Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [256,8,Mod(1,256)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(256, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("256.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 256 = 2^{8} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 256.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: | \(79.9705665239\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{3}, \sqrt{601})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 305x^{2} + 306x + 21606 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{11} \) |
| Twist minimal: | no (minimal twist has level 64) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 305x^{2} + 306x + 21606 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -8\nu^{3} + 12\nu^{2} + 3624\nu - 1814 ) / 589 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 128\nu^{3} - 192\nu^{2} - 20288\nu + 10176 ) / 589 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -56\nu^{3} + 2440\nu^{2} + 6520\nu - 364920 ) / 589 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 16\beta _1 + 32 ) / 64 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} + \beta_{2} + 2\beta _1 + 1228 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 24\beta_{3} + 465\beta_{2} + 2560\beta _1 + 14720 ) / 64 \)
|
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 | −29.0306 | 0 | −107.493 | 0 | 534.108 | 0 | −1344.22 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −29.0306 | 0 | 107.493 | 0 | −534.108 | 0 | −1344.22 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 69.0306 | 0 | −232.201 | 0 | 1504.06 | 0 | 2578.22 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 69.0306 | 0 | 232.201 | 0 | −1504.06 | 0 | 2578.22 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 256.8.a.p | 4 | |
| 4.b | odd | 2 | 1 | 256.8.a.j | 4 | ||
| 8.b | even | 2 | 1 | 256.8.a.j | 4 | ||
| 8.d | odd | 2 | 1 | inner | 256.8.a.p | 4 | |
| 16.e | even | 4 | 2 | 64.8.b.c | ✓ | 8 | |
| 16.f | odd | 4 | 2 | 64.8.b.c | ✓ | 8 | |
| 48.i | odd | 4 | 2 | 576.8.d.f | 8 | ||
| 48.k | even | 4 | 2 | 576.8.d.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 64.8.b.c | ✓ | 8 | 16.e | even | 4 | 2 | |
| 64.8.b.c | ✓ | 8 | 16.f | odd | 4 | 2 | |
| 256.8.a.j | 4 | 4.b | odd | 2 | 1 | ||
| 256.8.a.j | 4 | 8.b | even | 2 | 1 | ||
| 256.8.a.p | 4 | 1.a | even | 1 | 1 | trivial | |
| 256.8.a.p | 4 | 8.d | odd | 2 | 1 | inner | |
| 576.8.d.f | 8 | 48.i | odd | 4 | 2 | ||
| 576.8.d.f | 8 | 48.k | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(256))\):
|
\( T_{3}^{2} - 40T_{3} - 2004 \)
|
|
\( T_{5}^{4} - 65472T_{5}^{2} + 623001600 \)
|
|
\( T_{7}^{4} - 2547456T_{7}^{2} + 645335875584 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} - 40 T - 2004)^{2} \)
|
| $5$ |
\( T^{4} - 65472 T^{2} + 623001600 \)
|
| $7$ |
\( T^{4} + \cdots + 645335875584 \)
|
| $11$ |
\( (T^{2} - 4344 T - 8804916)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 55\!\cdots\!04 \)
|
| $17$ |
\( (T^{2} - 22740 T + 61426404)^{2} \)
|
| $19$ |
\( (T^{2} + 18712 T - 34167764)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 38\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 91\!\cdots\!64 \)
|
| $31$ |
\( T^{4} + \cdots + 29\!\cdots\!64 \)
|
| $37$ |
\( T^{4} + \cdots + 33\!\cdots\!04 \)
|
| $41$ |
\( (T^{2} - 209388 T - 538628183964)^{2} \)
|
| $43$ |
\( (T^{2} - 677240 T + 102395697676)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 34\!\cdots\!04 \)
|
| $53$ |
\( T^{4} + \cdots + 99\!\cdots\!00 \)
|
| $59$ |
\( (T^{2} - 123576 T - 24547579956)^{2} \)
|
| $61$ |
\( T^{4} + \cdots + 42\!\cdots\!04 \)
|
| $67$ |
\( (T^{2} + \cdots + 2125980170476)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 13\!\cdots\!84 \)
|
| $73$ |
\( (T^{2} + \cdots + 2806911930244)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 23\!\cdots\!84 \)
|
| $83$ |
\( (T^{2} + \cdots + 27336427713900)^{2} \)
|
| $89$ |
\( (T^{2} + \cdots - 32517358628796)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots - 30423027470684)^{2} \)
|
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