Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7942,2,Mod(1,7942)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7942, 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("7942.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7942 = 2 \cdot 11 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7942.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: | \(63.4171892853\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.485125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 4x^{4} + 8x^{3} + 2x^{2} - 5x + 1 \)
|
| 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
\( x^{6} - 2x^{5} - 4x^{4} + 8x^{3} + 2x^{2} - 5x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 4\nu^{3} + 3\nu^{2} + 2\nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{5} + \nu^{4} + 5\nu^{3} - 3\nu^{2} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 5\nu^{3} + 4\nu^{2} + 5\nu - 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + 2\nu^{4} + 4\nu^{3} - 7\nu^{2} - 3\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 3\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 4\beta_{4} + 4\beta_{3} + \beta_{2} + \beta _1 + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 5\beta_{3} + 6\beta_{2} + 11\beta _1 + 5 \)
|
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 | −3.13985 | 1.00000 | 2.31095 | −3.13985 | 1.69668 | 1.00000 | 6.85866 | 2.31095 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.83865 | 1.00000 | −2.78675 | −1.83865 | 2.06568 | 1.00000 | 0.380639 | −2.78675 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.55021 | 1.00000 | −0.456268 | −1.55021 | −4.63156 | 1.00000 | −0.596855 | −0.456268 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.779746 | 1.00000 | 3.63013 | −0.779746 | 0.544350 | 1.00000 | −2.39200 | 3.63013 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0.947988 | 1.00000 | −1.17386 | 0.947988 | 0.851145 | 1.00000 | −2.10132 | −1.17386 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.36047 | 1.00000 | 2.47580 | 1.36047 | −2.52629 | 1.00000 | −1.14913 | 2.47580 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(1\) |
| \(19\) | \(-1\) |
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 | 7942.2.a.bl | yes | 6 |
| 19.b | odd | 2 | 1 | 7942.2.a.bk | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7942.2.a.bk | ✓ | 6 | 19.b | odd | 2 | 1 | |
| 7942.2.a.bl | yes | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):
|
\( T_{3}^{6} + 5T_{3}^{5} + 3T_{3}^{4} - 14T_{3}^{3} - 14T_{3}^{2} + 9T_{3} + 9 \)
|
|
\( T_{5}^{6} - 4T_{5}^{5} - 9T_{5}^{4} + 40T_{5}^{3} + 13T_{5}^{2} - 71T_{5} - 31 \)
|
|
\( T_{13}^{6} - 11T_{13}^{4} - 2T_{13}^{3} + 29T_{13}^{2} + T_{13} - 19 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
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| $3$ |
\( T^{6} + 5 T^{5} + \cdots + 9 \)
|
| $5$ |
\( T^{6} - 4 T^{5} + \cdots - 31 \)
|
| $7$ |
\( T^{6} + 2 T^{5} + \cdots + 19 \)
|
| $11$ |
\( (T + 1)^{6} \)
|
| $13$ |
\( T^{6} - 11 T^{4} + \cdots - 19 \)
|
| $17$ |
\( T^{6} + 3 T^{5} + \cdots + 101 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} + 5 T^{5} + \cdots - 779 \)
|
| $29$ |
\( T^{6} + 17 T^{5} + \cdots - 6811 \)
|
| $31$ |
\( T^{6} + 9 T^{5} + \cdots + 10691 \)
|
| $37$ |
\( T^{6} + 3 T^{5} + \cdots - 23279 \)
|
| $41$ |
\( T^{6} + 12 T^{5} + \cdots + 128659 \)
|
| $43$ |
\( T^{6} + 2 T^{5} + \cdots - 27689 \)
|
| $47$ |
\( T^{6} - 6 T^{5} + \cdots - 23971 \)
|
| $53$ |
\( T^{6} + 2 T^{5} + \cdots + 9619 \)
|
| $59$ |
\( T^{6} + 19 T^{5} + \cdots - 39539 \)
|
| $61$ |
\( T^{6} + 7 T^{5} + \cdots + 389 \)
|
| $67$ |
\( T^{6} + 15 T^{5} + \cdots + 40769 \)
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| $71$ |
\( T^{6} + 7 T^{5} + \cdots - 183089 \)
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| $73$ |
\( T^{6} + 21 T^{5} + \cdots + 2939 \)
|
| $79$ |
\( T^{6} + 28 T^{5} + \cdots + 1744961 \)
|
| $83$ |
\( T^{6} + 11 T^{5} + \cdots - 58319 \)
|
| $89$ |
\( T^{6} + 17 T^{5} + \cdots - 959969 \)
|
| $97$ |
\( T^{6} - 21 T^{5} + \cdots - 1699 \)
|
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