Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [980,2,Mod(117,980)]
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([0, 3, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("980.117");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 980.v (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: | \(48\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
117.1 | 0 | −0.769551 | + | 2.87200i | 0 | −2.17592 | + | 0.515161i | 0 | 0 | 0 | −5.05812 | − | 2.92031i | 0 | ||||||||||||
117.2 | 0 | −0.595923 | + | 2.22401i | 0 | −1.87454 | + | 1.21905i | 0 | 0 | 0 | −1.99304 | − | 1.15068i | 0 | ||||||||||||
117.3 | 0 | −0.521820 | + | 1.94746i | 0 | 0.430599 | − | 2.19422i | 0 | 0 | 0 | −0.922223 | − | 0.532446i | 0 | ||||||||||||
117.4 | 0 | −0.238078 | + | 0.888521i | 0 | −0.258173 | + | 2.22111i | 0 | 0 | 0 | 1.86529 | + | 1.07692i | 0 | ||||||||||||
117.5 | 0 | −0.206041 | + | 0.768955i | 0 | 1.99002 | − | 1.01971i | 0 | 0 | 0 | 2.04924 | + | 1.18313i | 0 | ||||||||||||
117.6 | 0 | −0.144854 | + | 0.540603i | 0 | 1.50707 | + | 1.65189i | 0 | 0 | 0 | 2.32681 | + | 1.34338i | 0 | ||||||||||||
117.7 | 0 | 0.144854 | − | 0.540603i | 0 | −1.50707 | − | 1.65189i | 0 | 0 | 0 | 2.32681 | + | 1.34338i | 0 | ||||||||||||
117.8 | 0 | 0.206041 | − | 0.768955i | 0 | −1.99002 | + | 1.01971i | 0 | 0 | 0 | 2.04924 | + | 1.18313i | 0 | ||||||||||||
117.9 | 0 | 0.238078 | − | 0.888521i | 0 | 0.258173 | − | 2.22111i | 0 | 0 | 0 | 1.86529 | + | 1.07692i | 0 | ||||||||||||
117.10 | 0 | 0.521820 | − | 1.94746i | 0 | −0.430599 | + | 2.19422i | 0 | 0 | 0 | −0.922223 | − | 0.532446i | 0 | ||||||||||||
117.11 | 0 | 0.595923 | − | 2.22401i | 0 | 1.87454 | − | 1.21905i | 0 | 0 | 0 | −1.99304 | − | 1.15068i | 0 | ||||||||||||
117.12 | 0 | 0.769551 | − | 2.87200i | 0 | 2.17592 | − | 0.515161i | 0 | 0 | 0 | −5.05812 | − | 2.92031i | 0 | ||||||||||||
313.1 | 0 | −2.87200 | − | 0.769551i | 0 | −0.641815 | − | 2.14198i | 0 | 0 | 0 | 5.05812 | + | 2.92031i | 0 | ||||||||||||
313.2 | 0 | −2.22401 | − | 0.595923i | 0 | 0.118459 | − | 2.23293i | 0 | 0 | 0 | 1.99304 | + | 1.15068i | 0 | ||||||||||||
313.3 | 0 | −1.94746 | − | 0.521820i | 0 | −1.68495 | + | 1.47002i | 0 | 0 | 0 | 0.922223 | + | 0.532446i | 0 | ||||||||||||
313.4 | 0 | −0.888521 | − | 0.238078i | 0 | 1.79445 | − | 1.33414i | 0 | 0 | 0 | −1.86529 | − | 1.07692i | 0 | ||||||||||||
313.5 | 0 | −0.768955 | − | 0.206041i | 0 | 0.111918 | + | 2.23327i | 0 | 0 | 0 | −2.04924 | − | 1.18313i | 0 | ||||||||||||
313.6 | 0 | −0.540603 | − | 0.144854i | 0 | 2.18411 | + | 0.479220i | 0 | 0 | 0 | −2.32681 | − | 1.34338i | 0 | ||||||||||||
313.7 | 0 | 0.540603 | + | 0.144854i | 0 | −2.18411 | − | 0.479220i | 0 | 0 | 0 | −2.32681 | − | 1.34338i | 0 | ||||||||||||
313.8 | 0 | 0.768955 | + | 0.206041i | 0 | −0.111918 | − | 2.23327i | 0 | 0 | 0 | −2.04924 | − | 1.18313i | 0 | ||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
7.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
7.d | odd | 6 | 1 | inner |
35.f | even | 4 | 1 | inner |
35.k | even | 12 | 1 | inner |
35.l | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 980.2.v.c | 48 | |
5.c | odd | 4 | 1 | inner | 980.2.v.c | 48 | |
7.b | odd | 2 | 1 | inner | 980.2.v.c | 48 | |
7.c | even | 3 | 1 | 980.2.m.b | ✓ | 24 | |
7.c | even | 3 | 1 | inner | 980.2.v.c | 48 | |
7.d | odd | 6 | 1 | 980.2.m.b | ✓ | 24 | |
7.d | odd | 6 | 1 | inner | 980.2.v.c | 48 | |
35.f | even | 4 | 1 | inner | 980.2.v.c | 48 | |
35.k | even | 12 | 1 | 980.2.m.b | ✓ | 24 | |
35.k | even | 12 | 1 | inner | 980.2.v.c | 48 | |
35.l | odd | 12 | 1 | 980.2.m.b | ✓ | 24 | |
35.l | odd | 12 | 1 | inner | 980.2.v.c | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
980.2.m.b | ✓ | 24 | 7.c | even | 3 | 1 | |
980.2.m.b | ✓ | 24 | 7.d | odd | 6 | 1 | |
980.2.m.b | ✓ | 24 | 35.k | even | 12 | 1 | |
980.2.m.b | ✓ | 24 | 35.l | odd | 12 | 1 | |
980.2.v.c | 48 | 1.a | even | 1 | 1 | trivial | |
980.2.v.c | 48 | 5.c | odd | 4 | 1 | inner | |
980.2.v.c | 48 | 7.b | odd | 2 | 1 | inner | |
980.2.v.c | 48 | 7.c | even | 3 | 1 | inner | |
980.2.v.c | 48 | 7.d | odd | 6 | 1 | inner | |
980.2.v.c | 48 | 35.f | even | 4 | 1 | inner | |
980.2.v.c | 48 | 35.k | even | 12 | 1 | inner | |
980.2.v.c | 48 | 35.l | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 124 T_{3}^{44} + 11274 T_{3}^{40} - 426352 T_{3}^{36} + 11678355 T_{3}^{32} + \cdots + 1048576 \) acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\).