Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [431,2,Mod(1,431)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(431, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("431.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 431 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 431.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: | \(3.44155232712\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.725.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 3x^{2} + x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} - x^{3} - 3x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 2\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 3\beta_1 \)
|
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 |
|
−2.09529 | −1.47726 | 2.39026 | −0.522740 | 3.09529 | −0.294963 | −0.817703 | −0.817703 | 1.09529 | ||||||||||||||||||||||||||||||
| 1.2 | −0.737640 | −2.35567 | −1.45589 | 0.355674 | 1.73764 | 2.19353 | 2.54920 | 2.54920 | −0.262360 | |||||||||||||||||||||||||||||||
| 1.3 | 0.477260 | 1.09529 | −1.77222 | −3.09529 | 0.522740 | 1.29496 | −1.80033 | −1.80033 | −1.47726 | |||||||||||||||||||||||||||||||
| 1.4 | 1.35567 | −0.262360 | −0.162147 | −1.73764 | −0.355674 | −1.19353 | −2.93117 | −2.93117 | −2.35567 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(431\) | \(1\) |
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 | 431.2.a.e | ✓ | 4 |
| 3.b | odd | 2 | 1 | 3879.2.a.m | 4 | ||
| 4.b | odd | 2 | 1 | 6896.2.a.o | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 431.2.a.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 3879.2.a.m | 4 | 3.b | odd | 2 | 1 | ||
| 6896.2.a.o | 4 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(431))\):
|
\( T_{2}^{4} + T_{2}^{3} - 3T_{2}^{2} - T_{2} + 1 \)
|
|
\( T_{3}^{4} + 3T_{3}^{3} - 4T_{3} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + T^{3} - 3 T^{2} + \cdots + 1 \)
|
| $3$ |
\( T^{4} + 3 T^{3} + \cdots - 1 \)
|
| $5$ |
\( T^{4} + 5 T^{3} + \cdots - 1 \)
|
| $7$ |
\( T^{4} - 2 T^{3} + \cdots + 1 \)
|
| $11$ |
\( T^{4} - T^{3} + \cdots - 19 \)
|
| $13$ |
\( T^{4} + 5 T^{3} + \cdots - 1 \)
|
| $17$ |
\( T^{4} - 2 T^{3} + \cdots + 101 \)
|
| $19$ |
\( T^{4} + 6 T^{3} + \cdots - 49 \)
|
| $23$ |
\( T^{4} - 4 T^{3} + \cdots + 1 \)
|
| $29$ |
\( T^{4} + 5 T^{3} + \cdots + 25 \)
|
| $31$ |
\( T^{4} + 25 T^{3} + \cdots + 1399 \)
|
| $37$ |
\( T^{4} + T^{3} + \cdots + 109 \)
|
| $41$ |
\( T^{4} - 2 T^{3} + \cdots - 509 \)
|
| $43$ |
\( T^{4} + 16 T^{3} + \cdots - 19 \)
|
| $47$ |
\( T^{4} + 4 T^{3} + \cdots - 169 \)
|
| $53$ |
\( T^{4} - 4 T^{3} + \cdots + 1301 \)
|
| $59$ |
\( T^{4} - 5 T^{3} + \cdots + 79 \)
|
| $61$ |
\( T^{4} - 110 T^{2} + \cdots - 275 \)
|
| $67$ |
\( T^{4} - 9 T^{3} + \cdots + 941 \)
|
| $71$ |
\( T^{4} - 10 T^{3} + \cdots + 304 \)
|
| $73$ |
\( T^{4} + 50 T^{3} + \cdots + 23869 \)
|
| $79$ |
\( T^{4} - 8 T^{3} + \cdots + 7211 \)
|
| $83$ |
\( T^{4} + 15 T^{3} + \cdots - 341 \)
|
| $89$ |
\( T^{4} + 35 T^{3} + \cdots - 2441 \)
|
| $97$ |
\( T^{4} + 6 T^{3} + \cdots + 6359 \)
|
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