Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8037,2,Mod(1,8037)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8037, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("8037.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8037 = 3^{2} \cdot 19 \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8037.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: | \(64.1757681045\) |
| 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: |
\( x^{7} - 2x^{6} - 9x^{5} + 14x^{4} + 23x^{3} - 19x^{2} - 14x + 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2679) |
| 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
\( x^{7} - 2x^{6} - 9x^{5} + 14x^{4} + 23x^{3} - 19x^{2} - 14x + 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 5\nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} - 2\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 7\nu^{3} - \nu^{2} + 6\nu - 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + 7\nu^{3} + 2\nu^{2} - 7\nu - 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 8\nu^{4} - \nu^{3} + 13\nu^{2} - \nu - 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{5} + 6\beta_{4} + \beta_{3} + 8\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 7\beta_{2} + 30\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 35\beta_{5} + 35\beta_{4} + 8\beta_{3} + \beta_{2} + 57\beta _1 + 87 \)
|
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 |
|
−2.26927 | 0 | 3.14959 | 3.56052 | 0 | −0.858342 | −2.60873 | 0 | −8.07978 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.37483 | 0 | −0.109849 | −0.874519 | 0 | −1.13950 | 2.90068 | 0 | 1.20231 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.811152 | 0 | −1.34203 | 2.60918 | 0 | 4.14006 | 2.71090 | 0 | −2.11644 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.289043 | 0 | −1.91645 | −2.68919 | 0 | 0.516308 | −1.13202 | 0 | −0.777292 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.04097 | 0 | −0.916372 | 0.554273 | 0 | 3.51162 | −3.03587 | 0 | 0.576984 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.53902 | 0 | 4.44664 | 3.17758 | 0 | 2.26996 | 6.21208 | 0 | 8.06795 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.58621 | 0 | 4.68848 | −0.337843 | 0 | −1.44011 | 6.95296 | 0 | −0.873733 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(19\) | \(1\) |
| \(47\) | \(1\) |
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 | 8037.2.a.k | 7 | |
| 3.b | odd | 2 | 1 | 2679.2.a.l | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2679.2.a.l | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 8037.2.a.k | 7 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):
|
\( T_{2}^{7} - 2T_{2}^{6} - 9T_{2}^{5} + 14T_{2}^{4} + 23T_{2}^{3} - 19T_{2}^{2} - 14T_{2} + 5 \)
|
|
\( T_{5}^{7} - 6T_{5}^{6} - T_{5}^{5} + 53T_{5}^{4} - 48T_{5}^{3} - 71T_{5}^{2} + 22T_{5} + 13 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 2 T^{6} + \cdots + 5 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} - 6 T^{6} + \cdots + 13 \)
|
| $7$ |
\( T^{7} - 7 T^{6} + \cdots + 24 \)
|
| $11$ |
\( T^{7} - 9 T^{6} + \cdots + 3 \)
|
| $13$ |
\( T^{7} + 2 T^{6} + \cdots + 243 \)
|
| $17$ |
\( T^{7} - 6 T^{6} + \cdots - 3 \)
|
| $19$ |
\( (T + 1)^{7} \)
|
| $23$ |
\( (T - 5)^{7} \)
|
| $29$ |
\( T^{7} - 26 T^{6} + \cdots + 14263 \)
|
| $31$ |
\( T^{7} + 11 T^{6} + \cdots - 26489 \)
|
| $37$ |
\( T^{7} + 3 T^{6} + \cdots + 10795 \)
|
| $41$ |
\( T^{7} - 20 T^{6} + \cdots - 7373 \)
|
| $43$ |
\( T^{7} + 5 T^{6} + \cdots + 25 \)
|
| $47$ |
\( (T + 1)^{7} \)
|
| $53$ |
\( T^{7} + 3 T^{6} + \cdots + 13335 \)
|
| $59$ |
\( T^{7} - 15 T^{6} + \cdots - 12495 \)
|
| $61$ |
\( T^{7} + 9 T^{6} + \cdots + 1835 \)
|
| $67$ |
\( T^{7} + 4 T^{6} + \cdots + 2525 \)
|
| $71$ |
\( T^{7} - 4 T^{6} + \cdots + 2649 \)
|
| $73$ |
\( T^{7} + 8 T^{6} + \cdots - 114181 \)
|
| $79$ |
\( T^{7} + 11 T^{6} + \cdots - 15912 \)
|
| $83$ |
\( T^{7} - 4 T^{6} + \cdots - 233449 \)
|
| $89$ |
\( T^{7} - 23 T^{6} + \cdots + 132445 \)
|
| $97$ |
\( T^{7} + 7 T^{6} + \cdots - 519433 \)
|
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