Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6889,2,Mod(1,6889)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6889, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6889.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6889 = 83^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6889.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: | \(55.0089419525\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.9059636.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 8x^{4} + 11x^{3} + 4x^{2} - 4x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 83) |
| 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
\( x^{6} - x^{5} - 8x^{4} + 11x^{3} + 4x^{2} - 4x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 4\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 12\nu^{2} - 2\nu - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} + \nu^{4} + 8\nu^{3} - 11\nu^{2} - 4\nu + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( -2\nu^{5} + 2\nu^{4} + 15\nu^{3} - 22\nu^{2} - \nu + 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{3} - \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + 2\beta_{4} + 7\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{5} - 7\beta_{4} + 7\beta_{3} + \beta_{2} - 11\beta _1 + 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( -12\beta_{5} + 19\beta_{4} - 4\beta_{3} + \beta_{2} + 52\beta _1 - 27 \)
|
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 |
|
−2.62359 | −1.49589 | 4.88322 | 3.47366 | 3.92460 | −1.39854 | −7.56440 | −0.762314 | −9.11346 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.59835 | −1.32239 | 0.554733 | −4.05060 | 2.11364 | −2.22837 | 2.31005 | −1.25129 | 6.47429 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.16417 | 1.13227 | −0.644717 | 1.45742 | −1.31815 | 3.35950 | 3.07889 | −1.71797 | −1.69668 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.429349 | 2.76735 | −1.81566 | −1.71413 | 1.18816 | −3.46533 | −1.63825 | 4.65821 | −0.735961 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.66658 | −2.04941 | 0.777479 | −2.79494 | −3.41549 | 3.61008 | −2.03743 | 1.20006 | −4.65798 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.29018 | 1.96807 | 3.24494 | 1.62860 | 4.50724 | 3.12266 | 2.85114 | 0.873293 | 3.72978 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(83\) | \(-1\) |
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 | 6889.2.a.e | 6 | |
| 83.b | odd | 2 | 1 | 83.2.a.b | ✓ | 6 | |
| 249.d | even | 2 | 1 | 747.2.a.j | 6 | ||
| 332.b | even | 2 | 1 | 1328.2.a.l | 6 | ||
| 415.d | odd | 2 | 1 | 2075.2.a.g | 6 | ||
| 581.b | even | 2 | 1 | 4067.2.a.d | 6 | ||
| 664.e | odd | 2 | 1 | 5312.2.a.bn | 6 | ||
| 664.h | even | 2 | 1 | 5312.2.a.bo | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 83.2.a.b | ✓ | 6 | 83.b | odd | 2 | 1 | |
| 747.2.a.j | 6 | 249.d | even | 2 | 1 | ||
| 1328.2.a.l | 6 | 332.b | even | 2 | 1 | ||
| 2075.2.a.g | 6 | 415.d | odd | 2 | 1 | ||
| 4067.2.a.d | 6 | 581.b | even | 2 | 1 | ||
| 5312.2.a.bn | 6 | 664.e | odd | 2 | 1 | ||
| 5312.2.a.bo | 6 | 664.h | even | 2 | 1 | ||
| 6889.2.a.e | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + T_{2}^{5} - 9T_{2}^{4} - 7T_{2}^{3} + 20T_{2}^{2} + 12T_{2} - 8 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6889))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 9 T^{4} + \cdots - 8 \)
|
| $3$ |
\( T^{6} - T^{5} + \cdots - 25 \)
|
| $5$ |
\( T^{6} + 2 T^{5} + \cdots - 160 \)
|
| $7$ |
\( T^{6} - 3 T^{5} + \cdots - 409 \)
|
| $11$ |
\( T^{6} + 3 T^{5} + \cdots - 113 \)
|
| $13$ |
\( T^{6} + 14 T^{5} + \cdots + 992 \)
|
| $17$ |
\( T^{6} + 5 T^{5} + \cdots - 275 \)
|
| $19$ |
\( T^{6} - 4 T^{5} + \cdots + 6176 \)
|
| $23$ |
\( T^{6} + 5 T^{5} + \cdots + 10912 \)
|
| $29$ |
\( T^{6} + T^{5} + \cdots - 55 \)
|
| $31$ |
\( T^{6} - 3 T^{5} + \cdots - 313 \)
|
| $37$ |
\( T^{6} - 39 T^{5} + \cdots - 91499 \)
|
| $41$ |
\( T^{6} + T^{5} + \cdots - 248 \)
|
| $43$ |
\( T^{6} - 8 T^{5} + \cdots + 6400 \)
|
| $47$ |
\( T^{6} - 12 T^{5} + \cdots + 25952 \)
|
| $53$ |
\( T^{6} + 14 T^{5} + \cdots - 64 \)
|
| $59$ |
\( T^{6} + 17 T^{5} + \cdots + 3527 \)
|
| $61$ |
\( T^{6} + 5 T^{5} + \cdots - 47347 \)
|
| $67$ |
\( T^{6} + 16 T^{5} + \cdots + 264256 \)
|
| $71$ |
\( T^{6} - 26 T^{5} + \cdots + 7232 \)
|
| $73$ |
\( T^{6} - 6 T^{5} + \cdots - 39136 \)
|
| $79$ |
\( T^{6} - 12 T^{5} + \cdots - 160 \)
|
| $83$ |
\( T^{6} \)
|
| $89$ |
\( T^{6} - 22 T^{5} + \cdots + 144896 \)
|
| $97$ |
\( T^{6} + 6 T^{5} + \cdots - 101120 \)
|
| show more | |
| show less |