Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8008,2,Mod(1,8008)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8008, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("8008.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8008.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: | \(63.9442019386\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.668973.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 9x^{3} - x^{2} + 7x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{5} - x^{4} - 9x^{3} - x^{2} + 7x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 8\nu^{2} - 3\nu + 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 10\nu^{2} - 10\nu + 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 9\nu^{2} + 2\nu - 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( 3\nu^{4} - \nu^{3} - 29\nu^{2} - 19\nu + 15 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - 2\beta_{2} + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + 4\beta_{3} - 2\beta_{2} + 3\beta _1 + 13 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{4} + 5\beta_{3} - 5\beta_{2} + \beta _1 + 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 20\beta_{4} + 50\beta_{3} - 37\beta_{2} + 30\beta _1 + 125 ) / 3 \)
|
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 |
|
0 | −3.08441 | 0 | −0.239305 | 0 | 1.00000 | 0 | 6.51357 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.830554 | 0 | 3.93077 | 0 | 1.00000 | 0 | −2.31018 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.293231 | 0 | −3.94142 | 0 | 1.00000 | 0 | −2.91402 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.18752 | 0 | −2.05962 | 0 | 1.00000 | 0 | 1.78523 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.43421 | 0 | 1.30958 | 0 | 1.00000 | 0 | 2.92540 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
| \(11\) | \(1\) |
| \(13\) | \(1\) |
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 | 8008.2.a.j | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8008.2.a.j | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):
|
\( T_{3}^{5} - T_{3}^{4} - 10T_{3}^{3} + 12T_{3}^{2} + 11T_{3} - 4 \)
|
|
\( T_{5}^{5} + T_{5}^{4} - 18T_{5}^{3} - 16T_{5}^{2} + 39T_{5} + 10 \)
|
|
\( T_{17}^{5} - 19T_{17}^{4} + 126T_{17}^{3} - 344T_{17}^{2} + 327T_{17} - 18 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - T^{4} - 10 T^{3} + \cdots - 4 \)
|
| $5$ |
\( T^{5} + T^{4} + \cdots + 10 \)
|
| $7$ |
\( (T - 1)^{5} \)
|
| $11$ |
\( (T + 1)^{5} \)
|
| $13$ |
\( (T + 1)^{5} \)
|
| $17$ |
\( T^{5} - 19 T^{4} + \cdots - 18 \)
|
| $19$ |
\( T^{5} + 5 T^{4} + \cdots + 452 \)
|
| $23$ |
\( T^{5} - 5 T^{4} + \cdots + 64 \)
|
| $29$ |
\( T^{5} + 17 T^{4} + \cdots - 11558 \)
|
| $31$ |
\( T^{5} - 4 T^{4} + \cdots - 332 \)
|
| $37$ |
\( T^{5} - 10 T^{4} + \cdots + 1646 \)
|
| $41$ |
\( T^{5} - 10 T^{4} + \cdots - 13050 \)
|
| $43$ |
\( T^{5} + 25 T^{4} + \cdots - 7232 \)
|
| $47$ |
\( T^{5} - 3 T^{4} + \cdots + 1396 \)
|
| $53$ |
\( T^{5} - 22 T^{4} + \cdots + 4022 \)
|
| $59$ |
\( T^{5} - 21 T^{4} + \cdots - 156600 \)
|
| $61$ |
\( T^{5} - 26 T^{4} + \cdots - 1530 \)
|
| $67$ |
\( T^{5} - 28 T^{4} + \cdots + 524 \)
|
| $71$ |
\( T^{5} - 28 T^{4} + \cdots - 4608 \)
|
| $73$ |
\( T^{5} + 4 T^{4} + \cdots + 1494 \)
|
| $79$ |
\( T^{5} + 11 T^{4} + \cdots + 117980 \)
|
| $83$ |
\( T^{5} - 222 T^{3} + \cdots - 13108 \)
|
| $89$ |
\( T^{5} + 37 T^{4} + \cdots - 162 \)
|
| $97$ |
\( T^{5} - 18 T^{4} + \cdots - 302 \)
|
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