Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [187,2,Mod(69,187)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(187, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([8, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("187.69");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 187 = 11 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 187.g (of order \(5\), degree \(4\), 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: | \(1.49320251780\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.324000000.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} ) / 9 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} ) / 27 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} ) / 27 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{4} \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{5} \)
|
| \(\nu^{6}\) | \(=\) |
\( 27\beta_{6} \)
|
| \(\nu^{7}\) | \(=\) |
\( 27\beta_{7} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/187\mathbb{Z}\right)^\times\).
| \(n\) | \(35\) | \(122\) |
| \(\chi(n)\) | \(\beta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 69.1 |
|
−0.500000 | + | 0.363271i | −0.344250 | − | 1.05949i | −0.500000 | + | 1.53884i | −0.901259 | − | 0.654803i | 0.557008 | + | 0.404690i | 1.34425 | − | 4.13718i | −0.690983 | − | 2.12663i | 1.42303 | − | 1.03389i | 0.688500 | ||||||||||||||||||||||||||
| 69.2 | −0.500000 | + | 0.363271i | 0.726216 | + | 2.23506i | −0.500000 | + | 1.53884i | 1.90126 | + | 1.38135i | −1.17504 | − | 0.853718i | 0.273784 | − | 0.842620i | −0.690983 | − | 2.12663i | −2.04107 | + | 1.48292i | −1.45243 | |||||||||||||||||||||||||||
| 86.1 | −0.500000 | + | 1.53884i | −0.0922415 | + | 0.0670174i | −0.500000 | − | 0.363271i | −0.0352331 | − | 0.108436i | −0.0570084 | − | 0.175454i | 1.09224 | + | 0.793560i | −1.80902 | + | 1.31433i | −0.923034 | + | 2.84081i | 0.184483 | |||||||||||||||||||||||||||
| 86.2 | −0.500000 | + | 1.53884i | 2.71028 | − | 1.96913i | −0.500000 | − | 0.363271i | 1.03523 | + | 3.18612i | 1.67504 | + | 5.15525i | −1.71028 | − | 1.24259i | −1.80902 | + | 1.31433i | 2.54107 | − | 7.82060i | −5.42055 | |||||||||||||||||||||||||||
| 103.1 | −0.500000 | − | 0.363271i | −0.344250 | + | 1.05949i | −0.500000 | − | 1.53884i | −0.901259 | + | 0.654803i | 0.557008 | − | 0.404690i | 1.34425 | + | 4.13718i | −0.690983 | + | 2.12663i | 1.42303 | + | 1.03389i | 0.688500 | |||||||||||||||||||||||||||
| 103.2 | −0.500000 | − | 0.363271i | 0.726216 | − | 2.23506i | −0.500000 | − | 1.53884i | 1.90126 | − | 1.38135i | −1.17504 | + | 0.853718i | 0.273784 | + | 0.842620i | −0.690983 | + | 2.12663i | −2.04107 | − | 1.48292i | −1.45243 | |||||||||||||||||||||||||||
| 137.1 | −0.500000 | − | 1.53884i | −0.0922415 | − | 0.0670174i | −0.500000 | + | 0.363271i | −0.0352331 | + | 0.108436i | −0.0570084 | + | 0.175454i | 1.09224 | − | 0.793560i | −1.80902 | − | 1.31433i | −0.923034 | − | 2.84081i | 0.184483 | |||||||||||||||||||||||||||
| 137.2 | −0.500000 | − | 1.53884i | 2.71028 | + | 1.96913i | −0.500000 | + | 0.363271i | 1.03523 | − | 3.18612i | 1.67504 | − | 5.15525i | −1.71028 | + | 1.24259i | −1.80902 | − | 1.31433i | 2.54107 | + | 7.82060i | −5.42055 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 187.2.g.d | ✓ | 8 |
| 11.c | even | 5 | 1 | inner | 187.2.g.d | ✓ | 8 |
| 11.c | even | 5 | 1 | 2057.2.a.q | 4 | ||
| 11.d | odd | 10 | 1 | 2057.2.a.t | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 187.2.g.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 187.2.g.d | ✓ | 8 | 11.c | even | 5 | 1 | inner |
| 2057.2.a.q | 4 | 11.c | even | 5 | 1 | ||
| 2057.2.a.t | 4 | 11.d | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(187, [\chi])\):
|
\( T_{2}^{4} + 2T_{2}^{3} + 4T_{2}^{2} + 3T_{2} + 1 \)
|
|
\( T_{3}^{8} - 6T_{3}^{7} + 20T_{3}^{6} - 34T_{3}^{5} + 54T_{3}^{4} - 4T_{3}^{3} + 75T_{3}^{2} + 14T_{3} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{8} - 6 T^{7} + \cdots + 1 \)
|
| $5$ |
\( T^{8} - 4 T^{7} + \cdots + 1 \)
|
| $7$ |
\( T^{8} - 2 T^{7} + \cdots + 121 \)
|
| $11$ |
\( T^{8} + 2 T^{7} + \cdots + 14641 \)
|
| $13$ |
\( (T^{4} - 5 T^{3} + 15 T^{2} + \cdots + 25)^{2} \)
|
| $17$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $19$ |
\( T^{8} + 4 T^{7} + \cdots + 11881 \)
|
| $23$ |
\( (T^{4} - 2 T^{3} - 10 T^{2} + \cdots - 11)^{2} \)
|
| $29$ |
\( T^{8} + 53 T^{6} + \cdots + 121 \)
|
| $31$ |
\( T^{8} + 4 T^{7} + \cdots + 256 \)
|
| $37$ |
\( T^{8} - 10 T^{7} + \cdots + 7230721 \)
|
| $41$ |
\( T^{8} + 10 T^{7} + \cdots + 591361 \)
|
| $43$ |
\( (T^{2} - 4 T - 56)^{4} \)
|
| $47$ |
\( T^{8} - 10 T^{7} + \cdots + 3200521 \)
|
| $53$ |
\( T^{8} - 4 T^{7} + \cdots + 256 \)
|
| $59$ |
\( (T^{4} - 3 T^{3} + 9 T^{2} + \cdots + 81)^{2} \)
|
| $61$ |
\( T^{8} - 30 T^{7} + \cdots + 251254201 \)
|
| $67$ |
\( (T^{4} + 10 T^{3} + \cdots - 11)^{2} \)
|
| $71$ |
\( T^{8} - 18 T^{7} + \cdots + 3031081 \)
|
| $73$ |
\( T^{8} + 6 T^{7} + \cdots + 421201 \)
|
| $79$ |
\( T^{8} + 4 T^{7} + \cdots + 156150016 \)
|
| $83$ |
\( T^{8} + 20 T^{7} + \cdots + 121 \)
|
| $89$ |
\( (T^{4} + 24 T^{3} + \cdots - 23216)^{2} \)
|
| $97$ |
\( T^{8} - 12 T^{7} + \cdots + 331776 \)
|
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