Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [176,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("176.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 176 = 2^{4} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(28.2275522871\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1784453.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 368x - 2705 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 88) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 368x - 2705 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 11\nu - 245 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{2} + 11\beta _1 + 980 ) / 4 \)
|
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 | −25.3952 | 0 | −2.46146 | 0 | −36.9338 | 0 | 401.918 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −3.36210 | 0 | 105.922 | 0 | −123.284 | 0 | −231.696 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 14.7573 | 0 | −47.4604 | 0 | 48.2178 | 0 | −25.2213 | 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 | 176.6.a.j | 3 | |
| 4.b | odd | 2 | 1 | 88.6.a.b | ✓ | 3 | |
| 8.b | even | 2 | 1 | 704.6.a.s | 3 | ||
| 8.d | odd | 2 | 1 | 704.6.a.r | 3 | ||
| 12.b | even | 2 | 1 | 792.6.a.f | 3 | ||
| 44.c | even | 2 | 1 | 968.6.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 88.6.a.b | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 176.6.a.j | 3 | 1.a | even | 1 | 1 | trivial | |
| 704.6.a.r | 3 | 8.d | odd | 2 | 1 | ||
| 704.6.a.s | 3 | 8.b | even | 2 | 1 | ||
| 792.6.a.f | 3 | 12.b | even | 2 | 1 | ||
| 968.6.a.c | 3 | 44.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 14T_{3}^{2} - 339T_{3} - 1260 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(176))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 14 T^{2} + \cdots - 1260 \)
|
| $5$ |
\( T^{3} - 56 T^{2} + \cdots - 12374 \)
|
| $7$ |
\( T^{3} + 112 T^{2} + \cdots - 219552 \)
|
| $11$ |
\( (T + 121)^{3} \)
|
| $13$ |
\( T^{3} - 450 T^{2} + \cdots + 51255776 \)
|
| $17$ |
\( T^{3} - 1274 T^{2} + \cdots - 54141048 \)
|
| $19$ |
\( T^{3} + \cdots - 4934825280 \)
|
| $23$ |
\( T^{3} + 4042 T^{2} + \cdots + 645194920 \)
|
| $29$ |
\( T^{3} + \cdots + 34612290944 \)
|
| $31$ |
\( T^{3} + \cdots - 2443760880 \)
|
| $37$ |
\( T^{3} + 2916 T^{2} + \cdots - 548244166 \)
|
| $41$ |
\( T^{3} + \cdots + 19244597856 \)
|
| $43$ |
\( T^{3} + \cdots + 106822549552 \)
|
| $47$ |
\( T^{3} + \cdots - 584697794560 \)
|
| $53$ |
\( T^{3} + \cdots - 16689929059320 \)
|
| $59$ |
\( T^{3} + \cdots + 16036070192844 \)
|
| $61$ |
\( T^{3} + \cdots - 29532796072736 \)
|
| $67$ |
\( T^{3} + \cdots + 84021494680468 \)
|
| $71$ |
\( T^{3} + \cdots + 75746574829848 \)
|
| $73$ |
\( T^{3} + \cdots - 348170414631776 \)
|
| $79$ |
\( T^{3} + \cdots + 118332099622976 \)
|
| $83$ |
\( T^{3} + \cdots + 82355829648 \)
|
| $89$ |
\( T^{3} + \cdots - 28304140091626 \)
|
| $97$ |
\( T^{3} + \cdots - 19\!\cdots\!26 \)
|
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