Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2583,2,Mod(1,2583)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2583, 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("2583.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2583 = 3^{2} \cdot 7 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2583.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: | \(20.6253588421\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{7} - 3x^{6} - 6x^{5} + 16x^{4} + 14x^{3} - 20x^{2} - 10x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 861) |
| 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
\( x^{7} - 3x^{6} - 6x^{5} + 16x^{4} + 14x^{3} - 20x^{2} - 10x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 3\nu^{5} - 4\nu^{4} + 12\nu^{3} + 4\nu^{2} - 8\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 3\nu^{5} - 6\nu^{4} + 16\nu^{3} + 12\nu^{2} - 16\nu - 4 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} + 5\nu^{5} - 2\nu^{4} - 18\nu^{3} + 14\nu^{2} + 14\nu - 6 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 2\beta_{3} + 8\beta_{2} + 10\beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} - 3\beta_{5} + 4\beta_{4} + 9\beta_{3} + 21\beta_{2} + 33\beta _1 + 33 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3\beta_{6} - 13\beta_{5} + 18\beta_{4} + 23\beta_{3} + 67\beta_{2} + 83\beta _1 + 115 \)
|
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 |
|
−2.69463 | 0 | 5.26102 | 2.08082 | 0 | 1.00000 | −8.78724 | 0 | −5.60704 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.32604 | 0 | 3.41045 | −0.0978008 | 0 | 1.00000 | −3.28076 | 0 | 0.227488 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.76298 | 0 | 1.10809 | −3.51322 | 0 | 1.00000 | 1.57241 | 0 | 6.19372 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.718443 | 0 | −1.48384 | 3.61951 | 0 | 1.00000 | 2.50294 | 0 | −2.60041 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.190092 | 0 | −1.96387 | −2.28332 | 0 | 1.00000 | −0.753499 | 0 | −0.434041 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.35311 | 0 | −0.169102 | −1.31916 | 0 | 1.00000 | −2.93503 | 0 | −1.78497 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.95889 | 0 | 1.83724 | 0.513172 | 0 | 1.00000 | −0.318823 | 0 | 1.00525 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(41\) | \(1\) |
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 | 2583.2.a.u | 7 | |
| 3.b | odd | 2 | 1 | 861.2.a.m | ✓ | 7 | |
| 21.c | even | 2 | 1 | 6027.2.a.y | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 861.2.a.m | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 2583.2.a.u | 7 | 1.a | even | 1 | 1 | trivial | |
| 6027.2.a.y | 7 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2583))\):
|
\( T_{2}^{7} + 4T_{2}^{6} - 3T_{2}^{5} - 24T_{2}^{4} - 7T_{2}^{3} + 34T_{2}^{2} + 15T_{2} - 4 \)
|
|
\( T_{5}^{7} + T_{5}^{6} - 18T_{5}^{5} - 18T_{5}^{4} + 69T_{5}^{3} + 57T_{5}^{2} - 36T_{5} - 4 \)
|
|
\( T_{11}^{7} + 11T_{11}^{6} + 16T_{11}^{5} - 176T_{11}^{4} - 592T_{11}^{3} - 208T_{11}^{2} + 464T_{11} - 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 4 T^{6} + \cdots - 4 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} + T^{6} - 18 T^{5} + \cdots - 4 \)
|
| $7$ |
\( (T - 1)^{7} \)
|
| $11$ |
\( T^{7} + 11 T^{6} + \cdots - 64 \)
|
| $13$ |
\( T^{7} + 7 T^{6} + \cdots + 72 \)
|
| $17$ |
\( T^{7} + 11 T^{6} + \cdots + 2168 \)
|
| $19$ |
\( T^{7} - 4 T^{6} + \cdots - 1440 \)
|
| $23$ |
\( T^{7} + 7 T^{6} + \cdots - 30592 \)
|
| $29$ |
\( T^{7} + 4 T^{6} + \cdots - 10 \)
|
| $31$ |
\( T^{7} + 7 T^{6} + \cdots - 288 \)
|
| $37$ |
\( T^{7} - 141 T^{5} + \cdots + 32738 \)
|
| $41$ |
\( (T + 1)^{7} \)
|
| $43$ |
\( T^{7} - T^{6} + \cdots - 7632 \)
|
| $47$ |
\( T^{7} + 14 T^{6} + \cdots + 3632 \)
|
| $53$ |
\( T^{7} + 23 T^{6} + \cdots - 102708 \)
|
| $59$ |
\( T^{7} + 8 T^{6} + \cdots + 320 \)
|
| $61$ |
\( T^{7} + 3 T^{6} + \cdots - 39128 \)
|
| $67$ |
\( T^{7} - 3 T^{6} + \cdots + 227592 \)
|
| $71$ |
\( T^{7} + 7 T^{6} + \cdots + 33696 \)
|
| $73$ |
\( T^{7} + 11 T^{6} + \cdots + 1448 \)
|
| $79$ |
\( T^{7} + T^{6} + \cdots - 89440 \)
|
| $83$ |
\( T^{7} - 388 T^{5} + \cdots - 79616 \)
|
| $89$ |
\( T^{7} + 32 T^{6} + \cdots + 659520 \)
|
| $97$ |
\( T^{7} + 25 T^{6} + \cdots - 783908 \)
|
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