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Properties

Label 980.2.x.k
Level $980$
Weight $2$
Character orbit 980.x
Analytic conductor $7.825$
Analytic rank $0$
Dimension $72$
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(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: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 140)
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 16 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 16 q^{6} + 16 q^{10} + 16 q^{12} - 8 q^{13} + 8 q^{16} + 20 q^{17} - 28 q^{18} + 40 q^{20} + 8 q^{22} + 20 q^{25} + 32 q^{26} + 4 q^{30} - 20 q^{37} + 36 q^{40} - 20 q^{45} - 16 q^{46} - 48 q^{48} + 80 q^{50} - 16 q^{52} + 44 q^{53} - 32 q^{57} + 4 q^{58} - 40 q^{60} + 64 q^{61} + 80 q^{62} - 4 q^{65} - 32 q^{66} - 80 q^{68} - 80 q^{72} - 52 q^{73} + 16 q^{76} - 152 q^{78} + 20 q^{80} + 36 q^{81} - 56 q^{82} - 40 q^{85} - 56 q^{86} + 40 q^{88} - 32 q^{90} - 112 q^{92} - 32 q^{93} - 120 q^{96} + 40 q^{97}+O(q^{100}) \) 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 −1.40976 − 0.112153i −2.40053 + 0.643219i 1.97484 + 0.316219i 2.21679 − 0.292984i 3.45630 − 0.637557i 0 −2.74859 − 0.667278i 2.75072 − 1.58813i −3.15800 + 0.164416i
67.2 −1.39406 − 0.237896i 1.37693 − 0.368946i 1.88681 + 0.663284i −0.867347 − 2.06100i −2.00729 + 0.186768i 0 −2.47254 − 1.37352i −0.838273 + 0.483977i 0.718831 + 3.07949i
67.3 −1.34619 + 0.433322i −0.188355 + 0.0504696i 1.62446 − 1.16667i −1.63812 + 1.52203i 0.231693 − 0.149560i 0 −1.68130 + 2.27447i −2.56515 + 1.48099i 1.54570 − 2.75877i
67.4 −1.17392 − 0.788610i 2.86329 − 0.767216i 0.756188 + 1.85153i 1.62505 + 1.53598i −3.96631 − 1.35737i 0 0.572433 − 2.76990i 5.01173 − 2.89352i −0.696393 − 3.08465i
67.5 −1.02198 − 0.977529i −1.29197 + 0.346182i 0.0888759 + 1.99802i 0.0222296 + 2.23596i 1.65877 + 0.909146i 0 1.86230 − 2.12881i −1.04873 + 0.605487i 2.16299 − 2.30683i
67.6 −1.00345 + 0.996542i −1.72922 + 0.463343i 0.0138066 − 1.99995i −0.0513098 − 2.23548i 1.27344 − 2.18818i 0 1.97918 + 2.02060i 0.177443 − 0.102447i 2.27924 + 2.19205i
67.7 −0.763272 + 1.19055i 3.11677 − 0.835136i −0.834830 − 1.81743i −1.95767 + 1.08052i −1.38467 + 4.34811i 0 2.80095 + 0.393286i 6.41872 − 3.70585i 0.207827 − 3.15544i
67.8 −0.282803 − 1.38565i 0.542165 − 0.145273i −1.84004 + 0.783732i 2.00174 − 0.996518i −0.354623 − 0.710167i 0 1.60635 + 2.32801i −2.32524 + 1.34248i −1.94692 − 2.49189i
67.9 −0.144404 − 1.40682i 1.74026 − 0.466302i −1.95830 + 0.406301i −2.21738 − 0.288497i −0.907305 − 2.38090i 0 0.854378 + 2.69630i 0.213002 − 0.122977i −0.0856655 + 3.16112i
67.10 0.0657371 + 1.41268i −3.11677 + 0.835136i −1.99136 + 0.185732i −1.95767 + 1.08052i −1.38467 − 4.34811i 0 −0.393286 − 2.80095i 6.41872 − 3.70585i −1.65512 − 2.69455i
67.11 0.370738 + 1.36475i 1.72922 − 0.463343i −1.72511 + 1.01193i −0.0513098 − 2.23548i 1.27344 + 2.18818i 0 −2.02060 − 1.97918i 0.177443 − 0.102447i 3.03186 − 0.898803i
67.12 0.828468 − 1.14614i −1.74026 + 0.466302i −0.627281 − 1.89908i −2.21738 − 0.288497i −0.907305 + 2.38090i 0 −2.69630 − 0.854378i 0.213002 − 0.122977i −2.16769 + 2.30242i
67.13 0.937739 − 1.05861i −0.542165 + 0.145273i −0.241290 − 1.98539i 2.00174 − 0.996518i −0.354623 + 0.710167i 0 −2.32801 − 1.60635i −2.32524 + 1.34248i 0.822188 − 3.05352i
67.14 0.949176 + 1.04836i 0.188355 − 0.0504696i −0.198132 + 1.99016i −1.63812 + 1.52203i 0.231693 + 0.149560i 0 −2.27447 + 1.68130i −2.56515 + 1.48099i −3.15050 − 0.272675i
