Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4020,2,Mod(841,4020)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4020, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4020.841");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4020.q (of order \(3\), 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: | \(32.0998616126\) |
Analytic rank: | \(0\) |
Dimension: | \(22\) |
Relative dimension: | \(11\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
841.1 | 0 | −1.00000 | 0 | −1.00000 | 0 | −2.18326 | + | 3.78152i | 0 | 1.00000 | 0 | ||||||||||||||||
841.2 | 0 | −1.00000 | 0 | −1.00000 | 0 | −1.77038 | + | 3.06640i | 0 | 1.00000 | 0 | ||||||||||||||||
841.3 | 0 | −1.00000 | 0 | −1.00000 | 0 | −1.49766 | + | 2.59403i | 0 | 1.00000 | 0 | ||||||||||||||||
841.4 | 0 | −1.00000 | 0 | −1.00000 | 0 | −0.775137 | + | 1.34258i | 0 | 1.00000 | 0 | ||||||||||||||||
841.5 | 0 | −1.00000 | 0 | −1.00000 | 0 | −0.533315 | + | 0.923729i | 0 | 1.00000 | 0 | ||||||||||||||||
841.6 | 0 | −1.00000 | 0 | −1.00000 | 0 | 0.278615 | − | 0.482575i | 0 | 1.00000 | 0 | ||||||||||||||||
841.7 | 0 | −1.00000 | 0 | −1.00000 | 0 | 0.479055 | − | 0.829747i | 0 | 1.00000 | 0 | ||||||||||||||||
841.8 | 0 | −1.00000 | 0 | −1.00000 | 0 | 0.637931 | − | 1.10493i | 0 | 1.00000 | 0 | ||||||||||||||||
841.9 | 0 | −1.00000 | 0 | −1.00000 | 0 | 1.82121 | − | 3.15443i | 0 | 1.00000 | 0 | ||||||||||||||||
841.10 | 0 | −1.00000 | 0 | −1.00000 | 0 | 1.92862 | − | 3.34047i | 0 | 1.00000 | 0 | ||||||||||||||||
841.11 | 0 | −1.00000 | 0 | −1.00000 | 0 | 2.11433 | − | 3.66213i | 0 | 1.00000 | 0 | ||||||||||||||||
3781.1 | 0 | −1.00000 | 0 | −1.00000 | 0 | −2.18326 | − | 3.78152i | 0 | 1.00000 | 0 | ||||||||||||||||
3781.2 | 0 | −1.00000 | 0 | −1.00000 | 0 | −1.77038 | − | 3.06640i | 0 | 1.00000 | 0 | ||||||||||||||||
3781.3 | 0 | −1.00000 | 0 | −1.00000 | 0 | −1.49766 | − | 2.59403i | 0 | 1.00000 | 0 | ||||||||||||||||
3781.4 | 0 | −1.00000 | 0 | −1.00000 | 0 | −0.775137 | − | 1.34258i | 0 | 1.00000 | 0 | ||||||||||||||||
3781.5 | 0 | −1.00000 | 0 | −1.00000 | 0 | −0.533315 | − | 0.923729i | 0 | 1.00000 | 0 | ||||||||||||||||
3781.6 | 0 | −1.00000 | 0 | −1.00000 | 0 | 0.278615 | + | 0.482575i | 0 | 1.00000 | 0 | ||||||||||||||||
3781.7 | 0 | −1.00000 | 0 | −1.00000 | 0 | 0.479055 | + | 0.829747i | 0 | 1.00000 | 0 | ||||||||||||||||
3781.8 | 0 | −1.00000 | 0 | −1.00000 | 0 | 0.637931 | + | 1.10493i | 0 | 1.00000 | 0 | ||||||||||||||||
3781.9 | 0 | −1.00000 | 0 | −1.00000 | 0 | 1.82121 | + | 3.15443i | 0 | 1.00000 | 0 | ||||||||||||||||
See all 22 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
67.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 4020.2.q.l | ✓ | 22 |
67.c | even | 3 | 1 | inner | 4020.2.q.l | ✓ | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4020.2.q.l | ✓ | 22 | 1.a | even | 1 | 1 | trivial |
4020.2.q.l | ✓ | 22 | 67.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4020, [\chi])\):
\( T_{7}^{22} - T_{7}^{21} + 47 T_{7}^{20} - 30 T_{7}^{19} + 1420 T_{7}^{18} - 729 T_{7}^{17} + \cdots + 9603801 \) |
\( T_{11}^{22} + 6 T_{11}^{21} + 74 T_{11}^{20} + 406 T_{11}^{19} + 3337 T_{11}^{18} + 16360 T_{11}^{17} + \cdots + 20729809 \) |
\( T_{17}^{22} - 4 T_{17}^{21} + 120 T_{17}^{20} - 476 T_{17}^{19} + 9450 T_{17}^{18} - 38405 T_{17}^{17} + \cdots + 6233260401 \) |