Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6256,2,Mod(1,6256)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6256, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6256.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6256 = 2^{4} \cdot 17 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6256.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: | \(49.9544115045\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{8} - 4x^{7} - 5x^{6} + 28x^{5} + 7x^{4} - 60x^{3} + 36x - 7 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 3128) |
| 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
\( x^{8} - 4x^{7} - 5x^{6} + 28x^{5} + 7x^{4} - 60x^{3} + 36x - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 4\nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + 5\nu^{6} - 24\nu^{4} + 9\nu^{3} + 31\nu^{2} - 7\nu - 7 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 5\nu^{6} + 2\nu^{5} + 20\nu^{4} - 21\nu^{3} - 15\nu^{2} + 23\nu - 5 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{6} - 4\nu^{5} - \nu^{4} + 16\nu^{3} - 7\nu^{2} - 14\nu + 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 8\beta_{2} + 10\beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 2\beta_{4} + 10\beta_{3} + 20\beta_{2} + 34\beta _1 + 32 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} + 4\beta_{6} + 4\beta_{5} + 9\beta_{4} + 26\beta_{3} + 63\beta_{2} + 87\beta _1 + 110 \)
|
| \(\nu^{7}\) | \(=\) |
\( 5\beta_{7} + 20\beta_{6} + 18\beta_{5} + 21\beta_{4} + 91\beta_{3} + 172\beta_{2} + 264\beta _1 + 279 \)
|
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 |
|
0 | −2.46321 | 0 | −0.636094 | 0 | 0.158335 | 0 | 3.06743 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.29480 | 0 | −3.16531 | 0 | 3.00304 | 0 | 2.26612 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.79210 | 0 | 2.47517 | 0 | 4.93872 | 0 | 0.211605 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.257502 | 0 | 1.72654 | 0 | 0.584070 | 0 | −2.93369 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.369837 | 0 | −3.12932 | 0 | 1.77233 | 0 | −2.86322 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.26686 | 0 | 1.10318 | 0 | 1.24406 | 0 | −1.39508 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.21289 | 0 | 1.45444 | 0 | −4.07876 | 0 | 1.89690 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.95803 | 0 | −1.82860 | 0 | 1.37821 | 0 | 5.74994 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(17\) | \( -1 \) |
| \(23\) | \( -1 \) |
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 | 6256.2.a.ba | 8 | |
| 4.b | odd | 2 | 1 | 3128.2.a.e | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3128.2.a.e | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 6256.2.a.ba | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6256))\):
|
\( T_{3}^{8} - 15T_{3}^{6} - 2T_{3}^{5} + 68T_{3}^{4} + 8T_{3}^{3} - 92T_{3}^{2} + 8T_{3} + 8 \)
|
|
\( T_{5}^{8} + 2T_{5}^{7} - 16T_{5}^{6} - 20T_{5}^{5} + 91T_{5}^{4} + 40T_{5}^{3} - 187T_{5}^{2} + 14T_{5} + 79 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 15 T^{6} + \cdots + 8 \)
|
| $5$ |
\( T^{8} + 2 T^{7} + \cdots + 79 \)
|
| $7$ |
\( T^{8} - 9 T^{7} + \cdots - 17 \)
|
| $11$ |
\( T^{8} + 7 T^{7} + \cdots - 143 \)
|
| $13$ |
\( T^{8} + 16 T^{7} + \cdots + 769 \)
|
| $17$ |
\( (T - 1)^{8} \)
|
| $19$ |
\( T^{8} + 6 T^{7} + \cdots + 1240 \)
|
| $23$ |
\( (T - 1)^{8} \)
|
| $29$ |
\( T^{8} + 16 T^{7} + \cdots - 303640 \)
|
| $31$ |
\( T^{8} + T^{7} + \cdots - 61544 \)
|
| $37$ |
\( T^{8} + 25 T^{7} + \cdots + 127721 \)
|
| $41$ |
\( T^{8} - 6 T^{7} + \cdots + 7816 \)
|
| $43$ |
\( T^{8} + 4 T^{7} + \cdots - 9656 \)
|
| $47$ |
\( T^{8} - 13 T^{7} + \cdots - 38605 \)
|
| $53$ |
\( T^{8} + 32 T^{7} + \cdots + 20127520 \)
|
| $59$ |
\( T^{8} - 11 T^{7} + \cdots + 9479 \)
|
| $61$ |
\( T^{8} + 23 T^{7} + \cdots + 3224179 \)
|
| $67$ |
\( T^{8} + 17 T^{7} + \cdots + 28136 \)
|
| $71$ |
\( T^{8} - 9 T^{7} + \cdots + 5817896 \)
|
| $73$ |
\( T^{8} + T^{7} + \cdots + 4216 \)
|
| $79$ |
\( T^{8} + 30 T^{7} + \cdots - 4883405 \)
|
| $83$ |
\( T^{8} - 23 T^{7} + \cdots - 6493208 \)
|
| $89$ |
\( T^{8} + 6 T^{7} + \cdots - 17403560 \)
|
| $97$ |
\( T^{8} - 10 T^{7} + \cdots - 13622756 \)
|
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