67.15 1.27696 + 0.607752i 2.40053 − 0.643219i 1.26127 + 1.55216i 2.21679 − 0.292984i 3.45630 + 0.637557i 0 0.667278 + 2.74859i 2.75072 − 1.58813i 3.00882 + 0.973129i
67.16 1.32624 + 0.491006i −1.37693 + 0.368946i 1.51783 + 1.30238i −0.867347 − 2.06100i −2.00729 − 0.186768i 0 1.37352 + 2.47254i −0.838273 + 0.483977i −0.138349 − 3.15925i
67.17 1.37382 − 0.335576i 1.29197 − 0.346182i 1.77478 − 0.922043i 0.0222296 + 2.23596i 1.65877 − 0.909146i 0 2.12881 − 1.86230i −1.04873 + 0.605487i 0.780873 + 3.06435i
67.18 1.41095 − 0.0959952i −2.86329 + 0.767216i 1.98157 − 0.270889i 1.62505 + 1.53598i −3.96631 + 1.35737i 0 2.76990 − 0.572433i 5.01173 − 2.89352i 2.44031 + 2.01119i
263.1 −1.40682 + 0.144404i 0.466302 + 1.74026i 1.95830 − 0.406301i 0.858844 − 2.06455i −0.907305 − 2.38090i 0 −2.69630 + 0.854378i −0.213002 + 0.122977i −0.910111 + 3.02848i
263.2 −1.38565 + 0.282803i 0.145273 + 0.542165i 1.84004 − 0.783732i −1.86388 + 1.23530i −0.354623 − 0.710167i 0 −2.32801 + 1.60635i 2.32524 − 1.34248i 2.23334 − 2.23880i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.18
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.c even 3 1 inner
20.e even 4 1 inner
28.g odd 6 1 inner
35.l odd 12 1 inner
140.w even 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.k 72
4.b odd 2 1 inner 980.2.x.k 72
5.c odd 4 1 inner 980.2.x.k 72
7.b odd 2 1 980.2.x.l 72
7.c even 3 1 140.2.k.a ✓ 36
7.c even 3 1 inner 980.2.x.k 72
7.d odd 6 1 980.2.k.l 36
7.d odd 6 1 980.2.x.l 72
20.e even 4 1 inner 980.2.x.k 72
28.d even 2 1 980.2.x.l 72
28.f even 6 1 980.2.k.l 36
28.f even 6 1 980.2.x.l 72
28.g odd 6 1 140.2.k.a ✓ 36
28.g odd 6 1 inner 980.2.x.k 72
35.f even 4 1 980.2.x.l 72
35.j even 6 1 700.2.k.b 36
35.k even 12 1 980.2.k.l 36
35.k even 12 1 980.2.x.l 72
35.l odd 12 1 140.2.k.a ✓ 36
35.l odd 12 1 700.2.k.b 36
35.l odd 12 1 inner 980.2.x.k 72
140.j odd 4 1 980.2.x.l 72
140.p odd 6 1 700.2.k.b 36
140.w even 12 1 140.2.k.a ✓ 36
140.w even 12 1 700.2.k.b 36
140.w even 12 1 inner 980.2.x.k 72
140.x odd 12 1 980.2.k.l 36
140.x odd 12 1 980.2.x.l 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.k.a ✓ 36 7.c even 3 1
140.2.k.a ✓ 36 28.g odd 6 1
140.2.k.a ✓ 36 35.l odd 12 1
140.2.k.a ✓ 36 140.w even 12 1
700.2.k.b 36 35.j even 6 1
700.2.k.b 36 35.l odd 12 1
700.2.k.b 36 140.p odd 6 1
700.2.k.b 36 140.w even 12 1
980.2.k.l 36 7.d odd 6 1
980.2.k.l 36 28.f even 6 1
980.2.k.l 36 35.k even 12 1
980.2.k.l 36 140.x odd 12 1
980.2.x.k 72 1.a even 1 1 trivial
980.2.x.k 72 4.b odd 2 1 inner
980.2.x.k 72 5.c odd 4 1 inner
980.2.x.k 72 7.c even 3 1 inner
980.2.x.k 72 20.e even 4 1 inner
980.2.x.k 72 28.g odd 6 1 inner
980.2.x.k 72 35.l odd 12 1 inner
980.2.x.k 72 140.w even 12 1 inner
980.2.x.l 72 7.b odd 2 1
980.2.x.l 72 7.d odd 6 1
980.2.x.l 72 28.d even 2 1
980.2.x.l 72 28.f even 6 1
980.2.x.l 72 35.f even 4 1
980.2.x.l 72 35.k even 12 1
980.2.x.l 72 140.j odd 4 1
980.2.x.l 72 140.x odd 12 1

Hecke kernels

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

\( T_{3}^{72} - 252 T_{3}^{68} + 41458 T_{3}^{64} - 3918368 T_{3}^{60} + 266196595 T_{3}^{56} + \cdots + 4294967296 \) Copy content Toggle raw display
\( T_{11}^{36} - 100 T_{11}^{34} + 6018 T_{11}^{32} - 236528 T_{11}^{30} + 6882011 T_{11}^{28} + \cdots + 17179869184 \) Copy content Toggle raw display
\( T_{13}^{18} + 2 T_{13}^{17} + 2 T_{13}^{16} + 40 T_{13}^{15} + 1510 T_{13}^{14} + 4524 T_{13}^{13} + \cdots + 1468603208 \) Copy content Toggle raw display