Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6039,2,Mod(1,6039)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6039, 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("6039.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6039 = 3^{2} \cdot 11 \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6039.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: | \(48.2216577807\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.2661761.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 6x^{4} + 3x^{3} + 9x^{2} - x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 671) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 6x^{4} + 3x^{3} + 9x^{2} - x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 5\nu + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 5\nu^{3} + 2\nu^{2} + 5\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 5\beta_{2} + 6\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 7\beta_{3} + 8\beta_{2} + 19\beta _1 + 8 \)
|
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 |
|
−1.78997 | 0 | 1.20400 | 3.29725 | 0 | −1.30328 | 1.42482 | 0 | −5.90199 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.54773 | 0 | 0.395474 | −0.422675 | 0 | 1.36588 | 2.48338 | 0 | 0.654188 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.782747 | 0 | −1.38731 | 0.742408 | 0 | −2.34697 | 2.65140 | 0 | −0.581118 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.255699 | 0 | −1.93462 | −2.57819 | 0 | −0.612130 | −1.00608 | 0 | −0.659240 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.57580 | 0 | 0.483130 | 0.593176 | 0 | −2.68584 | −2.39028 | 0 | 0.934724 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.28896 | 0 | 3.23932 | −0.631975 | 0 | 0.582341 | 2.83675 | 0 | −1.44656 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(11\) | \(-1\) |
| \(61\) | \(1\) |
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 | 6039.2.a.b | 6 | |
| 3.b | odd | 2 | 1 | 671.2.a.b | ✓ | 6 | |
| 33.d | even | 2 | 1 | 7381.2.a.h | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 671.2.a.b | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 6039.2.a.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 7381.2.a.h | 6 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 7T_{2}^{4} - 2T_{2}^{3} + 12T_{2}^{2} + 5T_{2} - 2 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 7 T^{4} + \cdots - 2 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - T^{5} - 9 T^{4} + \cdots - 1 \)
|
| $7$ |
\( T^{6} + 5 T^{5} + \cdots + 4 \)
|
| $11$ |
\( (T - 1)^{6} \)
|
| $13$ |
\( T^{6} + 4 T^{5} + \cdots + 2 \)
|
| $17$ |
\( T^{6} - 5 T^{5} + \cdots - 926 \)
|
| $19$ |
\( T^{6} + 3 T^{5} + \cdots - 2 \)
|
| $23$ |
\( T^{6} - 3 T^{5} + \cdots + 131 \)
|
| $29$ |
\( T^{6} - T^{5} - 9 T^{4} + \cdots - 2 \)
|
| $31$ |
\( T^{6} + 10 T^{5} + \cdots + 7 \)
|
| $37$ |
\( T^{6} + 19 T^{5} + \cdots - 2993 \)
|
| $41$ |
\( T^{6} - 7 T^{5} + \cdots + 724 \)
|
| $43$ |
\( T^{6} + 2 T^{5} + \cdots - 2 \)
|
| $47$ |
\( T^{6} + 5 T^{5} + \cdots - 26800 \)
|
| $53$ |
\( T^{6} - 9 T^{5} + \cdots - 1714 \)
|
| $59$ |
\( T^{6} - 5 T^{5} + \cdots - 4793 \)
|
| $61$ |
\( (T + 1)^{6} \)
|
| $67$ |
\( T^{6} + 14 T^{5} + \cdots - 27503 \)
|
| $71$ |
\( T^{6} - 14 T^{5} + \cdots - 10877 \)
|
| $73$ |
\( T^{6} + 14 T^{5} + \cdots + 467006 \)
|
| $79$ |
\( T^{6} - 5 T^{5} + \cdots - 80764 \)
|
| $83$ |
\( T^{6} + 17 T^{5} + \cdots - 59872 \)
|
| $89$ |
\( T^{6} - 25 T^{5} + \cdots - 547 \)
|
| $97$ |
\( T^{6} + 24 T^{5} + \cdots + 24269 \)
|
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