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: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.37485.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 8x^{2} + 12x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7} \) |
| Twist minimal: | no (minimal twist has level 160) |
| 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)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - x^{3} - 8x^{2} + 12x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{3} - 18\nu + 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{3} - 28\nu + 18 \)
|
| \(\beta_{3}\) | \(=\) |
\( 8\nu^{2} + 8\nu - 36 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta _1 + 2 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 2\beta _1 + 34 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 9\beta_{2} - 14\beta _1 - 22 ) / 8 \)
|
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 | −8.57295 | 0 | 0 | 0 | −22.1403 | 0 | 46.4955 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −4.30169 | 0 | 0 | 0 | 28.3162 | 0 | −8.49545 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 4.30169 | 0 | 0 | 0 | −28.3162 | 0 | −8.49545 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 8.57295 | 0 | 0 | 0 | 22.1403 | 0 | 46.4955 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1600.4.a.cv | 4 | |
| 4.b | odd | 2 | 1 | inner | 1600.4.a.cv | 4 | |
| 5.b | even | 2 | 1 | 1600.4.a.cu | 4 | ||
| 5.c | odd | 4 | 2 | 320.4.c.j | 8 | ||
| 8.b | even | 2 | 1 | 800.4.a.y | 4 | ||
| 8.d | odd | 2 | 1 | 800.4.a.y | 4 | ||
| 20.d | odd | 2 | 1 | 1600.4.a.cu | 4 | ||
| 20.e | even | 4 | 2 | 320.4.c.j | 8 | ||
| 40.e | odd | 2 | 1 | 800.4.a.z | 4 | ||
| 40.f | even | 2 | 1 | 800.4.a.z | 4 | ||
| 40.i | odd | 4 | 2 | 160.4.c.d | ✓ | 8 | |
| 40.k | even | 4 | 2 | 160.4.c.d | ✓ | 8 | |
| 120.q | odd | 4 | 2 | 1440.4.f.k | 8 | ||
| 120.w | even | 4 | 2 | 1440.4.f.k | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 160.4.c.d | ✓ | 8 | 40.i | odd | 4 | 2 | |
| 160.4.c.d | ✓ | 8 | 40.k | even | 4 | 2 | |
| 320.4.c.j | 8 | 5.c | odd | 4 | 2 | ||
| 320.4.c.j | 8 | 20.e | even | 4 | 2 | ||
| 800.4.a.y | 4 | 8.b | even | 2 | 1 | ||
| 800.4.a.y | 4 | 8.d | odd | 2 | 1 | ||
| 800.4.a.z | 4 | 40.e | odd | 2 | 1 | ||
| 800.4.a.z | 4 | 40.f | even | 2 | 1 | ||
| 1440.4.f.k | 8 | 120.q | odd | 4 | 2 | ||
| 1440.4.f.k | 8 | 120.w | even | 4 | 2 | ||
| 1600.4.a.cu | 4 | 5.b | even | 2 | 1 | ||
| 1600.4.a.cu | 4 | 20.d | odd | 2 | 1 | ||
| 1600.4.a.cv | 4 | 1.a | even | 1 | 1 | trivial | |
| 1600.4.a.cv | 4 | 4.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1600))\):
|
\( T_{3}^{4} - 92T_{3}^{2} + 1360 \)
|
|
\( T_{7}^{4} - 1292T_{7}^{2} + 393040 \)
|
|
\( T_{11}^{4} - 4992T_{11}^{2} + 3133440 \)
|
|
\( T_{13}^{2} - 104T_{13} + 2368 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 92T^{2} + 1360 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 1292 T^{2} + 393040 \)
|
| $11$ |
\( T^{4} - 4992 T^{2} + 3133440 \)
|
| $13$ |
\( (T^{2} - 104 T + 2368)^{2} \)
|
| $17$ |
\( (T^{2} - 5376)^{2} \)
|
| $19$ |
\( T^{4} - 30080 T^{2} + 217600000 \)
|
| $23$ |
\( T^{4} - 11852 T^{2} + 2514640 \)
|
| $29$ |
\( (T^{2} - 156 T - 15420)^{2} \)
|
| $31$ |
\( T^{4} - 23552 T^{2} + 89128960 \)
|
| $37$ |
\( (T^{2} - 136 T - 11840)^{2} \)
|
| $41$ |
\( (T^{2} + 48 T - 13620)^{2} \)
|
| $43$ |
\( T^{4} - 251708 T^{2} + 57154000 \)
|
| $47$ |
\( T^{4} + \cdots + 3485953360 \)
|
| $53$ |
\( (T^{2} - 24 T - 75456)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 4289679360 \)
|
| $61$ |
\( (T^{2} + 528 T + 8460)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 27064708560 \)
|
| $71$ |
\( T^{4} - 46592 T^{2} + 272957440 \)
|
| $73$ |
\( (T^{2} + 720 T - 33024)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 196885872640 \)
|
| $83$ |
\( T^{4} + \cdots + 739915650000 \)
|
| $89$ |
\( (T^{2} + 20 T - 537500)^{2} \)
|
| $97$ |
\( (T + 1136)^{4} \)
|
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