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: | 6.0.19298288.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 3x^{4} - 2x^{3} + 9x^{2} - 9x + 27 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 3x^{4} - 2x^{3} + 9x^{2} - 9x + 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + \nu^{3} - \nu - 3 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 6\nu^{3} + 7\nu^{2} - 9\nu ) / 18 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{4} + 3\nu^{3} + 11\nu^{2} - 6\nu + 27 ) / 18 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} - 3\nu^{3} + 2\nu^{2} - 9\nu + 9 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{5} + \beta_{4} - \beta_{3} - 5\beta_{2} - 2\beta _1 - 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{5} + 4\beta_{3} - 6\beta_{2} - 6\beta _1 + 4 \)
|
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 |
|
−1.93543 | −1.34067 | − | 1.09664i | 1.74590 | − | 3.21911i | 2.59477 | + | 2.12247i | 0.254102 | 0.491797 | 0.594767 | + | 2.94045i | 6.23037i | |||||||||||||||||||||||||||||
| 176.2 | −1.93543 | −1.34067 | + | 1.09664i | 1.74590 | 3.21911i | 2.59477 | − | 2.12247i | 0.254102 | 0.491797 | 0.594767 | − | 2.94045i | − | 6.23037i | ||||||||||||||||||||||||||||||
| 176.3 | 1.46260 | 1.16170 | − | 1.28470i | 0.139194 | 0.594299i | 1.69910 | − | 1.87900i | 1.86081 | −2.72161 | −0.300896 | − | 2.98487i | 0.869221i | |||||||||||||||||||||||||||||||
| 176.4 | 1.46260 | 1.16170 | + | 1.28470i | 0.139194 | − | 0.594299i | 1.69910 | + | 1.87900i | 1.86081 | −2.72161 | −0.300896 | + | 2.98487i | − | 0.869221i | |||||||||||||||||||||||||||||
| 176.5 | 2.47283 | −0.321037 | − | 1.70204i | 4.11491 | 2.50682i | −0.793871 | − | 4.20886i | −2.11491 | 5.22982 | −2.79387 | + | 1.09283i | 6.19895i | |||||||||||||||||||||||||||||||
| 176.6 | 2.47283 | −0.321037 | + | 1.70204i | 4.11491 | − | 2.50682i | −0.793871 | + | 4.20886i | −2.11491 | 5.22982 | −2.79387 | − | 1.09283i | − | 6.19895i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 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.c | yes | 6 |
| 3.b | odd | 2 | 1 | 177.2.d.a | ✓ | 6 | |
| 59.b | odd | 2 | 1 | 177.2.d.a | ✓ | 6 | |
| 177.d | even | 2 | 1 | inner | 177.2.d.c | yes | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 177.2.d.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 177.2.d.a | ✓ | 6 | 59.b | odd | 2 | 1 | |
| 177.2.d.c | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 177.2.d.c | yes | 6 | 177.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 2T_{2}^{2} - 4T_{2} + 7 \)
acting on \(S_{2}^{\mathrm{new}}(177, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} - 2 T^{2} - 4 T + 7)^{2} \)
|
| $3$ |
\( T^{6} + T^{5} + \cdots + 27 \)
|
| $5$ |
\( T^{6} + 17 T^{4} + \cdots + 23 \)
|
| $7$ |
\( (T^{3} - 4 T + 1)^{2} \)
|
| $11$ |
\( (T^{3} + 10 T^{2} + \cdots + 14)^{2} \)
|
| $13$ |
\( T^{6} + 58 T^{4} + \cdots + 4508 \)
|
| $17$ |
\( T^{6} + 27 T^{4} + \cdots + 368 \)
|
| $19$ |
\( (T^{3} - 2 T^{2} - 26 T - 37)^{2} \)
|
| $23$ |
\( (T^{3} + 9 T^{2} + 23 T + 14)^{2} \)
|
| $29$ |
\( T^{6} + 54 T^{4} + \cdots + 1127 \)
|
| $31$ |
\( T^{6} + 115 T^{4} + \cdots + 18032 \)
|
| $37$ |
\( T^{6} + 223 T^{4} + \cdots + 220892 \)
|
| $41$ |
\( T^{6} + 58 T^{4} + \cdots + 23 \)
|
| $43$ |
\( T^{6} + 206 T^{4} + \cdots + 288512 \)
|
| $47$ |
\( (T^{3} - 23 T^{2} + \cdots - 406)^{2} \)
|
| $53$ |
\( T^{6} + 121 T^{4} + \cdots + 3887 \)
|
| $59$ |
\( T^{6} + 10 T^{5} + \cdots + 205379 \)
|
| $61$ |
\( T^{6} + 199 T^{4} + \cdots + 4508 \)
|
| $67$ |
\( T^{6} + 58 T^{4} + \cdots + 4508 \)
|
| $71$ |
\( T^{6} + 306 T^{4} + \cdots + 67068 \)
|
| $73$ |
\( T^{6} + 247 T^{4} + \cdots + 72128 \)
|
| $79$ |
\( (T^{3} + 7 T^{2} + \cdots - 347)^{2} \)
|
| $83$ |
\( (T^{3} + 25 T^{2} + \cdots + 14)^{2} \)
|
| $89$ |
\( (T^{3} - 13 T^{2} + \cdots + 518)^{2} \)
|
| $97$ |
\( T^{6} + 342 T^{4} + \cdots + 18032 \)
|
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