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Properties

Label 980.2.k.m
Level $980$
Weight $2$
Character orbit 980.k
Analytic conductor $7.825$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,2,Mod(687,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.687");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.k (of order \(4\), 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: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 4 q^{2} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 4 q^{2} - 16 q^{8} + 32 q^{16} + 20 q^{22} + 32 q^{25} - 28 q^{30} - 64 q^{32} + 16 q^{36} - 8 q^{37} - 184 q^{46} - 12 q^{50} + 96 q^{53} - 8 q^{57} + 124 q^{58} + 8 q^{60} - 120 q^{65} - 80 q^{72} - 36 q^{78} - 72 q^{81} + 96 q^{85} + 104 q^{86} + 48 q^{88} - 152 q^{92} - 176 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
687.1 −1.40945 + 0.115993i −2.30756 − 2.30756i 1.97309 − 0.326974i −2.05338 − 0.885223i 3.52006 + 2.98473i 0 −2.74304 + 0.689718i 7.64971i 2.99682 + 1.00950i
687.2 −1.40945 + 0.115993i 2.30756 + 2.30756i 1.97309 − 0.326974i 2.05338 + 0.885223i −3.52006 − 2.98473i 0 −2.74304 + 0.689718i 7.64971i −2.99682 − 1.00950i
687.3 −1.40919 − 0.119147i −0.457835 − 0.457835i 1.97161 + 0.335800i 1.87486 + 1.21856i 0.590624 + 0.699723i 0 −2.73835 − 0.708115i − 2.58077i −2.49685 − 1.94056i
687.4 −1.40919 − 0.119147i 0.457835 + 0.457835i 1.97161 + 0.335800i −1.87486 − 1.21856i −0.590624 − 0.699723i 0 −2.73835 − 0.708115i − 2.58077i 2.49685 + 1.94056i
687.5 −1.35483 − 0.405491i −1.41506 − 1.41506i 1.67115 + 1.09875i 1.52518 − 1.63518i 1.34338 + 2.49096i 0 −1.81861 − 2.16626i 1.00477i −2.72942 + 1.59696i
687.6 −1.35483 − 0.405491i 1.41506 + 1.41506i 1.67115 + 1.09875i −1.52518 + 1.63518i −1.34338 − 2.49096i 0 −1.81861 − 2.16626i 1.00477i 2.72942 − 1.59696i
687.7 −1.04231 + 0.955823i −0.609124 − 0.609124i 0.172803 − 1.99252i 1.66096 − 1.49706i 1.21711 + 0.0526786i 0 1.72438 + 2.24199i − 2.25794i −0.300304 + 3.14799i
687.8 −1.04231 + 0.955823i 0.609124 + 0.609124i 0.172803 − 1.99252i −1.66096 + 1.49706i −1.21711 − 0.0526786i 0 1.72438 + 2.24199i − 2.25794i 0.300304 − 3.14799i
687.9 −0.839356 + 1.13819i −2.01474 − 2.01474i −0.590963 − 1.91070i −1.27678 − 1.83571i 3.98424 − 0.602077i 0 2.67077 + 0.931125i 5.11832i 3.16106 + 0.0875860i
687.10 −0.839356 + 1.13819i 2.01474 + 2.01474i −0.590963 − 1.91070i 1.27678 + 1.83571i −3.98424 + 0.602077i 0 2.67077 + 0.931125i 5.11832i −3.16106 − 0.0875860i
687.11 −0.405491 − 1.35483i −1.41506 − 1.41506i −1.67115 + 1.09875i −1.52518 + 1.63518i −1.34338 + 2.49096i 0 2.16626 + 1.81861i 1.00477i 2.83385 + 1.40332i
687.12 −0.405491 − 1.35483i 1.41506 + 1.41506i −1.67115 + 1.09875i 1.52518 − 1.63518i 1.34338 − 2.49096i 0 2.16626 + 1.81861i 1.00477i −2.83385 − 1.40332i
