Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1600,4,Mod(1,1600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1600, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1600.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1600 = 2^{6} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1600.a (trivial) |
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: | yes |
| Analytic conductor: | \(94.4030560092\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 5) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −2.00000 | 0 | 0 | 0 | 6.00000 | 0 | −23.0000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1600.4.a.s | 1 | |
| 4.b | odd | 2 | 1 | 1600.4.a.bi | 1 | ||
| 5.b | even | 2 | 1 | 320.4.a.h | 1 | ||
| 8.b | even | 2 | 1 | 400.4.a.m | 1 | ||
| 8.d | odd | 2 | 1 | 25.4.a.c | 1 | ||
| 20.d | odd | 2 | 1 | 320.4.a.g | 1 | ||
| 24.f | even | 2 | 1 | 225.4.a.b | 1 | ||
| 40.e | odd | 2 | 1 | 5.4.a.a | ✓ | 1 | |
| 40.f | even | 2 | 1 | 80.4.a.d | 1 | ||
| 40.i | odd | 4 | 2 | 400.4.c.k | 2 | ||
| 40.k | even | 4 | 2 | 25.4.b.a | 2 | ||
| 56.e | even | 2 | 1 | 1225.4.a.k | 1 | ||
| 80.k | odd | 4 | 2 | 1280.4.d.e | 2 | ||
| 80.q | even | 4 | 2 | 1280.4.d.l | 2 | ||
| 120.i | odd | 2 | 1 | 720.4.a.u | 1 | ||
| 120.m | even | 2 | 1 | 45.4.a.d | 1 | ||
| 120.q | odd | 4 | 2 | 225.4.b.c | 2 | ||
| 280.n | even | 2 | 1 | 245.4.a.a | 1 | ||
| 280.ba | even | 6 | 2 | 245.4.e.g | 2 | ||
| 280.bi | odd | 6 | 2 | 245.4.e.f | 2 | ||
| 360.z | odd | 6 | 2 | 405.4.e.l | 2 | ||
| 360.bd | even | 6 | 2 | 405.4.e.c | 2 | ||
| 440.c | even | 2 | 1 | 605.4.a.d | 1 | ||
| 520.b | odd | 2 | 1 | 845.4.a.b | 1 | ||
| 680.k | odd | 2 | 1 | 1445.4.a.a | 1 | ||
| 760.p | even | 2 | 1 | 1805.4.a.h | 1 | ||
| 840.b | odd | 2 | 1 | 2205.4.a.q | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.4.a.a | ✓ | 1 | 40.e | odd | 2 | 1 | |
| 25.4.a.c | 1 | 8.d | odd | 2 | 1 | ||
| 25.4.b.a | 2 | 40.k | even | 4 | 2 | ||
| 45.4.a.d | 1 | 120.m | even | 2 | 1 | ||
| 80.4.a.d | 1 | 40.f | even | 2 | 1 | ||
| 225.4.a.b | 1 | 24.f | even | 2 | 1 | ||
| 225.4.b.c | 2 | 120.q | odd | 4 | 2 | ||
| 245.4.a.a | 1 | 280.n | even | 2 | 1 | ||
| 245.4.e.f | 2 | 280.bi | odd | 6 | 2 | ||
| 245.4.e.g | 2 | 280.ba | even | 6 | 2 | ||
| 320.4.a.g | 1 | 20.d | odd | 2 | 1 | ||
| 320.4.a.h | 1 | 5.b | even | 2 | 1 | ||
| 400.4.a.m | 1 | 8.b | even | 2 | 1 | ||
| 400.4.c.k | 2 | 40.i | odd | 4 | 2 | ||
| 405.4.e.c | 2 | 360.bd | even | 6 | 2 | ||
| 405.4.e.l | 2 | 360.z | odd | 6 | 2 | ||
| 605.4.a.d | 1 | 440.c | even | 2 | 1 | ||
| 720.4.a.u | 1 | 120.i | odd | 2 | 1 | ||
| 845.4.a.b | 1 | 520.b | odd | 2 | 1 | ||
| 1225.4.a.k | 1 | 56.e | even | 2 | 1 | ||
| 1280.4.d.e | 2 | 80.k | odd | 4 | 2 | ||
| 1280.4.d.l | 2 | 80.q | even | 4 | 2 | ||
| 1445.4.a.a | 1 | 680.k | odd | 2 | 1 | ||
| 1600.4.a.s | 1 | 1.a | even | 1 | 1 | trivial | |
| 1600.4.a.bi | 1 | 4.b | odd | 2 | 1 | ||
| 1805.4.a.h | 1 | 760.p | even | 2 | 1 | ||
| 2205.4.a.q | 1 | 840.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1600))\):
|
\( T_{3} + 2 \)
|
|
\( T_{7} - 6 \)
|
|
\( T_{11} - 32 \)
|
|
\( T_{13} + 38 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T + 2 \)
|
| $5$ |
\( T \)
|
| $7$ |
\( T - 6 \)
|
| $11$ |
\( T - 32 \)
|
| $13$ |
\( T + 38 \)
|
| $17$ |
\( T + 26 \)
|
| $19$ |
\( T - 100 \)
|
| $23$ |
\( T + 78 \)
|
| $29$ |
\( T - 50 \)
|
| $31$ |
\( T - 108 \)
|
| $37$ |
\( T - 266 \)
|
| $41$ |
\( T - 22 \)
|
| $43$ |
\( T + 442 \)
|
| $47$ |
\( T + 514 \)
|
| $53$ |
\( T - 2 \)
|
| $59$ |
\( T - 500 \)
|
| $61$ |
\( T - 518 \)
|
| $67$ |
\( T + 126 \)
|
| $71$ |
\( T + 412 \)
|
| $73$ |
\( T - 878 \)
|
| $79$ |
\( T + 600 \)
|
| $83$ |
\( T + 282 \)
|
| $89$ |
\( T + 150 \)
|
| $97$ |
\( T + 386 \)
|
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