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: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 24x^{4} - 10x^{3} - 18x^{2} - 2x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 24x^{4} - 10x^{3} - 18x^{2} - 2x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 10\nu^{2} + 9\nu - 3 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 9\nu^{4} + 12\nu^{3} + 21\nu^{2} - 9\nu - 6 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 2\nu^{6} - 9\nu^{5} + 12\nu^{4} + 23\nu^{3} - 9\nu^{2} - 12\nu - 2 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{7} - 4\nu^{6} - 18\nu^{5} + 25\nu^{4} + 47\nu^{3} - 26\nu^{2} - 34\nu - 1 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{7} + 8\nu^{6} + 22\nu^{5} - 51\nu^{4} - 41\nu^{3} + 59\nu^{2} + 19\nu - 8 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{7} + 8\nu^{6} + 22\nu^{5} - 50\nu^{4} - 42\nu^{3} + 51\nu^{2} + 27\nu - 1 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -4\nu^{7} + 10\nu^{6} + 31\nu^{5} - 64\nu^{4} - 63\nu^{3} + 76\nu^{2} + 31\nu - 15 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} - 2\beta_{5} + \beta_{3} + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{7} - \beta_{5} + \beta_{4} + 3\beta_{3} - 2\beta_{2} + \beta _1 + 10 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 9\beta_{7} + 5\beta_{6} - 14\beta_{5} + 4\beta_{4} + \beta_{3} + 17 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 17\beta_{7} + 5\beta_{6} - 14\beta_{5} + 12\beta_{4} + 17\beta_{3} - 16\beta_{2} + 8\beta _1 + 61 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 34\beta_{7} + 18\beta_{6} - 49\beta_{5} + 21\beta_{4} + \beta_{3} - 6\beta_{2} + 7\beta _1 + 72 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 148\beta_{7} + 66\beta_{6} - 151\beta_{5} + 123\beta_{4} + 91\beta_{3} - 118\beta_{2} + 79\beta _1 + 456 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 527\beta_{7} + 293\beta_{6} - 727\beta_{5} + 397\beta_{4} + 20\beta_{3} - 170\beta_{2} + 197\beta _1 + 1195 ) / 4 \)
|
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 | −3.17170 | 0 | −2.42750 | 0 | 1.00000 | 0 | 7.05967 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.59521 | 0 | 0.486965 | 0 | 1.00000 | 0 | 3.73512 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.02769 | 0 | 2.88924 | 0 | 1.00000 | 0 | 1.11151 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.867389 | 0 | −3.42583 | 0 | 1.00000 | 0 | −2.24764 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.837498 | 0 | −2.01769 | 0 | 1.00000 | 0 | −2.29860 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.425730 | 0 | 2.69163 | 0 | 1.00000 | 0 | −2.81875 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.75319 | 0 | −1.19008 | 0 | 1.00000 | 0 | 0.0736754 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.32056 | 0 | −4.00673 | 0 | 1.00000 | 0 | 2.38501 | 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.l | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8008.2.a.l | ✓ | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):
|
\( T_{3}^{8} + 5T_{3}^{7} - 3T_{3}^{6} - 43T_{3}^{5} - 30T_{3}^{4} + 87T_{3}^{3} + 97T_{3}^{2} - 4T_{3} - 21 \)
|
|
\( T_{5}^{8} + 7T_{5}^{7} - 3T_{5}^{6} - 107T_{5}^{5} - 140T_{5}^{4} + 387T_{5}^{3} + 835T_{5}^{2} + 146T_{5} - 303 \)
|
|
\( T_{17}^{8} - 3T_{17}^{7} - 43T_{17}^{6} + 105T_{17}^{5} + 596T_{17}^{4} - 997T_{17}^{3} - 2773T_{17}^{2} + 2052T_{17} + 729 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 5 T^{7} + \cdots - 21 \)
|
| $5$ |
\( T^{8} + 7 T^{7} + \cdots - 303 \)
|
| $7$ |
\( (T - 1)^{8} \)
|
| $11$ |
\( (T - 1)^{8} \)
|
| $13$ |
\( (T - 1)^{8} \)
|
| $17$ |
\( T^{8} - 3 T^{7} + \cdots + 729 \)
|
| $19$ |
\( T^{8} + 19 T^{7} + \cdots + 31333 \)
|
| $23$ |
\( T^{8} - T^{7} + \cdots - 12176 \)
|
| $29$ |
\( T^{8} + 21 T^{7} + \cdots + 551568 \)
|
| $31$ |
\( T^{8} - 2 T^{7} + \cdots + 411088 \)
|
| $37$ |
\( T^{8} + 12 T^{7} + \cdots - 380048 \)
|
| $41$ |
\( T^{8} + 6 T^{7} + \cdots + 198544 \)
|
| $43$ |
\( T^{8} + 19 T^{7} + \cdots + 716063 \)
|
| $47$ |
\( T^{8} + 7 T^{7} + \cdots + 7632 \)
|
| $53$ |
\( T^{8} + 26 T^{7} + \cdots + 51687 \)
|
| $59$ |
\( T^{8} + 17 T^{7} + \cdots - 6400 \)
|
| $61$ |
\( T^{8} - 371 T^{6} + \cdots + 24798519 \)
|
| $67$ |
\( T^{8} + 24 T^{7} + \cdots - 57137 \)
|
| $71$ |
\( T^{8} - 2 T^{7} + \cdots - 21456 \)
|
| $73$ |
\( T^{8} - 2 T^{7} + \cdots + 7209936 \)
|
| $79$ |
\( T^{8} - 7 T^{7} + \cdots + 238457 \)
|
| $83$ |
\( T^{8} + 6 T^{7} + \cdots + 11321 \)
|
| $89$ |
\( T^{8} - 13 T^{7} + \cdots + 2919537 \)
|
| $97$ |
\( T^{8} - 18 T^{7} + \cdots - 820368 \)
|
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