Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [392,4,Mod(1,392)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(392, 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("392.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 392.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: | \(23.1287487223\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1929.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 10x + 13 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 56) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 10x + 13 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} + 2\nu - 15 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} - \beta _1 + 29 ) / 4 \)
|
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 | −8.51203 | 0 | −8.47688 | 0 | 0 | 0 | 45.4547 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −0.138208 | 0 | 10.0858 | 0 | 0 | 0 | −26.9809 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 7.65024 | 0 | −14.6089 | 0 | 0 | 0 | 31.5262 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
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 | 392.4.a.j | 3 | |
| 4.b | odd | 2 | 1 | 784.4.a.bd | 3 | ||
| 7.b | odd | 2 | 1 | 392.4.a.k | 3 | ||
| 7.c | even | 3 | 2 | 392.4.i.n | 6 | ||
| 7.d | odd | 6 | 2 | 56.4.i.a | ✓ | 6 | |
| 21.g | even | 6 | 2 | 504.4.s.i | 6 | ||
| 28.d | even | 2 | 1 | 784.4.a.bc | 3 | ||
| 28.f | even | 6 | 2 | 112.4.i.f | 6 | ||
| 56.j | odd | 6 | 2 | 448.4.i.l | 6 | ||
| 56.m | even | 6 | 2 | 448.4.i.k | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.4.i.a | ✓ | 6 | 7.d | odd | 6 | 2 | |
| 112.4.i.f | 6 | 28.f | even | 6 | 2 | ||
| 392.4.a.j | 3 | 1.a | even | 1 | 1 | trivial | |
| 392.4.a.k | 3 | 7.b | odd | 2 | 1 | ||
| 392.4.i.n | 6 | 7.c | even | 3 | 2 | ||
| 448.4.i.k | 6 | 56.m | even | 6 | 2 | ||
| 448.4.i.l | 6 | 56.j | odd | 6 | 2 | ||
| 504.4.s.i | 6 | 21.g | even | 6 | 2 | ||
| 784.4.a.bc | 3 | 28.d | even | 2 | 1 | ||
| 784.4.a.bd | 3 | 4.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(392))\):
|
\( T_{3}^{3} + T_{3}^{2} - 65T_{3} - 9 \)
|
|
\( T_{5}^{3} + 13T_{5}^{2} - 109T_{5} - 1249 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + T^{2} - 65T - 9 \)
|
| $5$ |
\( T^{3} + 13 T^{2} + \cdots - 1249 \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} + 11 T^{2} + \cdots - 8099 \)
|
| $13$ |
\( T^{3} + 70 T^{2} + \cdots - 17752 \)
|
| $17$ |
\( T^{3} + 97 T^{2} + \cdots - 586557 \)
|
| $19$ |
\( T^{3} - 81 T^{2} + \cdots + 123377 \)
|
| $23$ |
\( T^{3} - 191 T^{2} + \cdots + 3492279 \)
|
| $29$ |
\( T^{3} + 162 T^{2} + \cdots - 1324296 \)
|
| $31$ |
\( T^{3} + 597 T^{2} + \cdots + 4663139 \)
|
| $37$ |
\( T^{3} - 217 T^{2} + \cdots + 15110373 \)
|
| $41$ |
\( T^{3} + 698 T^{2} + \cdots - 6465192 \)
|
| $43$ |
\( T^{3} + 308 T^{2} + \cdots + 38848 \)
|
| $47$ |
\( T^{3} + 139 T^{2} + \cdots - 18709731 \)
|
| $53$ |
\( T^{3} - 197 T^{2} + \cdots - 2914839 \)
|
| $59$ |
\( T^{3} + 353 T^{2} + \cdots - 37291113 \)
|
| $61$ |
\( T^{3} + 449 T^{2} + \cdots + 9942147 \)
|
| $67$ |
\( T^{3} + 519 T^{2} + \cdots - 21323807 \)
|
| $71$ |
\( T^{3} - 224 T^{2} + \cdots + 7551488 \)
|
| $73$ |
\( T^{3} + 1701 T^{2} + \cdots + 72721831 \)
|
| $79$ |
\( T^{3} - 1143 T^{2} + \cdots + 297161663 \)
|
| $83$ |
\( T^{3} - 1380 T^{2} + \cdots - 4797376 \)
|
| $89$ |
\( T^{3} + 1749 T^{2} + \cdots - 155620521 \)
|
| $97$ |
\( T^{3} + 2602 T^{2} + \cdots + 528741144 \)
|
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