Properties

Label 980.2.k.l
Level $980$
Weight $2$
Character orbit 980.k
Analytic conductor $7.825$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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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: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 140)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 8 q^{6} + 16 q^{10} + 16 q^{12} + 4 q^{13} - 8 q^{16} + 20 q^{17} + 28 q^{18} - 20 q^{20} + 4 q^{22} - 20 q^{25} + 32 q^{26} - 4 q^{30} + 20 q^{37} + 36 q^{40} - 20 q^{45} + 16 q^{46} + 24 q^{48} + 40 q^{50} - 16 q^{52} - 44 q^{53} - 16 q^{57} - 4 q^{58} + 40 q^{60} + 64 q^{61} - 40 q^{62} + 4 q^{65} - 32 q^{66} - 80 q^{68} + 80 q^{72} - 52 q^{73} - 8 q^{76} - 76 q^{78} + 20 q^{80} - 36 q^{81} - 56 q^{82} - 20 q^{85} + 56 q^{86} - 40 q^{88} + 16 q^{90} - 56 q^{92} + 32 q^{93} - 120 q^{96} - 20 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
687.1 −1.38250 + 0.297828i 0.137886 + 0.137886i 1.82260 0.823494i 0.499053 + 2.17967i −0.231693 0.149560i 0 −2.27447 + 1.68130i 2.96198i −1.33911 2.86475i
687.2 −1.36728 0.361308i 1.26588 + 1.26588i 1.73891 + 0.988019i −1.96164 1.07330i −1.27344 2.18818i 0 −2.02060 1.97918i 0.204893i 2.29431 + 2.17626i
687.3 −1.25629 0.649412i −2.28163 2.28163i 1.15653 + 1.63170i −0.0430826 + 2.23565i 1.38467 + 4.34811i 0 −0.393286 2.80095i 7.41170i 1.50599 2.78065i
687.4 −1.16481 + 0.802007i 1.75731 + 1.75731i 0.713568 1.86837i 0.854664 2.06629i −3.45630 0.637557i 0 0.667278 + 2.74859i 3.17626i 0.661657 + 3.09228i
687.5 −1.08834 + 0.903055i −1.00798 1.00798i 0.368985 1.96567i −2.21855 0.279354i 2.00729 + 0.186768i 0 1.37352 + 2.47254i 0.967954i 2.66682 1.69944i
687.6 −0.649412 1.25629i 2.28163 + 2.28163i −1.15653 + 1.63170i −0.0430826 + 2.23565i 1.38467 4.34811i 0 2.80095 + 0.393286i 7.41170i 2.83661 1.39774i
687.7 −0.622342 + 1.26992i −2.09607 2.09607i −1.22538 1.58065i 2.14272 0.639344i 3.96631 1.35737i 0 2.76990 0.572433i 5.78704i −0.521588 + 3.11897i
687.8 −0.396294 + 1.35755i 0.945787 + 0.945787i −1.68590 1.07598i 1.94751 + 1.09873i −1.65877 + 0.909146i 0 2.12881 1.86230i 1.21097i −2.26337 + 2.20843i
687.9 −0.361308 1.36728i −1.26588 1.26588i −1.73891 + 0.988019i −1.96164 1.07330i −1.27344 + 2.18818i 0 1.97918 + 2.02060i 0.204893i −0.758752 + 3.06990i
687.10 0.297828 1.38250i −0.137886 0.137886i −1.82260 0.823494i 0.499053 + 2.17967i −0.231693 + 0.149560i 0 −1.68130 + 2.27447i 2.96198i 3.16201 0.0407730i
687.11 0.447909 + 1.34141i −0.396892 0.396892i −1.59875 + 1.20166i 0.137858 2.23181i 0.354623 0.710167i 0 −2.32801 1.60635i 2.68495i 3.05552 0.814726i
687.12 0.578354 + 1.29055i −1.27396 1.27396i −1.33101 + 1.49278i −1.35854 + 1.77606i 0.907305 2.38090i 0 −2.69630 0.854378i 0.245954i −3.07780 0.726061i
687.13 0.802007 1.16481i −1.75731 1.75731i −0.713568 1.86837i 0.854664 2.06629i −3.45630 + 0.637557i 0 −2.74859 0.667278i 3.17626i −1.72139 2.65270i
