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.
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 |
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 \(S_{2}^{\mathrm{new}}(980, [\chi])\):
\( T_{3}^{36} + 252 T_{3}^{32} + 22046 T_{3}^{28} + 818612 T_{3}^{24} + 13539297 T_{3}^{20} + 105649592 T_{3}^{16} + \cdots + 65536 \) |
\( T_{13}^{18} - 2 T_{13}^{17} + 2 T_{13}^{16} - 40 T_{13}^{15} + 1510 T_{13}^{14} - 4524 T_{13}^{13} + \cdots + 1468603208 \) |