Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [406,2,Mod(1,406)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(406, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("406.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 406 = 2 \cdot 7 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 406.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: | \(3.24192632206\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.11348.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 5x^{2} + x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| 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
\( x^{4} - x^{3} - 5x^{2} + x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{3} - 3\beta_{2} + 7\beta _1 + 9 ) / 2 \)
|
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 |
|
1.00000 | −2.76342 | 1.00000 | −2.47620 | −2.76342 | 1.00000 | 1.00000 | 4.63646 | −2.47620 | ||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −0.757235 | 1.00000 | 3.97587 | −0.757235 | 1.00000 | 1.00000 | −2.42659 | 3.97587 | |||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 1.12631 | 1.00000 | 0.153156 | 1.12631 | 1.00000 | 1.00000 | −1.73143 | 0.153156 | |||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 3.39434 | 1.00000 | −2.65282 | 3.39434 | 1.00000 | 1.00000 | 8.52156 | −2.65282 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(29\) | \(-1\) |
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 | 406.2.a.g | ✓ | 4 |
| 3.b | odd | 2 | 1 | 3654.2.a.bg | 4 | ||
| 4.b | odd | 2 | 1 | 3248.2.a.x | 4 | ||
| 7.b | odd | 2 | 1 | 2842.2.a.r | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 406.2.a.g | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 2842.2.a.r | 4 | 7.b | odd | 2 | 1 | ||
| 3248.2.a.x | 4 | 4.b | odd | 2 | 1 | ||
| 3654.2.a.bg | 4 | 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(406))\):
|
\( T_{3}^{4} - T_{3}^{3} - 10T_{3}^{2} + 4T_{3} + 8 \)
|
|
\( T_{5}^{4} + T_{5}^{3} - 14T_{5}^{2} - 24T_{5} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{4} \)
|
| $3$ |
\( T^{4} - T^{3} - 10 T^{2} + \cdots + 8 \)
|
| $5$ |
\( T^{4} + T^{3} - 14 T^{2} + \cdots + 4 \)
|
| $7$ |
\( (T - 1)^{4} \)
|
| $11$ |
\( T^{4} - 7 T^{3} + \cdots - 16 \)
|
| $13$ |
\( T^{4} + 7 T^{3} + \cdots + 28 \)
|
| $17$ |
\( T^{4} - 20 T^{2} + \cdots - 16 \)
|
| $19$ |
\( T^{4} - 2 T^{3} + \cdots + 32 \)
|
| $23$ |
\( T^{4} - 6 T^{3} + \cdots - 32 \)
|
| $29$ |
\( (T - 1)^{4} \)
|
| $31$ |
\( T^{4} + 7 T^{3} + \cdots + 356 \)
|
| $37$ |
\( T^{4} + 12 T^{3} + \cdots - 16 \)
|
| $41$ |
\( T^{4} - 4 T^{3} + \cdots - 448 \)
|
| $43$ |
\( T^{4} + 5 T^{3} + \cdots + 256 \)
|
| $47$ |
\( T^{4} + 11 T^{3} + \cdots - 188 \)
|
| $53$ |
\( T^{4} - 5 T^{3} + \cdots - 248 \)
|
| $59$ |
\( T^{4} - 16 T^{3} + \cdots - 2752 \)
|
| $61$ |
\( T^{4} + 34 T^{3} + \cdots - 1376 \)
|
| $67$ |
\( T^{4} - 2 T^{3} + \cdots + 24832 \)
|
| $71$ |
\( T^{4} - 24 T^{3} + \cdots - 28544 \)
|
| $73$ |
\( T^{4} + 24 T^{3} + \cdots - 64 \)
|
| $79$ |
\( T^{4} + 9 T^{3} + \cdots - 544 \)
|
| $83$ |
\( T^{4} + 8 T^{3} + \cdots + 17024 \)
|
| $89$ |
\( T^{4} - 2 T^{3} + \cdots + 824 \)
|
| $97$ |
\( T^{4} + 4 T^{3} + \cdots - 448 \)
|
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