Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4008,2,Mod(1,4008)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4008, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("4008.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4008 = 2^{3} \cdot 3 \cdot 167 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4008.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: | \(32.0040411301\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 7x^{5} + 11x^{4} + 12x^{3} - 14x^{2} - 6x + 3 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 7x^{5} + 11x^{4} + 12x^{3} - 14x^{2} - 6x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{5} + \nu^{4} + 7\nu^{3} - 2\nu^{2} - 8\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 3\nu^{2} + 8\nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{6} + \nu^{5} + 7\nu^{4} - 2\nu^{3} - 9\nu^{2} - \nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 7\nu^{4} + 2\nu^{3} + 9\nu^{2} + 3\nu - 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{6} + \nu^{5} + 8\nu^{4} - 3\nu^{3} - 15\nu^{2} + 6 \)
|
| \(\beta_{6}\) | \(=\) |
\( 2\nu^{6} - 2\nu^{5} - 15\nu^{4} + 6\nu^{3} + 23\nu^{2} - 2\nu - 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{6} + 2\beta_{5} + 3\beta_{4} + 5\beta_{3} + 2\beta_{2} + 2\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{3} + 7\beta_{2} + 7\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 16\beta_{6} + 18\beta_{5} + 15\beta_{4} + 29\beta_{3} + 24\beta_{2} + 22\beta _1 + 4 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 13\beta_{6} + 21\beta_{5} + 11\beta_{4} + 15\beta_{3} + 50\beta_{2} + 49\beta _1 + 34 \)
|
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 | −1.00000 | 0 | −3.20729 | 0 | 3.68779 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | −3.01562 | 0 | −1.84677 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | −1.68509 | 0 | −2.23045 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | −0.597616 | 0 | 2.60994 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.00000 | 0 | 0.0621653 | 0 | 0.782308 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.00000 | 0 | 1.82640 | 0 | 2.26944 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −1.00000 | 0 | 3.61705 | 0 | 2.72774 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(167\) | \(1\) |
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 | 4008.2.a.g | ✓ | 7 |
| 4.b | odd | 2 | 1 | 8016.2.a.w | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4008.2.a.g | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 8016.2.a.w | 7 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):
|
\( T_{5}^{7} + 3T_{5}^{6} - 15T_{5}^{5} - 50T_{5}^{4} + 23T_{5}^{3} + 133T_{5}^{2} + 56T_{5} - 4 \)
|
|
\( T_{7}^{7} - 8T_{7}^{6} + 11T_{7}^{5} + 55T_{7}^{4} - 147T_{7}^{3} - 37T_{7}^{2} + 336T_{7} - 192 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( (T + 1)^{7} \)
|
| $5$ |
\( T^{7} + 3 T^{6} + \cdots - 4 \)
|
| $7$ |
\( T^{7} - 8 T^{6} + \cdots - 192 \)
|
| $11$ |
\( T^{7} - T^{6} + \cdots - 16 \)
|
| $13$ |
\( T^{7} + 2 T^{6} + \cdots + 4 \)
|
| $17$ |
\( T^{7} - 11 T^{6} + \cdots + 4036 \)
|
| $19$ |
\( T^{7} - 2 T^{6} + \cdots - 1712 \)
|
| $23$ |
\( T^{7} - 17 T^{6} + \cdots + 928 \)
|
| $29$ |
\( T^{7} + 7 T^{6} + \cdots - 15036 \)
|
| $31$ |
\( T^{7} - 10 T^{6} + \cdots - 2656 \)
|
| $37$ |
\( T^{7} + 21 T^{6} + \cdots + 252 \)
|
| $41$ |
\( T^{7} - 8 T^{6} + \cdots + 9516 \)
|
| $43$ |
\( T^{7} + 12 T^{6} + \cdots - 399928 \)
|
| $47$ |
\( T^{7} - 25 T^{6} + \cdots + 54336 \)
|
| $53$ |
\( T^{7} + 7 T^{6} + \cdots + 740588 \)
|
| $59$ |
\( T^{7} - 3 T^{6} + \cdots - 1928 \)
|
| $61$ |
\( T^{7} + 14 T^{6} + \cdots + 144508 \)
|
| $67$ |
\( T^{7} - 4 T^{6} + \cdots + 3736 \)
|
| $71$ |
\( T^{7} - 27 T^{6} + \cdots + 3808 \)
|
| $73$ |
\( T^{7} + 12 T^{6} + \cdots + 1354996 \)
|
| $79$ |
\( T^{7} - 8 T^{6} + \cdots + 13328 \)
|
| $83$ |
\( T^{7} - 15 T^{6} + \cdots + 17907304 \)
|
| $89$ |
\( T^{7} - 14 T^{6} + \cdots - 8244 \)
|
| $97$ |
\( T^{7} - 3 T^{6} + \cdots + 19084 \)
|
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