Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [177,2,Mod(176,177)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(177, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("177.176");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 177 = 3 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 177.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: | \(1.41335211578\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.149721291.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 7x^{3} + 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 7x^{3} + 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 3\nu^{4} - 7\nu^{2} - 12\nu ) / 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} + 3\nu^{4} + 5\nu^{2} - 12\nu ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{3} - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + \beta_{2} + 4\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -3\beta_{4} + 3\beta_{3} + 4\beta_{2} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).
| \(n\) | \(61\) | \(119\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 176.1 |
|
0 | −1.24353 | − | 1.20567i | −2.00000 | 3.58579i | 0 | 5.15952 | 0 | 0.0927106 | + | 2.99857i | 0 | ||||||||||||||||||||||||||||||||
| 176.2 | 0 | −1.24353 | + | 1.20567i | −2.00000 | − | 3.58579i | 0 | 5.15952 | 0 | 0.0927106 | − | 2.99857i | 0 | ||||||||||||||||||||||||||||||||
| 176.3 | 0 | −0.422380 | − | 1.67976i | −2.00000 | − | 0.521533i | 0 | −3.59686 | 0 | −2.64319 | + | 1.41899i | 0 | ||||||||||||||||||||||||||||||||
| 176.4 | 0 | −0.422380 | + | 1.67976i | −2.00000 | 0.521533i | 0 | −3.59686 | 0 | −2.64319 | − | 1.41899i | 0 | |||||||||||||||||||||||||||||||||
| 176.5 | 0 | 1.66591 | − | 0.474089i | −2.00000 | − | 4.10732i | 0 | −1.56266 | 0 | 2.55048 | − | 1.57957i | 0 | ||||||||||||||||||||||||||||||||
| 176.6 | 0 | 1.66591 | + | 0.474089i | −2.00000 | 4.10732i | 0 | −1.56266 | 0 | 2.55048 | + | 1.57957i | 0 | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 59.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-59}) \) |
| 3.b | odd | 2 | 1 | inner |
| 177.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 177.2.d.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 177.2.d.b | ✓ | 6 |
| 59.b | odd | 2 | 1 | CM | 177.2.d.b | ✓ | 6 |
| 177.d | even | 2 | 1 | inner | 177.2.d.b | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 177.2.d.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 177.2.d.b | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
| 177.2.d.b | ✓ | 6 | 59.b | odd | 2 | 1 | CM |
| 177.2.d.b | ✓ | 6 | 177.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{2}^{\mathrm{new}}(177, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 7T^{3} + 27 \)
|
| $5$ |
\( T^{6} + 30 T^{4} + \cdots + 59 \)
|
| $7$ |
\( (T^{3} - 21 T - 29)^{2} \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( (T^{2} + 59)^{3} \)
|
| $19$ |
\( (T^{3} - 57 T - 119)^{2} \)
|
| $23$ |
\( T^{6} \)
|
| $29$ |
\( T^{6} + 174 T^{4} + \cdots + 72275 \)
|
| $31$ |
\( T^{6} \)
|
| $37$ |
\( T^{6} \)
|
| $41$ |
\( T^{6} + 246 T^{4} + \cdots + 59 \)
|
| $43$ |
\( T^{6} \)
|
| $47$ |
\( T^{6} \)
|
| $53$ |
\( T^{6} + 318 T^{4} + \cdots + 488579 \)
|
| $59$ |
\( (T^{2} + 59)^{3} \)
|
| $61$ |
\( T^{6} \)
|
| $67$ |
\( T^{6} \)
|
| $71$ |
\( (T^{2} + 59)^{3} \)
|
| $73$ |
\( T^{6} \)
|
| $79$ |
\( (T^{3} - 237 T + 835)^{2} \)
|
| $83$ |
\( T^{6} \)
|
| $89$ |
\( T^{6} \)
|
| $97$ |
\( T^{6} \)
|
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