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: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.8248384.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 5x^{4} + 14x^{3} - 2x^{2} - 6x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{6} - 3x^{5} - 5x^{4} + 14x^{3} - 2x^{2} - 6x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 10\nu^{3} + 3\nu^{2} + 16\nu - 4 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 7\nu^{2} + 5\nu - 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4\nu^{5} - 10\nu^{4} - 25\nu^{3} + 42\nu^{2} + 16\nu - 13 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 8\nu^{5} - 20\nu^{4} - 50\nu^{3} + 87\nu^{2} + 26\nu - 32 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 2\beta_{4} + 2\beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} - 5\beta_{4} + 2\beta_{3} - 2\beta_{2} + 10\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 10\beta_{5} - 23\beta_{4} + 5\beta_{3} - 3\beta_{2} + 27\beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( 27\beta_{5} - 67\beta_{4} + 25\beta_{3} - 20\beta_{2} + 105\beta _1 + 21 \)
|
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 | −3.16795 | 1.00000 | −0.274182 | −3.16795 | −0.754580 | 1.00000 | 7.03591 | −0.274182 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.69170 | 1.00000 | 4.26160 | −1.69170 | −4.19971 | 1.00000 | −0.138138 | 4.26160 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.591120 | 1.00000 | −1.19006 | −0.591120 | 2.48253 | 1.00000 | −2.65058 | −1.19006 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.315662 | 1.00000 | −2.37776 | −0.315662 | 0.238192 | 1.00000 | −2.90036 | −2.37776 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0.427555 | 1.00000 | −0.640209 | 0.427555 | −2.02656 | 1.00000 | −2.81720 | −0.640209 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.33888 | 1.00000 | −3.77939 | 2.33888 | −4.73988 | 1.00000 | 2.47036 | −3.77939 | |||||||||||||||||||||||||||||||||||||
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.l | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1342.2.a.l | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1342))\):
|
\( T_{3}^{6} + 3T_{3}^{5} - 5T_{3}^{4} - 16T_{3}^{3} - 5T_{3}^{2} + 3T_{3} + 1 \)
|
|
\( T_{5}^{6} + 4T_{5}^{5} - 12T_{5}^{4} - 72T_{5}^{3} - 102T_{5}^{2} - 52T_{5} - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} + 3 T^{5} + \cdots + 1 \)
|
| $5$ |
\( T^{6} + 4 T^{5} + \cdots - 8 \)
|
| $7$ |
\( T^{6} + 9 T^{5} + \cdots + 18 \)
|
| $11$ |
\( (T + 1)^{6} \)
|
| $13$ |
\( T^{6} + 10 T^{5} + \cdots + 2412 \)
|
| $17$ |
\( T^{6} + 10 T^{5} + \cdots + 1268 \)
|
| $19$ |
\( T^{6} + 17 T^{5} + \cdots - 5728 \)
|
| $23$ |
\( T^{6} + 19 T^{5} + \cdots - 6332 \)
|
| $29$ |
\( T^{6} + 7 T^{5} + \cdots + 7666 \)
|
| $31$ |
\( T^{6} - T^{5} + \cdots - 3951 \)
|
| $37$ |
\( T^{6} - 3 T^{5} + \cdots + 1823 \)
|
| $41$ |
\( T^{6} - 13 T^{5} + \cdots + 176872 \)
|
| $43$ |
\( T^{6} + 38 T^{5} + \cdots + 14204 \)
|
| $47$ |
\( T^{6} - 228 T^{4} + \cdots - 294568 \)
|
| $53$ |
\( T^{6} + 31 T^{5} + \cdots + 9439 \)
|
| $59$ |
\( T^{6} + 6 T^{5} + \cdots + 1952 \)
|
| $61$ |
\( (T - 1)^{6} \)
|
| $67$ |
\( T^{6} + 12 T^{5} + \cdots - 8192 \)
|
| $71$ |
\( T^{6} + 17 T^{5} + \cdots - 122413 \)
|
| $73$ |
\( T^{6} - 7 T^{5} + \cdots + 115578 \)
|
| $79$ |
\( T^{6} + 10 T^{5} + \cdots + 13856 \)
|
| $83$ |
\( T^{6} - 5 T^{5} + \cdots + 2 \)
|
| $89$ |
\( T^{6} - 332 T^{4} + \cdots - 940136 \)
|
| $97$ |
\( T^{6} - 5 T^{5} + \cdots - 275717 \)
|
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