Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6162,2,Mod(1,6162)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6162, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6162.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6162 = 2 \cdot 3 \cdot 13 \cdot 79 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6162.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: | \(49.2038177255\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.973904.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 2\nu - 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{4} + 4\nu^{3} + 2\nu^{2} - 12\nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 4\nu^{2} + 9\nu + 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + 3\nu^{3} + 5\nu^{2} - 9\nu - 10 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{4} + 9\beta_{3} + 2\beta_{2} - 3\beta _1 + 15 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 10\beta_{4} + 14\beta_{3} + 3\beta_{2} + 28 \)
|
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 | 1.00000 | 1.00000 | −3.91009 | 1.00000 | 2.18114 | 1.00000 | 1.00000 | −3.91009 | |||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 1.00000 | 1.00000 | −2.69675 | 1.00000 | 4.17082 | 1.00000 | 1.00000 | −2.69675 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 1.00000 | 1.00000 | −2.25774 | 1.00000 | −1.46588 | 1.00000 | 1.00000 | −2.25774 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.00000 | 1.00000 | 1.08699 | 1.00000 | −4.24209 | 1.00000 | 1.00000 | 1.08699 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.00000 | 1.00000 | 1.77758 | 1.00000 | −1.64399 | 1.00000 | 1.00000 | 1.77758 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(13\) | \( +1 \) |
| \(79\) | \( +1 \) |
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 | 6162.2.a.bd | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6162.2.a.bd | ✓ | 5 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6162))\):
|
\( T_{5}^{5} + 6T_{5}^{4} + 2T_{5}^{3} - 32T_{5}^{2} - 19T_{5} + 46 \)
|
|
\( T_{7}^{5} + T_{7}^{4} - 22T_{7}^{3} - 22T_{7}^{2} + 77T_{7} + 93 \)
|
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\( T_{11}^{5} + 10T_{11}^{4} + 22T_{11}^{3} - 24T_{11}^{2} - 47T_{11} + 6 \)
|
|
\( T_{17}^{5} + 5T_{17}^{4} - 40T_{17}^{3} - 116T_{17}^{2} + 440T_{17} - 92 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{5} \)
|
| $3$ |
\( (T - 1)^{5} \)
|
| $5$ |
\( T^{5} + 6 T^{4} + \cdots + 46 \)
|
| $7$ |
\( T^{5} + T^{4} + \cdots + 93 \)
|
| $11$ |
\( T^{5} + 10 T^{4} + \cdots + 6 \)
|
| $13$ |
\( (T + 1)^{5} \)
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| $17$ |
\( T^{5} + 5 T^{4} + \cdots - 92 \)
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| $19$ |
\( T^{5} + 15 T^{4} + \cdots - 16 \)
|
| $23$ |
\( T^{5} + 16 T^{4} + \cdots - 1236 \)
|
| $29$ |
\( T^{5} - 2 T^{4} + \cdots - 4092 \)
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| $31$ |
\( T^{5} - 10 T^{4} + \cdots + 88 \)
|
| $37$ |
\( T^{5} - 4 T^{4} + \cdots + 3072 \)
|
| $41$ |
\( T^{5} + 28 T^{4} + \cdots + 786 \)
|
| $43$ |
\( T^{5} - 11 T^{4} + \cdots + 29 \)
|
| $47$ |
\( T^{5} + 18 T^{4} + \cdots + 37728 \)
|
| $53$ |
\( T^{5} + 14 T^{4} + \cdots + 6092 \)
|
| $59$ |
\( T^{5} + 37 T^{4} + \cdots - 26944 \)
|
| $61$ |
\( T^{5} + 23 T^{4} + \cdots - 11488 \)
|
| $67$ |
\( T^{5} - 5 T^{4} + \cdots - 99961 \)
|
| $71$ |
\( T^{5} - 4 T^{4} + \cdots + 17118 \)
|
| $73$ |
\( T^{5} + 22 T^{4} + \cdots - 12968 \)
|
| $79$ |
\( (T + 1)^{5} \)
|
| $83$ |
\( T^{5} + 6 T^{4} + \cdots + 17378 \)
|
| $89$ |
\( T^{5} + 19 T^{4} + \cdots + 1004 \)
|
| $97$ |
\( T^{5} + 36 T^{4} + \cdots - 42816 \)
|
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