Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5780,2,Mod(5201,5780)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5780, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5780.5201");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 5780 = 2^{2} \cdot 5 \cdot 17^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 5780.c (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: | \(46.1535323683\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5201.1 | 0 | − | 3.40432i | 0 | − | 1.00000i | 0 | 3.34135i | 0 | −8.58938 | 0 | ||||||||||||||||
5201.2 | 0 | − | 3.00548i | 0 | 1.00000i | 0 | 0.0298224i | 0 | −6.03290 | 0 | |||||||||||||||||
5201.3 | 0 | − | 2.93491i | 0 | − | 1.00000i | 0 | 3.61113i | 0 | −5.61367 | 0 | ||||||||||||||||
5201.4 | 0 | − | 2.89759i | 0 | − | 1.00000i | 0 | − | 2.09483i | 0 | −5.39604 | 0 | |||||||||||||||
5201.5 | 0 | − | 2.69458i | 0 | 1.00000i | 0 | 0.697790i | 0 | −4.26077 | 0 | |||||||||||||||||
5201.6 | 0 | − | 2.55110i | 0 | 1.00000i | 0 | 5.11777i | 0 | −3.50814 | 0 | |||||||||||||||||
5201.7 | 0 | − | 1.40448i | 0 | − | 1.00000i | 0 | − | 4.82771i | 0 | 1.02744 | 0 | |||||||||||||||
5201.8 | 0 | − | 1.39405i | 0 | − | 1.00000i | 0 | − | 3.99249i | 0 | 1.05662 | 0 | |||||||||||||||
5201.9 | 0 | − | 1.11421i | 0 | 1.00000i | 0 | − | 1.48955i | 0 | 1.75853 | 0 | ||||||||||||||||
5201.10 | 0 | − | 0.575262i | 0 | − | 1.00000i | 0 | 4.22697i | 0 | 2.66907 | 0 | ||||||||||||||||
5201.11 | 0 | − | 0.323479i | 0 | 1.00000i | 0 | − | 4.16995i | 0 | 2.89536 | 0 | ||||||||||||||||
5201.12 | 0 | − | 0.0782500i | 0 | − | 1.00000i | 0 | − | 0.0785455i | 0 | 2.99388 | 0 | |||||||||||||||
5201.13 | 0 | 0.0782500i | 0 | 1.00000i | 0 | 0.0785455i | 0 | 2.99388 | 0 | ||||||||||||||||||
5201.14 | 0 | 0.323479i | 0 | − | 1.00000i | 0 | 4.16995i | 0 | 2.89536 | 0 | |||||||||||||||||
5201.15 | 0 | 0.575262i | 0 | 1.00000i | 0 | − | 4.22697i | 0 | 2.66907 | 0 | |||||||||||||||||
5201.16 | 0 | 1.11421i | 0 | − | 1.00000i | 0 | 1.48955i | 0 | 1.75853 | 0 | |||||||||||||||||
5201.17 | 0 | 1.39405i | 0 | 1.00000i | 0 | 3.99249i | 0 | 1.05662 | 0 | ||||||||||||||||||
5201.18 | 0 | 1.40448i | 0 | 1.00000i | 0 | 4.82771i | 0 | 1.02744 | 0 | ||||||||||||||||||
5201.19 | 0 | 2.55110i | 0 | − | 1.00000i | 0 | − | 5.11777i | 0 | −3.50814 | 0 | ||||||||||||||||
5201.20 | 0 | 2.69458i | 0 | − | 1.00000i | 0 | − | 0.697790i | 0 | −4.26077 | 0 | ||||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 5780.2.c.i | 24 | |
17.b | even | 2 | 1 | inner | 5780.2.c.i | 24 | |
17.c | even | 4 | 1 | 5780.2.a.p | ✓ | 12 | |
17.c | even | 4 | 1 | 5780.2.a.s | yes | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
5780.2.a.p | ✓ | 12 | 17.c | even | 4 | 1 | |
5780.2.a.s | yes | 12 | 17.c | even | 4 | 1 | |
5780.2.c.i | 24 | 1.a | even | 1 | 1 | trivial | |
5780.2.c.i | 24 | 17.b | 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 \(S_{2}^{\mathrm{new}}(5780, [\chi])\):
\( T_{3}^{24} + 57 T_{3}^{22} + 1392 T_{3}^{20} + 19013 T_{3}^{18} + 159021 T_{3}^{16} + 838506 T_{3}^{14} + \cdots + 361 \) |
\( T_{7}^{24} + 132 T_{7}^{22} + 7500 T_{7}^{20} + 239792 T_{7}^{18} + 4729386 T_{7}^{16} + 59253588 T_{7}^{14} + \cdots + 11449 \) |
\( T_{11}^{24} + 210 T_{11}^{22} + 19059 T_{11}^{20} + 979644 T_{11}^{18} + 31399539 T_{11}^{16} + \cdots + 55228760064 \) |