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Properties

Label 980.2.x.n
Level $980$
Weight $2$
Character orbit 980.x
Analytic conductor $7.825$
Analytic rank $0$
Dimension $112$
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: \(112\)
Relative dimension: \(28\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 4 q^{2} - 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 4 q^{2} - 32 q^{8} - 32 q^{16} + 40 q^{22} - 32 q^{25} + 28 q^{30} + 64 q^{32} + 32 q^{36} + 8 q^{37} + 184 q^{46} - 24 q^{50} - 96 q^{53} - 16 q^{57} - 124 q^{58} - 8 q^{60} + 120 q^{65} + 80 q^{72} - 72 q^{78} + 72 q^{81} + 192 q^{85} - 104 q^{86} - 48 q^{88} - 304 q^{92} + 176 q^{93}+O(q^{100}) \) 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 −1.38058 − 0.306614i −0.832079 + 0.222955i 1.81198 + 0.846608i −2.12698 − 0.689905i 1.21711 − 0.0526786i 0 −2.24199 − 1.72438i −1.95543 + 1.12897i 2.72492 + 1.60463i
67.2 −1.38058 − 0.306614i 0.832079 − 0.222955i 1.81198 + 0.846608i 2.12698 + 0.689905i −1.21711 + 0.0526786i 0 −2.24199 − 1.72438i −1.95543 + 1.12897i −2.72492 − 1.60463i
67.3 −1.29600 − 0.566026i −2.75218 + 0.737445i 1.35923 + 1.46714i −0.951377 + 2.02358i 3.98424 + 0.602077i 0 −0.931125 − 2.67077i 4.43260 − 2.55916i 2.37838 − 2.08406i
67.4 −1.29600 − 0.566026i 2.75218 − 0.737445i 1.35923 + 1.46714i 0.951377 − 2.02358i −3.98424 − 0.602077i 0 −0.931125 − 2.67077i 4.43260 − 2.55916i −2.37838 + 2.08406i
67.5 −1.27862 + 0.604271i −3.15219 + 0.844627i 1.26971 − 1.54526i 0.260066 + 2.22089i 3.52006 − 2.98473i 0 −0.689718 + 2.74304i 6.62484 − 3.82485i −1.67455 − 2.68252i
67.6 −1.27862 + 0.604271i 3.15219 − 0.844627i 1.26971 − 1.54526i −0.260066 − 2.22089i −3.52006 + 2.98473i 0 −0.689718 + 2.74304i 6.62484 − 3.82485i 1.67455 + 2.68252i
67.7 −1.16082 + 0.807777i −0.625414 + 0.167579i 0.694993 − 1.87536i 0.117869 − 2.23296i 0.590624 − 0.699723i 0 0.708115 + 2.73835i −2.23502 + 1.29039i 1.66691 + 2.68727i
67.8 −1.16082 + 0.807777i 0.625414 − 0.167579i 0.694993 − 1.87536i −0.117869 + 2.23296i −0.590624 + 0.699723i 0 0.708115 + 2.73835i −2.23502 + 1.29039i −1.66691 − 2.68727i
67.9 −0.970576 + 1.02858i −1.93300 + 0.517947i −0.115964 − 1.99664i −2.17870 − 0.503254i 1.34338 − 2.49096i 0 2.16626 + 1.81861i 0.870157 − 0.502386i 2.63223 − 1.75253i
67.10 −0.970576 + 1.02858i 1.93300 − 0.517947i −0.115964 − 1.99664i 2.17870 + 0.503254i −1.34338 + 2.49096i 0 2.16626 + 1.81861i 0.870157 − 0.502386i −2.63223 + 1.75253i
67.11 −0.682687 − 1.23852i −1.62424 + 0.435215i −1.06788 + 1.69105i 1.15295 − 1.91591i 1.64787 + 1.71455i 0 2.82343 + 0.168132i −0.149320 + 0.0862100i −3.16000 − 0.119985i
67.12 −0.682687 − 1.23852i 1.62424 − 0.435215i −1.06788 + 1.69105i −1.15295 + 1.91591i −1.64787 − 1.71455i 0 2.82343 + 0.168132i −0.149320 + 0.0862100i 3.16000 + 0.119985i
67.13 −0.0109473 − 1.41417i −2.41256 + 0.646443i −1.99976 + 0.0309628i −1.90112 − 1.17717i 0.940592 + 3.40469i 0 0.0656787 + 2.82766i 2.80447 − 1.61916i −1.64391 + 2.70140i
67.14 −0.0109473 − 1.41417i 2.41256 − 0.646443i −1.99976 + 0.0309628i 1.90112 + 1.17717i −0.940592 − 3.40469i 0 0.0656787 + 2.82766i 2.80447 − 1.61916i 1.64391 − 2.70140i
