Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [176,8,Mod(1,176)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(176, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("176.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 176 = 2^{4} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 176.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: | \(54.9797644852\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 341x^{2} + 1417x - 1412 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 11) |
| 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
\( x^{4} - x^{3} - 341x^{2} + 1417x - 1412 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} - \nu^{2} + 698\nu - 1619 ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{3} + 5\nu^{2} - 1019\nu + 1823 ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 9\nu^{3} + 8\nu^{2} - 2945\nu + 6631 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 3\beta _1 + 7 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 7\beta_{2} + 6\beta _1 + 512 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 353\beta_{3} - 377\beta_{2} + 939\beta _1 - 19033 ) / 24 \)
|
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 | −59.6211 | 0 | −60.1766 | 0 | −698.069 | 0 | 1367.68 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −12.2971 | 0 | 512.130 | 0 | −973.904 | 0 | −2035.78 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 29.4867 | 0 | 223.515 | 0 | 1410.66 | 0 | −1317.53 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 77.4315 | 0 | −138.468 | 0 | 91.3115 | 0 | 3808.63 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 176.8.a.j | 4 | |
| 4.b | odd | 2 | 1 | 11.8.a.b | ✓ | 4 | |
| 12.b | even | 2 | 1 | 99.8.a.g | 4 | ||
| 20.d | odd | 2 | 1 | 275.8.a.b | 4 | ||
| 28.d | even | 2 | 1 | 539.8.a.b | 4 | ||
| 44.c | even | 2 | 1 | 121.8.a.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.8.a.b | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 99.8.a.g | 4 | 12.b | even | 2 | 1 | ||
| 121.8.a.c | 4 | 44.c | even | 2 | 1 | ||
| 176.8.a.j | 4 | 1.a | even | 1 | 1 | trivial | |
| 275.8.a.b | 4 | 20.d | odd | 2 | 1 | ||
| 539.8.a.b | 4 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 35T_{3}^{3} - 4673T_{3}^{2} + 85815T_{3} + 1673964 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(176))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 35 T^{3} + \cdots + 1673964 \)
|
| $5$ |
\( T^{4} - 537 T^{3} + \cdots + 953818350 \)
|
| $7$ |
\( T^{4} + \cdots + 87571440704 \)
|
| $11$ |
\( (T - 1331)^{4} \)
|
| $13$ |
\( T^{4} + \cdots - 518474088880000 \)
|
| $17$ |
\( T^{4} + \cdots - 58\!\cdots\!56 \)
|
| $19$ |
\( T^{4} + \cdots - 31\!\cdots\!84 \)
|
| $23$ |
\( T^{4} + \cdots + 10\!\cdots\!44 \)
|
| $29$ |
\( T^{4} + \cdots - 41\!\cdots\!64 \)
|
| $31$ |
\( T^{4} + \cdots - 61\!\cdots\!36 \)
|
| $37$ |
\( T^{4} + \cdots + 41\!\cdots\!94 \)
|
| $41$ |
\( T^{4} + \cdots - 66\!\cdots\!84 \)
|
| $43$ |
\( T^{4} + \cdots + 28\!\cdots\!64 \)
|
| $47$ |
\( T^{4} + \cdots - 74\!\cdots\!04 \)
|
| $53$ |
\( T^{4} + \cdots + 18\!\cdots\!84 \)
|
| $59$ |
\( T^{4} + \cdots + 29\!\cdots\!64 \)
|
| $61$ |
\( T^{4} + \cdots + 10\!\cdots\!84 \)
|
| $67$ |
\( T^{4} + \cdots - 84\!\cdots\!64 \)
|
| $71$ |
\( T^{4} + \cdots + 59\!\cdots\!44 \)
|
| $73$ |
\( T^{4} + \cdots - 60\!\cdots\!76 \)
|
| $79$ |
\( T^{4} + \cdots + 25\!\cdots\!16 \)
|
| $83$ |
\( T^{4} + \cdots + 70\!\cdots\!96 \)
|
| $89$ |
\( T^{4} + \cdots - 24\!\cdots\!34 \)
|
| $97$ |
\( T^{4} + \cdots + 46\!\cdots\!46 \)
|
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