687.13 −0.119147 − 1.40919i −0.457835 − 0.457835i −1.97161 + 0.335800i −1.87486 − 1.21856i −0.590624 + 0.699723i 0 0.708115 + 2.73835i − 2.58077i −1.49379 + 2.78722i
687.14 −0.119147 − 1.40919i 0.457835 + 0.457835i −1.97161 + 0.335800i 1.87486 + 1.21856i 0.590624 − 0.699723i 0 0.708115 + 2.73835i − 2.58077i 1.49379 − 2.78722i
687.15 0.0280367 + 1.41394i −1.18903 − 1.18903i −1.99843 + 0.0792841i 1.08275 + 1.95644i 1.64787 − 1.71455i 0 −0.168132 − 2.82343i − 0.172420i −2.73592 + 1.58579i
687.16 0.0280367 + 1.41394i 1.18903 + 1.18903i −1.99843 + 0.0792841i −1.08275 − 1.95644i −1.64787 + 1.71455i 0 −0.168132 − 2.82343i − 0.172420i 2.73592 − 1.58579i
687.17 0.115993 − 1.40945i −2.30756 − 2.30756i −1.97309 − 0.326974i 2.05338 + 0.885223i −3.52006 + 2.98473i 0 −0.689718 + 2.74304i 7.64971i 1.48585 − 2.79146i
687.18 0.115993 − 1.40945i 2.30756 + 2.30756i −1.97309 − 0.326974i −2.05338 − 0.885223i 3.52006 − 2.98473i 0 −0.689718 + 2.74304i 7.64971i −1.48585 + 2.79146i
687.19 0.697605 + 1.23018i −1.76612 − 1.76612i −1.02669 + 1.71636i 1.97002 − 1.05783i 0.940592 − 3.40469i 0 −2.82766 − 0.0656787i 3.23833i 2.67562 + 1.68554i
687.20 0.697605 + 1.23018i 1.76612 + 1.76612i −1.02669 + 1.71636i −1.97002 + 1.05783i −0.940592 + 3.40469i 0 −2.82766 − 0.0656787i 3.23833i −2.67562 − 1.68554i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 687.28
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
20.e even 4 1 inner
28.d even 2 1 inner
35.f even 4 1 inner
140.j odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.k.m ✓ 56
4.b odd 2 1 inner 980.2.k.m ✓ 56
5.c odd 4 1 inner 980.2.k.m ✓ 56
7.b odd 2 1 inner 980.2.k.m ✓ 56
7.c even 3 2 980.2.x.n 112
7.d odd 6 2 980.2.x.n 112
20.e even 4 1 inner 980.2.k.m ✓ 56
28.d even 2 1 inner 980.2.k.m ✓ 56
28.f even 6 2 980.2.x.n 112
28.g odd 6 2 980.2.x.n 112
35.f even 4 1 inner 980.2.k.m ✓ 56
35.k even 12 2 980.2.x.n 112
35.l odd 12 2 980.2.x.n 112
140.j odd 4 1 inner 980.2.k.m ✓ 56
140.w even 12 2 980.2.x.n 112
140.x odd 12 2 980.2.x.n 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.k.m ✓ 56 1.a even 1 1 trivial
980.2.k.m ✓ 56 4.b odd 2 1 inner
980.2.k.m ✓ 56 5.c odd 4 1 inner
980.2.k.m ✓ 56 7.b odd 2 1 inner
980.2.k.m ✓ 56 20.e even 4 1 inner
980.2.k.m ✓ 56 28.d even 2 1 inner
980.2.k.m ✓ 56 35.f even 4 1 inner
980.2.k.m ✓ 56 140.j odd 4 1 inner
980.2.x.n 112 7.c even 3 2
980.2.x.n 112 7.d odd 6 2
980.2.x.n 112 28.f even 6 2
980.2.x.n 112 28.g odd 6 2
980.2.x.n 112 35.k even 12 2
980.2.x.n 112 35.l odd 12 2
980.2.x.n 112 140.w even 12 2
980.2.x.n 112 140.x odd 12 2

Hecke kernels

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

\( T_{3}^{28} + 243 T_{3}^{24} + 20003 T_{3}^{20} + 680681 T_{3}^{16} + 9330592 T_{3}^{12} + 43791144 T_{3}^{8} + \cdots + 3610000 \) 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