67.15 0.326252 + 1.37607i −1.93300 + 0.517947i −1.78712 + 0.897890i 2.17870 + 0.503254i −1.34338 − 2.49096i 0 −1.81861 − 2.16626i 0.870157 − 0.502386i 0.0182956 + 3.16222i
67.16 0.326252 + 1.37607i 1.93300 − 0.517947i −1.78712 + 0.897890i −2.17870 − 0.503254i 1.34338 + 2.49096i 0 −1.81861 − 2.16626i 0.870157 − 0.502386i −0.0182956 − 3.16222i
67.17 0.601409 + 1.27996i −0.625414 + 0.167579i −1.27662 + 1.53956i −0.117869 + 2.23296i −0.590624 − 0.699723i 0 −2.73835 − 0.708115i −2.23502 + 1.29039i −2.92899 + 1.19205i
67.18 0.601409 + 1.27996i 0.625414 − 0.167579i −1.27662 + 1.53956i 0.117869 − 2.23296i 0.590624 + 0.699723i 0 −2.73835 − 0.708115i −2.23502 + 1.29039i 2.92899 − 1.19205i
67.19 0.716566 − 1.21923i −2.41256 + 0.646443i −0.973066 − 1.74732i 1.90112 + 1.17717i −0.940592 + 3.40469i 0 −2.82766 − 0.0656787i 2.80447 − 1.61916i 2.79753 − 1.47439i
67.20 0.716566 − 1.21923i 2.41256 − 0.646443i −0.973066 − 1.74732i −1.90112 − 1.17717i 0.940592 − 3.40469i 0 −2.82766 − 0.0656787i 2.80447 − 1.61916i −2.79753 + 1.47439i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.28
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.n 112
4.b odd 2 1 inner 980.2.x.n 112
5.c odd 4 1 inner 980.2.x.n 112
7.b odd 2 1 inner 980.2.x.n 112
7.c even 3 1 980.2.k.m ✓ 56
7.c even 3 1 inner 980.2.x.n 112
7.d odd 6 1 980.2.k.m ✓ 56
7.d odd 6 1 inner 980.2.x.n 112
20.e even 4 1 inner 980.2.x.n 112
28.d even 2 1 inner 980.2.x.n 112
28.f even 6 1 980.2.k.m ✓ 56
28.f even 6 1 inner 980.2.x.n 112
28.g odd 6 1 980.2.k.m ✓ 56
28.g odd 6 1 inner 980.2.x.n 112
35.f even 4 1 inner 980.2.x.n 112
35.k even 12 1 980.2.k.m ✓ 56
35.k even 12 1 inner 980.2.x.n 112
35.l odd 12 1 980.2.k.m ✓ 56
35.l odd 12 1 inner 980.2.x.n 112
140.j odd 4 1 inner 980.2.x.n 112
140.w even 12 1 980.2.k.m ✓ 56
140.w even 12 1 inner 980.2.x.n 112
140.x odd 12 1 980.2.k.m ✓ 56
140.x odd 12 1 inner 980.2.x.n 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.k.m ✓ 56 7.c even 3 1
980.2.k.m ✓ 56 7.d odd 6 1
980.2.k.m ✓ 56 28.f even 6 1
980.2.k.m ✓ 56 28.g odd 6 1
980.2.k.m ✓ 56 35.k even 12 1
980.2.k.m ✓ 56 35.l odd 12 1
980.2.k.m ✓ 56 140.w even 12 1
980.2.k.m ✓ 56 140.x odd 12 1
980.2.x.n 112 1.a even 1 1 trivial
980.2.x.n 112 4.b odd 2 1 inner
980.2.x.n 112 5.c odd 4 1 inner
980.2.x.n 112 7.b odd 2 1 inner
980.2.x.n 112 7.c even 3 1 inner
980.2.x.n 112 7.d odd 6 1 inner
980.2.x.n 112 20.e even 4 1 inner
980.2.x.n 112 28.d even 2 1 inner
980.2.x.n 112 28.f even 6 1 inner
980.2.x.n 112 28.g odd 6 1 inner
980.2.x.n 112 35.f even 4 1 inner
980.2.x.n 112 35.k even 12 1 inner
980.2.x.n 112 35.l odd 12 1 inner
980.2.x.n 112 140.j odd 4 1 inner
980.2.x.n 112 140.w even 12 1 inner
980.2.x.n 112 140.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 \(S_{2}^{\mathrm{new}}(980, [\chi])\):

\( T_{3}^{56} - 243 T_{3}^{52} + 39046 T_{3}^{48} - 3499367 T_{3}^{44} + 225383934 T_{3}^{40} + \cdots + 13032100000000 \) Copy content Toggle raw display
\( T_{11}^{28} - 105 T_{11}^{26} + 6838 T_{11}^{24} - 281469 T_{11}^{22} + 8497430 T_{11}^{20} + \cdots + 577600000000 \) Copy content Toggle raw display
\( T_{13}^{28} + 2603 T_{13}^{24} + 1014243 T_{13}^{20} + 152453185 T_{13}^{16} + 9828930928 T_{13}^{12} + \cdots + 3321506250000 \) Copy content Toggle raw display