Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1388,2,Mod(1,1388)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1388, 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("1388.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1388 = 2^{2} \cdot 347 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1388.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: | \(11.0832358006\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.10149241.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 6x^{4} + 11x^{3} + 9x^{2} - 14x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 6x^{4} + 11x^{3} + 9x^{2} - 14x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 6\nu^{3} + 4\nu^{2} + 8\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 3\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 7\beta_{3} + 7\beta_{2} + 11\beta _1 + 8 \)
|
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 | −2.92662 | 0 | −3.59718 | 0 | −1.84417 | 0 | 5.56513 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.71008 | 0 | 3.46158 | 0 | 1.18724 | 0 | 4.34451 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.42006 | 0 | −0.705733 | 0 | 2.60182 | 0 | −0.983433 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.0611565 | 0 | 2.71856 | 0 | −1.58757 | 0 | −2.99626 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.608070 | 0 | −0.0938877 | 0 | 1.76864 | 0 | −2.63025 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.38753 | 0 | −1.78334 | 0 | −3.12595 | 0 | 2.70031 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(347\) | \( -1 \) |
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 | 1388.2.a.c | ✓ | 6 |
| 4.b | odd | 2 | 1 | 5552.2.a.g | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1388.2.a.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 5552.2.a.g | 6 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 4T_{3}^{5} - 4T_{3}^{4} - 26T_{3}^{3} - 9T_{3}^{2} + 17T_{3} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1388))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 4 T^{5} + \cdots - 1 \)
|
| $5$ |
\( T^{6} - 18 T^{4} + \cdots + 4 \)
|
| $7$ |
\( T^{6} + T^{5} + \cdots - 50 \)
|
| $11$ |
\( T^{6} + 8 T^{5} + \cdots + 49 \)
|
| $13$ |
\( T^{6} + 5 T^{5} + \cdots - 162 \)
|
| $17$ |
\( T^{6} + 16 T^{5} + \cdots + 92 \)
|
| $19$ |
\( T^{6} + 8 T^{5} + \cdots + 2416 \)
|
| $23$ |
\( T^{6} + 10 T^{5} + \cdots - 16 \)
|
| $29$ |
\( T^{6} - 20 T^{5} + \cdots + 2083 \)
|
| $31$ |
\( T^{6} + 27 T^{5} + \cdots + 904 \)
|
| $37$ |
\( T^{6} - 9 T^{5} + \cdots - 5596 \)
|
| $41$ |
\( T^{6} - 15 T^{5} + \cdots - 6124 \)
|
| $43$ |
\( T^{6} + 23 T^{5} + \cdots + 74669 \)
|
| $47$ |
\( T^{6} + 21 T^{5} + \cdots - 10358 \)
|
| $53$ |
\( T^{6} - 3 T^{5} + \cdots + 134 \)
|
| $59$ |
\( T^{6} - 9 T^{5} + \cdots - 3032 \)
|
| $61$ |
\( T^{6} + 8 T^{5} + \cdots + 95101 \)
|
| $67$ |
\( T^{6} + 18 T^{5} + \cdots - 17597 \)
|
| $71$ |
\( T^{6} + 2 T^{5} + \cdots + 24156 \)
|
| $73$ |
\( T^{6} + 8 T^{5} + \cdots - 25 \)
|
| $79$ |
\( T^{6} + 23 T^{5} + \cdots + 22 \)
|
| $83$ |
\( T^{6} + 22 T^{5} + \cdots - 1017 \)
|
| $89$ |
\( T^{6} - 21 T^{5} + \cdots - 8461 \)
|
| $97$ |
\( T^{6} + 5 T^{5} + \cdots + 12124 \)
|
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