Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [475,3,Mod(474,475)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("475.474");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 475 = 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 475.d (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: | \(12.9428125571\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
474.1 | −3.82982 | 0.102062 | 10.6675 | 0 | −0.390878 | − | 9.96826i | −25.5353 | −8.98958 | 0 | |||||||||||||||||
474.2 | −3.82982 | 0.102062 | 10.6675 | 0 | −0.390878 | 9.96826i | −25.5353 | −8.98958 | 0 | ||||||||||||||||||
474.3 | −3.01757 | −4.83172 | 5.10575 | 0 | 14.5801 | − | 6.15539i | −3.33669 | 14.3455 | 0 | |||||||||||||||||
474.4 | −3.01757 | −4.83172 | 5.10575 | 0 | 14.5801 | 6.15539i | −3.33669 | 14.3455 | 0 | ||||||||||||||||||
474.5 | −2.92865 | 2.02259 | 4.57698 | 0 | −5.92346 | − | 8.32815i | −1.68976 | −4.90912 | 0 | |||||||||||||||||
474.6 | −2.92865 | 2.02259 | 4.57698 | 0 | −5.92346 | 8.32815i | −1.68976 | −4.90912 | 0 | ||||||||||||||||||
474.7 | −2.34822 | 5.73703 | 1.51415 | 0 | −13.4718 | 9.56678i | 5.83734 | 23.9135 | 0 | ||||||||||||||||||
474.8 | −2.34822 | 5.73703 | 1.51415 | 0 | −13.4718 | − | 9.56678i | 5.83734 | 23.9135 | 0 | |||||||||||||||||
474.9 | −1.41833 | 1.48141 | −1.98835 | 0 | −2.10112 | − | 2.50491i | 8.49344 | −6.80543 | 0 | |||||||||||||||||
474.10 | −1.41833 | 1.48141 | −1.98835 | 0 | −2.10112 | 2.50491i | 8.49344 | −6.80543 | 0 | ||||||||||||||||||
474.11 | −1.40632 | −2.91656 | −2.02225 | 0 | 4.10162 | − | 11.7208i | 8.46924 | −0.493705 | 0 | |||||||||||||||||
474.12 | −1.40632 | −2.91656 | −2.02225 | 0 | 4.10162 | 11.7208i | 8.46924 | −0.493705 | 0 | ||||||||||||||||||
474.13 | −0.382406 | −3.15259 | −3.85377 | 0 | 1.20557 | 4.26742i | 3.00333 | 0.938823 | 0 | ||||||||||||||||||
474.14 | −0.382406 | −3.15259 | −3.85377 | 0 | 1.20557 | − | 4.26742i | 3.00333 | 0.938823 | 0 | |||||||||||||||||
474.15 | 0.382406 | 3.15259 | −3.85377 | 0 | 1.20557 | 4.26742i | −3.00333 | 0.938823 | 0 | ||||||||||||||||||
474.16 | 0.382406 | 3.15259 | −3.85377 | 0 | 1.20557 | − | 4.26742i | −3.00333 | 0.938823 | 0 | |||||||||||||||||
474.17 | 1.40632 | 2.91656 | −2.02225 | 0 | 4.10162 | − | 11.7208i | −8.46924 | −0.493705 | 0 | |||||||||||||||||
474.18 | 1.40632 | 2.91656 | −2.02225 | 0 | 4.10162 | 11.7208i | −8.46924 | −0.493705 | 0 | ||||||||||||||||||
474.19 | 1.41833 | −1.48141 | −1.98835 | 0 | −2.10112 | − | 2.50491i | −8.49344 | −6.80543 | 0 | |||||||||||||||||
474.20 | 1.41833 | −1.48141 | −1.98835 | 0 | −2.10112 | 2.50491i | −8.49344 | −6.80543 | 0 | ||||||||||||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
19.b | odd | 2 | 1 | inner |
95.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 475.3.d.d | 28 | |
5.b | even | 2 | 1 | inner | 475.3.d.d | 28 | |
5.c | odd | 4 | 1 | 475.3.c.h | ✓ | 14 | |
5.c | odd | 4 | 1 | 475.3.c.i | yes | 14 | |
19.b | odd | 2 | 1 | inner | 475.3.d.d | 28 | |
95.d | odd | 2 | 1 | inner | 475.3.d.d | 28 | |
95.g | even | 4 | 1 | 475.3.c.h | ✓ | 14 | |
95.g | even | 4 | 1 | 475.3.c.i | yes | 14 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
475.3.c.h | ✓ | 14 | 5.c | odd | 4 | 1 | |
475.3.c.h | ✓ | 14 | 95.g | even | 4 | 1 | |
475.3.c.i | yes | 14 | 5.c | odd | 4 | 1 | |
475.3.c.i | yes | 14 | 95.g | even | 4 | 1 | |
475.3.d.d | 28 | 1.a | even | 1 | 1 | trivial | |
475.3.d.d | 28 | 5.b | even | 2 | 1 | inner | |
475.3.d.d | 28 | 19.b | odd | 2 | 1 | inner | |
475.3.d.d | 28 | 95.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 42T_{2}^{12} + 677T_{2}^{10} - 5313T_{2}^{8} + 21125T_{2}^{6} - 40138T_{2}^{4} + 30565T_{2}^{2} - 3675 \) acting on \(S_{3}^{\mathrm{new}}(475, [\chi])\).