Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5040,2,Mod(1889,5040)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5040, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5040.1889");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 5040.k (of order \(2\), degree \(1\), 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: | \(40.2446026187\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | no (minimal twist has level 2520) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1889.1 | 0 | 0 | 0 | −2.22844 | − | 0.184526i | 0 | −0.741091 | + | 2.53984i | 0 | 0 | 0 | ||||||||||||||
1889.2 | 0 | 0 | 0 | −2.22844 | + | 0.184526i | 0 | −0.741091 | − | 2.53984i | 0 | 0 | 0 | ||||||||||||||
1889.3 | 0 | 0 | 0 | −1.99614 | − | 1.00769i | 0 | 2.58943 | + | 0.542983i | 0 | 0 | 0 | ||||||||||||||
1889.4 | 0 | 0 | 0 | −1.99614 | + | 1.00769i | 0 | 2.58943 | − | 0.542983i | 0 | 0 | 0 | ||||||||||||||
1889.5 | 0 | 0 | 0 | −1.52549 | − | 1.63489i | 0 | 0.114732 | − | 2.64326i | 0 | 0 | 0 | ||||||||||||||
1889.6 | 0 | 0 | 0 | −1.52549 | + | 1.63489i | 0 | 0.114732 | + | 2.64326i | 0 | 0 | 0 | ||||||||||||||
1889.7 | 0 | 0 | 0 | −0.873162 | − | 2.05854i | 0 | −2.52372 | − | 0.794245i | 0 | 0 | 0 | ||||||||||||||
1889.8 | 0 | 0 | 0 | −0.873162 | + | 2.05854i | 0 | −2.52372 | + | 0.794245i | 0 | 0 | 0 | ||||||||||||||
1889.9 | 0 | 0 | 0 | −0.727958 | − | 2.11426i | 0 | 1.21521 | + | 2.35016i | 0 | 0 | 0 | ||||||||||||||
1889.10 | 0 | 0 | 0 | −0.727958 | + | 2.11426i | 0 | 1.21521 | − | 2.35016i | 0 | 0 | 0 | ||||||||||||||
1889.11 | 0 | 0 | 0 | −0.655755 | − | 2.13775i | 0 | −2.09441 | + | 1.61662i | 0 | 0 | 0 | ||||||||||||||
1889.12 | 0 | 0 | 0 | −0.655755 | + | 2.13775i | 0 | −2.09441 | − | 1.61662i | 0 | 0 | 0 | ||||||||||||||
1889.13 | 0 | 0 | 0 | 0.655755 | − | 2.13775i | 0 | 2.09441 | − | 1.61662i | 0 | 0 | 0 | ||||||||||||||
1889.14 | 0 | 0 | 0 | 0.655755 | + | 2.13775i | 0 | 2.09441 | + | 1.61662i | 0 | 0 | 0 | ||||||||||||||
1889.15 | 0 | 0 | 0 | 0.727958 | − | 2.11426i | 0 | −1.21521 | − | 2.35016i | 0 | 0 | 0 | ||||||||||||||
1889.16 | 0 | 0 | 0 | 0.727958 | + | 2.11426i | 0 | −1.21521 | + | 2.35016i | 0 | 0 | 0 | ||||||||||||||
1889.17 | 0 | 0 | 0 | 0.873162 | − | 2.05854i | 0 | 2.52372 | + | 0.794245i | 0 | 0 | 0 | ||||||||||||||
1889.18 | 0 | 0 | 0 | 0.873162 | + | 2.05854i | 0 | 2.52372 | − | 0.794245i | 0 | 0 | 0 | ||||||||||||||
1889.19 | 0 | 0 | 0 | 1.52549 | − | 1.63489i | 0 | −0.114732 | + | 2.64326i | 0 | 0 | 0 | ||||||||||||||
1889.20 | 0 | 0 | 0 | 1.52549 | + | 1.63489i | 0 | −0.114732 | − | 2.64326i | 0 | 0 | 0 | ||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
105.g | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 5040.2.k.i | 24 | |
3.b | odd | 2 | 1 | 5040.2.k.h | 24 | ||
4.b | odd | 2 | 1 | 2520.2.k.a | ✓ | 24 | |
5.b | even | 2 | 1 | 5040.2.k.h | 24 | ||
7.b | odd | 2 | 1 | inner | 5040.2.k.i | 24 | |
12.b | even | 2 | 1 | 2520.2.k.b | yes | 24 | |
15.d | odd | 2 | 1 | inner | 5040.2.k.i | 24 | |
20.d | odd | 2 | 1 | 2520.2.k.b | yes | 24 | |
21.c | even | 2 | 1 | 5040.2.k.h | 24 | ||
28.d | even | 2 | 1 | 2520.2.k.a | ✓ | 24 | |
35.c | odd | 2 | 1 | 5040.2.k.h | 24 | ||
60.h | even | 2 | 1 | 2520.2.k.a | ✓ | 24 | |
84.h | odd | 2 | 1 | 2520.2.k.b | yes | 24 | |
105.g | even | 2 | 1 | inner | 5040.2.k.i | 24 | |
140.c | even | 2 | 1 | 2520.2.k.b | yes | 24 | |
420.o | odd | 2 | 1 | 2520.2.k.a | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2520.2.k.a | ✓ | 24 | 4.b | odd | 2 | 1 | |
2520.2.k.a | ✓ | 24 | 28.d | even | 2 | 1 | |
2520.2.k.a | ✓ | 24 | 60.h | even | 2 | 1 | |
2520.2.k.a | ✓ | 24 | 420.o | odd | 2 | 1 | |
2520.2.k.b | yes | 24 | 12.b | even | 2 | 1 | |
2520.2.k.b | yes | 24 | 20.d | odd | 2 | 1 | |
2520.2.k.b | yes | 24 | 84.h | odd | 2 | 1 | |
2520.2.k.b | yes | 24 | 140.c | even | 2 | 1 | |
5040.2.k.h | 24 | 3.b | odd | 2 | 1 | ||
5040.2.k.h | 24 | 5.b | even | 2 | 1 | ||
5040.2.k.h | 24 | 21.c | even | 2 | 1 | ||
5040.2.k.h | 24 | 35.c | odd | 2 | 1 | ||
5040.2.k.i | 24 | 1.a | even | 1 | 1 | trivial | |
5040.2.k.i | 24 | 7.b | odd | 2 | 1 | inner | |
5040.2.k.i | 24 | 15.d | odd | 2 | 1 | inner | |
5040.2.k.i | 24 | 105.g | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5040, [\chi])\):
\( T_{11}^{12} + 74T_{11}^{10} + 1832T_{11}^{8} + 20176T_{11}^{6} + 103248T_{11}^{4} + 217632T_{11}^{2} + 123904 \) |
\( T_{13}^{12} - 94T_{13}^{10} + 2680T_{13}^{8} - 21280T_{13}^{6} + 20160T_{13}^{4} - 6016T_{13}^{2} + 512 \) |
\( T_{23}^{6} - 2T_{23}^{5} - 68T_{23}^{4} + 272T_{23}^{3} + 252T_{23}^{2} - 1736T_{23} + 1312 \) |