Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8281,2,Mod(1,8281)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8281, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("8281.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8281 = 7^{2} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8281.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: | \(66.1241179138\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.6995813.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 6x^{4} + 4x^{3} + 7x^{2} - x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 91) |
| 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
\( x^{6} - x^{5} - 6x^{4} + 4x^{3} + 7x^{2} - x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 5\nu^{3} + 4\nu^{2} + 2\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 6\nu^{3} + 4\nu^{2} + 7\nu - 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + 2\nu^{4} + 5\nu^{3} - 9\nu^{2} - 3\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{3} + 5\beta_{2} + \beta _1 + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 5\beta_{4} + 7\beta_{3} + \beta_{2} + 24\beta _1 + 1 \)
|
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 |
|
−1.85816 | −2.29407 | 1.45276 | −0.197362 | 4.26275 | 0 | 1.01686 | 2.26275 | 0.366731 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.55469 | 0.489252 | 0.417051 | −1.19151 | −0.760633 | 0 | 2.46099 | −2.76063 | 1.85243 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.268125 | 1.14301 | −1.92811 | −2.56175 | 0.306470 | 0 | −1.05323 | −1.69353 | −0.686871 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.851125 | −0.661223 | −1.27559 | 3.44148 | −0.562784 | 0 | −2.78793 | −2.56278 | 2.92913 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.90556 | −0.428448 | 1.63116 | −1.47313 | −0.816433 | 0 | −0.702849 | −2.81643 | −2.80714 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.38804 | 2.75148 | 3.70272 | 0.982280 | 6.57063 | 0 | 4.06616 | 4.57063 | 2.34572 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(13\) | \(1\) |
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 | 8281.2.a.cf | 6 | |
| 7.b | odd | 2 | 1 | 8281.2.a.ce | 6 | ||
| 7.d | odd | 6 | 2 | 1183.2.e.g | 12 | ||
| 13.b | even | 2 | 1 | 8281.2.a.ca | 6 | ||
| 13.e | even | 6 | 2 | 637.2.f.j | 12 | ||
| 91.b | odd | 2 | 1 | 8281.2.a.bz | 6 | ||
| 91.k | even | 6 | 2 | 637.2.h.l | 12 | ||
| 91.l | odd | 6 | 2 | 91.2.h.b | yes | 12 | |
| 91.p | odd | 6 | 2 | 91.2.g.b | ✓ | 12 | |
| 91.s | odd | 6 | 2 | 1183.2.e.h | 12 | ||
| 91.t | odd | 6 | 2 | 637.2.f.k | 12 | ||
| 91.u | even | 6 | 2 | 637.2.g.l | 12 | ||
| 273.y | even | 6 | 2 | 819.2.n.d | 12 | ||
| 273.br | even | 6 | 2 | 819.2.s.d | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.2.g.b | ✓ | 12 | 91.p | odd | 6 | 2 | |
| 91.2.h.b | yes | 12 | 91.l | odd | 6 | 2 | |
| 637.2.f.j | 12 | 13.e | even | 6 | 2 | ||
| 637.2.f.k | 12 | 91.t | odd | 6 | 2 | ||
| 637.2.g.l | 12 | 91.u | even | 6 | 2 | ||
| 637.2.h.l | 12 | 91.k | even | 6 | 2 | ||
| 819.2.n.d | 12 | 273.y | even | 6 | 2 | ||
| 819.2.s.d | 12 | 273.br | even | 6 | 2 | ||
| 1183.2.e.g | 12 | 7.d | odd | 6 | 2 | ||
| 1183.2.e.h | 12 | 91.s | odd | 6 | 2 | ||
| 8281.2.a.bz | 6 | 91.b | odd | 2 | 1 | ||
| 8281.2.a.ca | 6 | 13.b | even | 2 | 1 | ||
| 8281.2.a.ce | 6 | 7.b | odd | 2 | 1 | ||
| 8281.2.a.cf | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):
|
\( T_{2}^{6} - 2T_{2}^{5} - 6T_{2}^{4} + 11T_{2}^{3} + 8T_{2}^{2} - 14T_{2} + 3 \)
|
|
\( T_{3}^{6} - T_{3}^{5} - 7T_{3}^{4} + 4T_{3}^{3} + 6T_{3}^{2} - T_{3} - 1 \)
|
|
\( T_{5}^{6} + T_{5}^{5} - 11T_{5}^{4} - 18T_{5}^{3} + 6T_{5}^{2} + 17T_{5} + 3 \)
|
|
\( T_{11}^{6} - 4T_{11}^{5} - 21T_{11}^{4} + 76T_{11}^{3} + 81T_{11}^{2} - 207T_{11} + 81 \)
|
|
\( T_{17}^{6} - 5T_{17}^{5} - 12T_{17}^{4} + 14T_{17}^{3} + 20T_{17}^{2} - 8T_{17} - 9 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 2 T^{5} + \cdots + 3 \)
|
| $3$ |
\( T^{6} - T^{5} - 7 T^{4} + \cdots - 1 \)
|
| $5$ |
\( T^{6} + T^{5} - 11 T^{4} + \cdots + 3 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} - 4 T^{5} + \cdots + 81 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} - 5 T^{5} + \cdots - 9 \)
|
| $19$ |
\( T^{6} - T^{5} + \cdots + 873 \)
|
| $23$ |
\( T^{6} - T^{5} + \cdots - 24387 \)
|
| $29$ |
\( T^{6} + 3 T^{5} + \cdots - 201 \)
|
| $31$ |
\( T^{6} + 16 T^{5} + \cdots - 2477 \)
|
| $37$ |
\( T^{6} + 13 T^{5} + \cdots - 13477 \)
|
| $41$ |
\( T^{6} - 8 T^{5} + \cdots + 2043 \)
|
| $43$ |
\( T^{6} - 11 T^{5} + \cdots + 37 \)
|
| $47$ |
\( T^{6} - T^{5} + \cdots - 17847 \)
|
| $53$ |
\( T^{6} - 2 T^{5} + \cdots - 69 \)
|
| $59$ |
\( T^{6} + 13 T^{5} + \cdots + 9123 \)
|
| $61$ |
\( T^{6} + 5 T^{5} + \cdots + 32481 \)
|
| $67$ |
\( T^{6} + 11 T^{5} + \cdots - 16623 \)
|
| $71$ |
\( T^{6} - 6 T^{5} + \cdots + 23043 \)
|
| $73$ |
\( T^{6} - 30 T^{5} + \cdots - 14029 \)
|
| $79$ |
\( T^{6} + 7 T^{5} + \cdots + 10529 \)
|
| $83$ |
\( T^{6} + 27 T^{5} + \cdots + 2673 \)
|
| $89$ |
\( T^{6} + 4 T^{5} + \cdots - 304479 \)
|
| $97$ |
\( T^{6} - 35 T^{5} + \cdots - 3899 \)
|
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