.k.i ✓ 32 28.f even 6 1
980.2.k.i ✓ 32 28.g odd 6 1
980.2.k.i ✓ 32 35.k even 12 1
980.2.k.i ✓ 32 35.l odd 12 1
980.2.k.i ✓ 32 140.w even 12 1
980.2.k.i ✓ 32 140.x odd 12 1
980.2.x.j 64 1.a even 1 1 trivial
980.2.x.j 64 4.b odd 2 1 inner
980.2.x.j 64 5.c odd 4 1 inner
980.2.x.j 64 7.b odd 2 1 inner
980.2.x.j 64 7.c even 3 1 inner
980.2.x.j 64 7.d odd 6 1 inner
980.2.x.j 64 20.e even 4 1 inner
980.2.x.j 64 28.d even 2 1 inner
980.2.x.j 64 28.f even 6 1 inner
980.2.x.j 64 28.g odd 6 1 inner
980.2.x.j 64 35.f even 4 1 inner
980.2.x.j 64 35.k even 12 1 inner
980.2.x.j 64 35.l odd 12 1 inner
980.2.x.j 64 140.j odd 4 1 inner
980.2.x.j 64 140.w even 12 1 inner
980.2.x.j 64 140.x odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\):

\( T_{3}^{32} - 126 T_{3}^{28} + 13015 T_{3}^{24} - 331966 T_{3}^{20} + 6371661 T_{3}^{16} + \cdots + 285610000 \) Copy content Toggle raw display
\( T_{11}^{16} - 58 T_{11}^{14} + 2375 T_{11}^{12} - 45890 T_{11}^{10} + 635241 T_{11}^{8} + \cdots + 103876864 \) Copy content Toggle raw display
\( T_{13}^{16} + 518T_{13}^{12} + 86869T_{13}^{8} + 5054260T_{13}^{4} + 53144100 \) Copy content Toggle raw display