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.
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 |
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 \(S_{2}^{\mathrm{new}}(980, [\chi])\):
\( T_{3}^{72} - 252 T_{3}^{68} + 41458 T_{3}^{64} - 3918368 T_{3}^{60} + 266196595 T_{3}^{56} + \cdots + 4294967296 \) |
\( T_{11}^{36} - 100 T_{11}^{34} + 6018 T_{11}^{32} - 236528 T_{11}^{30} + 6882011 T_{11}^{28} + \cdots + 17179869184 \) |
\( T_{13}^{18} + 2 T_{13}^{17} + 2 T_{13}^{16} + 40 T_{13}^{15} + 1510 T_{13}^{14} + 4524 T_{13}^{13} + \cdots + 1468603208 \) |