Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8722,2,Mod(1,8722)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8722, 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("8722.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8722 = 2 \cdot 7^{2} \cdot 89 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8722.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: | \(69.6455206430\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.135076.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 1246) |
| 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
\( x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 4\nu^{2} - 3\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + 3 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} - \beta_{2} + 2\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{4} + 5\beta_{2} + 5\beta _1 + 14 ) / 2 \)
|
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 | −1.14665 | 1.00000 | 4.24370 | −1.14665 | 0 | 1.00000 | −1.68519 | 4.24370 | |||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −0.283134 | 1.00000 | 1.52242 | −0.283134 | 0 | 1.00000 | −2.91984 | 1.52242 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 1.36831 | 1.00000 | 2.30176 | 1.36831 | 0 | 1.00000 | −1.12773 | 2.30176 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 2.60528 | 1.00000 | −1.72732 | 2.60528 | 0 | 1.00000 | 3.78749 | −1.72732 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 3.45619 | 1.00000 | 3.65944 | 3.45619 | 0 | 1.00000 | 8.94527 | 3.65944 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(89\) | \(-1\) |
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 | 8722.2.a.x | 5 | |
| 7.b | odd | 2 | 1 | 1246.2.a.n | ✓ | 5 | |
| 28.d | even | 2 | 1 | 9968.2.a.z | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1246.2.a.n | ✓ | 5 | 7.b | odd | 2 | 1 | |
| 8722.2.a.x | 5 | 1.a | even | 1 | 1 | trivial | |
| 9968.2.a.z | 5 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8722))\):
|
\( T_{3}^{5} - 6T_{3}^{4} + 7T_{3}^{3} + 10T_{3}^{2} - 12T_{3} - 4 \)
|
|
\( T_{5}^{5} - 10T_{5}^{4} + 29T_{5}^{3} - 2T_{5}^{2} - 96T_{5} + 94 \)
|
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\( T_{11}^{5} + 8T_{11}^{4} - 12T_{11}^{3} - 208T_{11}^{2} - 352T_{11} + 64 \)
|
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\( T_{13}^{5} - 8T_{13}^{4} - 32T_{13}^{3} + 332T_{13}^{2} - 368T_{13} - 176 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{5} \)
|
| $3$ |
\( T^{5} - 6 T^{4} + \cdots - 4 \)
|
| $5$ |
\( T^{5} - 10 T^{4} + \cdots + 94 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} + 8 T^{4} + \cdots + 64 \)
|
| $13$ |
\( T^{5} - 8 T^{4} + \cdots - 176 \)
|
| $17$ |
\( T^{5} - 8 T^{4} + \cdots - 536 \)
|
| $19$ |
\( T^{5} - 10 T^{4} + \cdots - 596 \)
|
| $23$ |
\( T^{5} + 2 T^{4} + \cdots + 808 \)
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| $29$ |
\( T^{5} + 10 T^{4} + \cdots + 1696 \)
|
| $31$ |
\( T^{5} - 10 T^{4} + \cdots + 292 \)
|
| $37$ |
\( T^{5} + 4 T^{4} + \cdots - 808 \)
|
| $41$ |
\( T^{5} - 28 T^{4} + \cdots - 64 \)
|
| $43$ |
\( T^{5} + 10 T^{4} + \cdots + 4334 \)
|
| $47$ |
\( T^{5} - 10 T^{4} + \cdots - 512 \)
|
| $53$ |
\( T^{5} - 4 T^{4} + \cdots - 3928 \)
|
| $59$ |
\( T^{5} - 10 T^{4} + \cdots + 1024 \)
|
| $61$ |
\( T^{5} - 16 T^{4} + \cdots + 16 \)
|
| $67$ |
\( T^{5} - 208 T^{3} + \cdots + 18688 \)
|
| $71$ |
\( T^{5} + 16 T^{4} + \cdots + 11408 \)
|
| $73$ |
\( T^{5} + 10 T^{4} + \cdots + 41504 \)
|
| $79$ |
\( T^{5} + 8 T^{4} + \cdots - 11728 \)
|
| $83$ |
\( T^{5} - 14 T^{4} + \cdots - 23552 \)
|
| $89$ |
\( (T - 1)^{5} \)
|
| $97$ |
\( T^{5} + 6 T^{4} + \cdots + 107344 \)
|
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