Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3627,2,Mod(1,3627)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3627, 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("3627.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3627 = 3^{2} \cdot 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3627.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: | \(28.9617408131\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{8} - 3x^{7} - 7x^{6} + 19x^{5} + 21x^{4} - 31x^{3} - 29x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 403) |
| 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
\( x^{8} - 3x^{7} - 7x^{6} + 19x^{5} + 21x^{4} - 31x^{3} - 29x^{2} - x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 3\nu^{2} + 9\nu + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 3\nu^{3} + 9\nu^{2} + 3\nu \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} - 4\nu^{6} - 3\nu^{5} + 21\nu^{4} + 3\nu^{3} - 30\nu^{2} - 9\nu + 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 20\nu^{4} + 18\nu^{3} - 34\nu^{2} - 20\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 3\beta_{3} + 6\beta_{2} + 9\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} + 12\beta_{3} + 12\beta_{2} + 30\beta _1 + 27 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} - \beta_{6} + 4\beta_{5} + 13\beta_{4} + 36\beta_{3} + 43\beta_{2} + 69\beta _1 + 90 \)
|
| \(\nu^{7}\) | \(=\) |
\( 4\beta_{7} - 3\beta_{6} + 19\beta_{5} + 40\beta_{4} + 114\beta_{3} + 109\beta_{2} + 201\beta _1 + 205 \)
|
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 |
|
−1.75938 | 0 | 1.09540 | 3.67909 | 0 | 4.14977 | 1.59152 | 0 | −6.47291 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.39479 | 0 | −0.0545724 | 0.0316546 | 0 | −2.20383 | 2.86569 | 0 | −0.0441514 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.933096 | 0 | −1.12933 | 3.66529 | 0 | −3.96149 | 2.91997 | 0 | −3.42006 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.841125 | 0 | −1.29251 | −0.443680 | 0 | 0.552392 | −2.76941 | 0 | −0.373191 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.24762 | 0 | −0.443454 | −0.848757 | 0 | −1.14028 | −3.04849 | 0 | −1.05892 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.75921 | 0 | 1.09483 | 3.30948 | 0 | −3.55859 | −1.59239 | 0 | 5.82207 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.57233 | 0 | 4.61688 | 4.29347 | 0 | 0.180190 | 6.73148 | 0 | 11.0442 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.66697 | 0 | 5.11275 | 1.31345 | 0 | 1.98184 | 8.30164 | 0 | 3.50294 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(13\) | \(1\) |
| \(31\) | \(1\) |
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 | 3627.2.a.q | 8 | |
| 3.b | odd | 2 | 1 | 403.2.a.d | ✓ | 8 | |
| 12.b | even | 2 | 1 | 6448.2.a.bf | 8 | ||
| 39.d | odd | 2 | 1 | 5239.2.a.j | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 403.2.a.d | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 3627.2.a.q | 8 | 1.a | even | 1 | 1 | trivial | |
| 5239.2.a.j | 8 | 39.d | odd | 2 | 1 | ||
| 6448.2.a.bf | 8 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3627))\):
|
\( T_{2}^{8} - 5T_{2}^{7} + 30T_{2}^{5} - 24T_{2}^{4} - 54T_{2}^{3} + 54T_{2}^{2} + 28T_{2} - 29 \)
|
|
\( T_{5}^{8} - 15T_{5}^{7} + 83T_{5}^{6} - 192T_{5}^{5} + 99T_{5}^{4} + 225T_{5}^{3} - 158T_{5}^{2} - 90T_{5} + 3 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 5 T^{7} + \cdots - 29 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 15 T^{7} + \cdots + 3 \)
|
| $7$ |
\( T^{8} + 4 T^{7} + \cdots + 29 \)
|
| $11$ |
\( T^{8} - 5 T^{7} + \cdots - 14580 \)
|
| $13$ |
\( (T + 1)^{8} \)
|
| $17$ |
\( T^{8} - 11 T^{7} + \cdots - 1940 \)
|
| $19$ |
\( T^{8} + 9 T^{7} + \cdots + 3557 \)
|
| $23$ |
\( T^{8} - 86 T^{6} + \cdots + 3452 \)
|
| $29$ |
\( T^{8} - 12 T^{7} + \cdots - 315444 \)
|
| $31$ |
\( (T + 1)^{8} \)
|
| $37$ |
\( T^{8} + 9 T^{7} + \cdots - 852812 \)
|
| $41$ |
\( T^{8} - 25 T^{7} + \cdots + 18859 \)
|
| $43$ |
\( T^{8} - 7 T^{7} + \cdots - 149092 \)
|
| $47$ |
\( T^{8} - 17 T^{7} + \cdots - 120144 \)
|
| $53$ |
\( T^{8} - 15 T^{7} + \cdots - 828 \)
|
| $59$ |
\( T^{8} - 15 T^{7} + \cdots - 1314177 \)
|
| $61$ |
\( T^{8} - 11 T^{7} + \cdots + 493348 \)
|
| $67$ |
\( T^{8} - 18 T^{7} + \cdots + 52672 \)
|
| $71$ |
\( T^{8} - 7 T^{7} + \cdots + 79141 \)
|
| $73$ |
\( T^{8} - 24 T^{7} + \cdots - 173100 \)
|
| $79$ |
\( T^{8} - 33 T^{7} + \cdots - 10460864 \)
|
| $83$ |
\( T^{8} - 13 T^{7} + \cdots + 3798748 \)
|
| $89$ |
\( T^{8} - 23 T^{7} + \cdots - 462276 \)
|
| $97$ |
\( T^{8} + 17 T^{7} + \cdots + 2246337 \)
|
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