687.14 0.903055 1.08834i 1.00798 + 1.00798i −0.368985 1.96567i −2.21855 0.279354i 2.00729 0.186768i 0 −2.47254 1.37352i 0.967954i −2.30750 + 2.16227i
687.15 1.26992 0.622342i 2.09607 + 2.09607i 1.22538 1.58065i 2.14272 0.639344i 3.96631 + 1.35737i 0 0.572433 2.76990i 5.78704i 2.32318 2.14542i
687.16 1.29055 + 0.578354i 1.27396 + 1.27396i 1.33101 + 1.49278i −1.35854 + 1.77606i 0.907305 + 2.38090i 0 0.854378 + 2.69630i 0.245954i −2.78044 + 1.50637i
687.17 1.34141 + 0.447909i 0.396892 + 0.396892i 1.59875 + 1.20166i 0.137858 2.23181i 0.354623 + 0.710167i 0 1.60635 + 2.32801i 2.68495i 1.18457 2.93203i
687.18 1.35755 0.396294i −0.945787 0.945787i 1.68590 1.07598i 1.94751 + 1.09873i −1.65877 0.909146i 0 1.86230 2.12881i 1.21097i 3.07927 + 0.719794i
883.1 −1.38250 0.297828i 0.137886 0.137886i 1.82260 + 0.823494i 0.499053 2.17967i −0.231693 + 0.149560i 0 −2.27447 1.68130i 2.96198i −1.33911 + 2.86475i
883.2 −1.36728 + 0.361308i 1.26588 1.26588i 1.73891 0.988019i −1.96164 + 1.07330i −1.27344 + 2.18818i 0 −2.02060 + 1.97918i 0.204893i 2.29431 2.17626i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 687.18
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
20.e even 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.l 36
4.b odd 2 1 inner 980.2.k.l 36
5.c odd 4 1 inner 980.2.k.l 36
7.b odd 2 1 140.2.k.a 36
7.c even 3 2 980.2.x.l 72
7.d odd 6 2 980.2.x.k 72
20.e even 4 1 inner 980.2.k.l 36
28.d even 2 1 140.2.k.a 36
28.f even 6 2 980.2.x.k 72
28.g odd 6 2 980.2.x.l 72
35.c odd 2 1 700.2.k.b 36
35.f even 4 1 140.2.k.a 36
35.f even 4 1 700.2.k.b 36
35.k even 12 2 980.2.x.k 72
35.l odd 12 2 980.2.x.l 72
140.c even 2 1 700.2.k.b 36
140.j odd 4 1 140.2.k.a 36
140.j odd 4 1 700.2.k.b 36
140.w even 12 2 980.2.x.l 72
140.x odd 12 2 980.2.x.k 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.k.a 36 7.b odd 2 1
140.2.k.a 36 28.d even 2 1
140.2.k.a 36 35.f even 4 1
140.2.k.a 36 140.j odd 4 1
700.2.k.b 36 35.c odd 2 1
700.2.k.b 36 35.f even 4 1
700.2.k.b 36 140.c even 2 1
700.2.k.b 36 140.j odd 4 1
980.2.k.l 36 1.a even 1 1 trivial
980.2.k.l 36 4.b odd 2 1 inner
980.2.k.l 36 5.c odd 4 1 inner
980.2.k.l 36 20.e even 4 1 inner
980.2.x.k 72 7.d odd 6 2
980.2.x.k 72 28.f even 6 2
980.2.x.k 72 35.k even 12 2
980.2.x.k 72 140.x odd 12 2
980.2.x.l 72 7.c even 3 2
980.2.x.l 72 28.g odd 6 2
980.2.x.l 72 35.l odd 12 2
980.2.x.l 72 140.w even 12 2

Hecke kernels

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

\( T_{3}^{36} + 252 T_{3}^{32} + 22046 T_{3}^{28} + 818612 T_{3}^{24} + 13539297 T_{3}^{20} + 105649592 T_{3}^{16} + 373675512 T_{3}^{12} + 493246368 T_{3}^{8} + 46038032 T_{3}^{4} + 65536 \) Copy content Toggle raw display
\( T_{13}^{18} - 2 T_{13}^{17} + 2 T_{13}^{16} - 40 T_{13}^{15} + 1510 T_{13}^{14} - 4524 T_{13}^{13} + 6828 T_{13}^{12} - 10528 T_{13}^{11} + 524869 T_{13}^{10} - 1691322 T_{13}^{9} + 2677034 T_{13}^{8} + 1157496 T_{13}^{7} + \cdots + 1468603208 \) Copy content Toggle raw display