Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8,7,Mod(3,8)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8.3");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8 = 2^{3} \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8.d (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(1.84043266896\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.3803625.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 6x^{2} - 16x + 256 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} - x^{3} + 6x^{2} - 16x + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} - 6\nu + 12 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 15\nu^{2} - 2\nu + 36 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + \beta _1 - 12 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} - 15\beta_{2} - 11\beta _1 + 36 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(7\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−4.62348 | − | 6.52867i | −32.4939 | −21.2470 | + | 60.3702i | − | 199.084i | 150.235 | + | 212.142i | 19.6656i | 492.372 | − | 140.406i | 326.854 | −1299.76 | + | 920.462i | |||||||||||||||||||
| 3.2 | −4.62348 | + | 6.52867i | −32.4939 | −21.2470 | − | 60.3702i | 199.084i | 150.235 | − | 212.142i | − | 19.6656i | 492.372 | + | 140.406i | 326.854 | −1299.76 | − | 920.462i | ||||||||||||||||||||
| 3.3 | 5.62348 | − | 5.69004i | 8.49390 | −0.753049 | − | 63.9956i | 59.7107i | 47.7652 | − | 48.3306i | 483.584i | −368.372 | − | 355.593i | −656.854 | 339.756 | + | 335.782i | |||||||||||||||||||||
| 3.4 | 5.62348 | + | 5.69004i | 8.49390 | −0.753049 | + | 63.9956i | − | 59.7107i | 47.7652 | + | 48.3306i | − | 483.584i | −368.372 | + | 355.593i | −656.854 | 339.756 | − | 335.782i | |||||||||||||||||||
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 | 8.7.d.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 72.7.b.b | 4 | ||
| 4.b | odd | 2 | 1 | 32.7.d.b | 4 | ||
| 8.b | even | 2 | 1 | 32.7.d.b | 4 | ||
| 8.d | odd | 2 | 1 | inner | 8.7.d.b | ✓ | 4 |
| 12.b | even | 2 | 1 | 288.7.b.b | 4 | ||
| 16.e | even | 4 | 2 | 256.7.c.l | 8 | ||
| 16.f | odd | 4 | 2 | 256.7.c.l | 8 | ||
| 24.f | even | 2 | 1 | 72.7.b.b | 4 | ||
| 24.h | odd | 2 | 1 | 288.7.b.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.7.d.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 8.7.d.b | ✓ | 4 | 8.d | odd | 2 | 1 | inner |
| 32.7.d.b | 4 | 4.b | odd | 2 | 1 | ||
| 32.7.d.b | 4 | 8.b | even | 2 | 1 | ||
| 72.7.b.b | 4 | 3.b | odd | 2 | 1 | ||
| 72.7.b.b | 4 | 24.f | even | 2 | 1 | ||
| 256.7.c.l | 8 | 16.e | even | 4 | 2 | ||
| 256.7.c.l | 8 | 16.f | odd | 4 | 2 | ||
| 288.7.b.b | 4 | 12.b | even | 2 | 1 | ||
| 288.7.b.b | 4 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 24T_{3} - 276 \)
acting on \(S_{7}^{\mathrm{new}}(8, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2 T^{3} + \cdots + 4096 \)
|
| $3$ |
\( (T^{2} + 24 T - 276)^{2} \)
|
| $5$ |
\( T^{4} + 43200 T^{2} + 141312000 \)
|
| $7$ |
\( T^{4} + 234240 T^{2} + 90439680 \)
|
| $11$ |
\( (T^{2} - 488 T - 1305044)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 28641121812480 \)
|
| $17$ |
\( (T^{2} - 2084 T - 15714236)^{2} \)
|
| $19$ |
\( (T^{2} + 728 T - 1642004)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 40\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 19\!\cdots\!80 \)
|
| $31$ |
\( T^{4} + \cdots + 41\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 10\!\cdots\!80 \)
|
| $41$ |
\( (T^{2} + 58972 T + 357521476)^{2} \)
|
| $43$ |
\( (T^{2} - 98728 T + 547375276)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 10\!\cdots\!80 \)
|
| $53$ |
\( T^{4} + \cdots + 29\!\cdots\!80 \)
|
| $59$ |
\( (T^{2} - 271016 T + 18317507884)^{2} \)
|
| $61$ |
\( T^{4} + \cdots + 33\!\cdots\!80 \)
|
| $67$ |
\( (T^{2} + 395096 T + 24071074924)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 11\!\cdots\!80 \)
|
| $73$ |
\( (T^{2} - 221956 T - 18286467836)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 31\!\cdots\!80 \)
|
| $83$ |
\( (T^{2} + 1732504 T + 746384920684)^{2} \)
|
| $89$ |
\( (T^{2} - 380612 T - 23540043644)^{2} \)
|
| $97$ |
\( (T^{2} + 463388 T - 711956225084)^{2} \)
|
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