Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1342,2,Mod(1,1342)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1342, 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("1342.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1342 = 2 \cdot 11 \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1342.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: | \(10.7159239513\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.136632976.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 10x^{4} + 14x^{3} + 33x^{2} - 22x - 36 \)
|
| 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
\( x^{6} - 2x^{5} - 10x^{4} + 14x^{3} + 33x^{2} - 22x - 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 8\nu^{2} + 3\nu + 13 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} + 8\nu^{3} - 12\nu^{2} - 15\nu + 14 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 4\nu^{4} - 4\nu^{3} + 26\nu^{2} - \nu - 38 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 6\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 2\beta_{3} + 9\beta_{2} + 11\beta _1 + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10\beta_{5} + 8\beta_{4} + 12\beta_{3} + 14\beta_{2} + 43\beta _1 + 34 \)
|
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 | −2.12006 | 1.00000 | −2.61473 | −2.12006 | −3.88496 | 1.00000 | 1.49467 | −2.61473 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.49617 | 1.00000 | 0.265317 | −1.49617 | 5.04198 | 1.00000 | −0.761484 | 0.265317 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.11055 | 1.00000 | 1.65614 | −1.11055 | −0.902918 | 1.00000 | −1.76668 | 1.65614 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.56551 | 1.00000 | 3.11469 | 1.56551 | 1.20763 | 1.00000 | −0.549178 | 3.11469 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.21778 | 1.00000 | 1.29925 | 2.21778 | 2.75713 | 1.00000 | 1.91853 | 1.29925 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.94349 | 1.00000 | −1.72066 | 2.94349 | 0.781146 | 1.00000 | 5.66415 | −1.72066 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(1\) |
| \(61\) | \(1\) |
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 | 1342.2.a.m | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1342.2.a.m | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1342))\):
|
\( T_{3}^{6} - 2T_{3}^{5} - 10T_{3}^{4} + 14T_{3}^{3} + 33T_{3}^{2} - 22T_{3} - 36 \)
|
|
\( T_{5}^{6} - 2T_{5}^{5} - 10T_{5}^{4} + 18T_{5}^{3} + 18T_{5}^{2} - 36T_{5} + 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots - 36 \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots + 8 \)
|
| $7$ |
\( T^{6} - 5 T^{5} + \cdots + 46 \)
|
| $11$ |
\( (T + 1)^{6} \)
|
| $13$ |
\( T^{6} - 3 T^{5} + \cdots - 1 \)
|
| $17$ |
\( T^{6} - 9 T^{5} + \cdots - 19 \)
|
| $19$ |
\( T^{6} - 7 T^{5} + \cdots + 96 \)
|
| $23$ |
\( T^{6} - 10 T^{5} + \cdots - 576 \)
|
| $29$ |
\( T^{6} - 4 T^{5} + \cdots - 744 \)
|
| $31$ |
\( T^{6} - 8 T^{5} + \cdots - 1588 \)
|
| $37$ |
\( T^{6} + 5 T^{5} + \cdots - 159 \)
|
| $41$ |
\( T^{6} - 2 T^{5} + \cdots + 22016 \)
|
| $43$ |
\( T^{6} - 8 T^{5} + \cdots + 292 \)
|
| $47$ |
\( T^{6} - 11 T^{5} + \cdots - 942 \)
|
| $53$ |
\( T^{6} - 7 T^{5} + \cdots + 11229 \)
|
| $59$ |
\( T^{6} + 5 T^{5} + \cdots - 143862 \)
|
| $61$ |
\( (T + 1)^{6} \)
|
| $67$ |
\( T^{6} + 5 T^{5} + \cdots - 10496 \)
|
| $71$ |
\( T^{6} - 10 T^{5} + \cdots + 55872 \)
|
| $73$ |
\( T^{6} + 16 T^{5} + \cdots + 8728 \)
|
| $79$ |
\( T^{6} - 3 T^{5} + \cdots - 5298 \)
|
| $83$ |
\( T^{6} - 12 T^{5} + \cdots - 3424 \)
|
| $89$ |
\( T^{6} - 4 T^{5} + \cdots - 744 \)
|
| $97$ |
\( T^{6} + 17 T^{5} + \cdots - 84449 \)
|
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