Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [87,6,Mod(1,87)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(87, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("87.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 87 = 3 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 87.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: | \(13.9533923237\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.8167381.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 31x^{2} + 14x + 160 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 31x^{2} + 14x + 160 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - 17\nu - 26 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} - \nu^{2} + 29\nu + 20 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 5\nu^{2} + 23\nu - 70 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} - \beta_{2} + \beta _1 + 31 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + 9\beta_{2} + 14\beta _1 + 19 \)
|
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 |
|
−10.9064 | 9.00000 | 86.9492 | −12.4731 | −98.1574 | −62.5499 | −599.297 | 81.0000 | 136.036 | ||||||||||||||||||||||||||||||
| 1.2 | −2.07657 | 9.00000 | −27.6879 | −43.9729 | −18.6891 | 237.973 | 123.946 | 81.0000 | 91.3126 | |||||||||||||||||||||||||||||||
| 1.3 | 3.59342 | 9.00000 | −19.0873 | 10.7422 | 32.3408 | −156.200 | −183.578 | 81.0000 | 38.6013 | |||||||||||||||||||||||||||||||
| 1.4 | 6.38952 | 9.00000 | 8.82600 | −90.2962 | 57.5057 | −47.2235 | −148.071 | 81.0000 | −576.950 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(29\) | \(-1\) |
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 | 87.6.a.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 261.6.a.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 87.6.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 261.6.a.b | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 3T_{2}^{3} - 84T_{2}^{2} + 72T_{2} + 520 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(87))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 3 T^{3} + \cdots + 520 \)
|
| $3$ |
\( (T - 9)^{4} \)
|
| $5$ |
\( T^{4} + 136 T^{3} + \cdots - 532012 \)
|
| $7$ |
\( T^{4} + 28 T^{3} + \cdots - 109797559 \)
|
| $11$ |
\( T^{4} + 872 T^{3} + \cdots + 884564356 \)
|
| $13$ |
\( T^{4} + \cdots + 89371959424 \)
|
| $17$ |
\( T^{4} + \cdots - 1122179651541 \)
|
| $19$ |
\( T^{4} + \cdots + 3270497775012 \)
|
| $23$ |
\( T^{4} + \cdots + 2597367582464 \)
|
| $29$ |
\( (T - 841)^{4} \)
|
| $31$ |
\( T^{4} + \cdots - 157683874879872 \)
|
| $37$ |
\( T^{4} + \cdots + 267222741589444 \)
|
| $41$ |
\( T^{4} + \cdots - 12\!\cdots\!32 \)
|
| $43$ |
\( T^{4} + \cdots - 156797028685120 \)
|
| $47$ |
\( T^{4} + \cdots - 18\!\cdots\!09 \)
|
| $53$ |
\( T^{4} + \cdots - 22\!\cdots\!24 \)
|
| $59$ |
\( T^{4} + \cdots - 88\!\cdots\!40 \)
|
| $61$ |
\( T^{4} + \cdots - 23\!\cdots\!40 \)
|
| $67$ |
\( T^{4} + \cdots - 66\!\cdots\!60 \)
|
| $71$ |
\( T^{4} + \cdots + 12\!\cdots\!76 \)
|
| $73$ |
\( T^{4} + \cdots - 11\!\cdots\!96 \)
|
| $79$ |
\( T^{4} + \cdots - 12\!\cdots\!60 \)
|
| $83$ |
\( T^{4} + \cdots - 34\!\cdots\!68 \)
|
| $89$ |
\( T^{4} + \cdots - 21\!\cdots\!40 \)
|
| $97$ |
\( T^{4} + \cdots + 15\!\cdots\!68 \)
|
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