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: | \(0\) |
| 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: |
\( x^{4} - x^{3} - 145x^{2} + 57x + 4950 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 145x^{2} + 57x + 4950 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 7\nu^{2} + 103\nu - 465 ) / 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 7\nu^{2} - 63\nu + 460 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 13\nu^{2} - 83\nu - 990 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 1 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} - \beta_{2} + 3\beta _1 + 581 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{3} + 48\beta_{2} + 105\beta _1 + 225 ) / 4 \)
|
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 | −28.4546 | 0 | 97.1664 | 0 | 3.39216 | 0 | 566.664 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −4.76524 | 0 | 18.8067 | 0 | −185.753 | 0 | −220.292 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 18.7555 | 0 | −105.263 | 0 | 75.9035 | 0 | 108.770 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 19.4643 | 0 | 82.2901 | 0 | 200.457 | 0 | 135.858 | 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.k | 4 | |
| 4.b | odd | 2 | 1 | 88.6.a.d | ✓ | 4 | |
| 8.b | even | 2 | 1 | 704.6.a.v | 4 | ||
| 8.d | odd | 2 | 1 | 704.6.a.w | 4 | ||
| 12.b | even | 2 | 1 | 792.6.a.k | 4 | ||
| 44.c | even | 2 | 1 | 968.6.a.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 88.6.a.d | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 176.6.a.k | 4 | 1.a | even | 1 | 1 | trivial | |
| 704.6.a.v | 4 | 8.b | even | 2 | 1 | ||
| 704.6.a.w | 4 | 8.d | odd | 2 | 1 | ||
| 792.6.a.k | 4 | 12.b | even | 2 | 1 | ||
| 968.6.a.e | 4 | 44.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 5T_{3}^{3} - 769T_{3}^{2} + 6945T_{3} + 49500 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(176))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 5 T^{3} + \cdots + 49500 \)
|
| $5$ |
\( T^{4} - 93 T^{3} + \cdots - 15828962 \)
|
| $7$ |
\( T^{4} - 94 T^{3} + \cdots - 9587264 \)
|
| $11$ |
\( (T - 121)^{4} \)
|
| $13$ |
\( T^{4} + \cdots - 2034425600 \)
|
| $17$ |
\( T^{4} + \cdots - 1245421621040 \)
|
| $19$ |
\( T^{4} + \cdots + 1456068701440 \)
|
| $23$ |
\( T^{4} + \cdots + 36632956434784 \)
|
| $29$ |
\( T^{4} + \cdots + 130248064000 \)
|
| $31$ |
\( T^{4} + \cdots + 10060442170840 \)
|
| $37$ |
\( T^{4} + \cdots - 25\!\cdots\!50 \)
|
| $41$ |
\( T^{4} + \cdots - 56\!\cdots\!40 \)
|
| $43$ |
\( T^{4} + \cdots + 67\!\cdots\!40 \)
|
| $47$ |
\( T^{4} + \cdots + 98\!\cdots\!08 \)
|
| $53$ |
\( T^{4} + \cdots + 37\!\cdots\!80 \)
|
| $59$ |
\( T^{4} + \cdots + 20\!\cdots\!28 \)
|
| $61$ |
\( T^{4} + \cdots + 13\!\cdots\!80 \)
|
| $67$ |
\( T^{4} + \cdots - 36\!\cdots\!60 \)
|
| $71$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots - 46\!\cdots\!20 \)
|
| $79$ |
\( T^{4} + \cdots + 83\!\cdots\!16 \)
|
| $83$ |
\( T^{4} + \cdots + 89\!\cdots\!80 \)
|
| $89$ |
\( T^{4} + \cdots - 34\!\cdots\!30 \)
|
| $97$ |
\( T^{4} + \cdots + 61\!\cdots\!30 \)
|
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