Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1568,3,Mod(1471,1568)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1568, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1568.1471");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1568 = 2^{5} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1568.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: | \(42.7249054517\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.9144576.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 12x^{4} + 36x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{7} \) |
| Twist minimal: | no (minimal twist has level 224) |
| 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
\( x^{6} + 12x^{4} + 36x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} + 6\nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{3} + 12\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 11\nu^{3} + 30\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} + 10\nu^{2} + 16 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - \beta_{2} - 16 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} - 6\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -3\beta_{5} + 5\beta_{2} + 48 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4\beta_{4} - 11\beta_{3} + 36\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(1471\) | \(1473\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1471.1 |
|
0 | − | 4.88448i | 0 | −2.32025 | 0 | 0 | 0 | −14.8582 | 0 | |||||||||||||||||||||||||||||||||||
| 1471.2 | 0 | − | 1.76873i | 0 | −8.20336 | 0 | 0 | 0 | 5.87158 | 0 | ||||||||||||||||||||||||||||||||||||
| 1471.3 | 0 | − | 0.115749i | 0 | 1.52360 | 0 | 0 | 0 | 8.98660 | 0 | ||||||||||||||||||||||||||||||||||||
| 1471.4 | 0 | 0.115749i | 0 | 1.52360 | 0 | 0 | 0 | 8.98660 | 0 | |||||||||||||||||||||||||||||||||||||
| 1471.5 | 0 | 1.76873i | 0 | −8.20336 | 0 | 0 | 0 | 5.87158 | 0 | |||||||||||||||||||||||||||||||||||||
| 1471.6 | 0 | 4.88448i | 0 | −2.32025 | 0 | 0 | 0 | −14.8582 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1568.3.d.i | 6 | |
| 4.b | odd | 2 | 1 | inner | 1568.3.d.i | 6 | |
| 7.b | odd | 2 | 1 | 1568.3.d.l | 6 | ||
| 7.c | even | 3 | 2 | 224.3.r.d | ✓ | 12 | |
| 28.d | even | 2 | 1 | 1568.3.d.l | 6 | ||
| 28.g | odd | 6 | 2 | 224.3.r.d | ✓ | 12 | |
| 56.k | odd | 6 | 2 | 448.3.r.f | 12 | ||
| 56.p | even | 6 | 2 | 448.3.r.f | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 224.3.r.d | ✓ | 12 | 7.c | even | 3 | 2 | |
| 224.3.r.d | ✓ | 12 | 28.g | odd | 6 | 2 | |
| 448.3.r.f | 12 | 56.k | odd | 6 | 2 | ||
| 448.3.r.f | 12 | 56.p | even | 6 | 2 | ||
| 1568.3.d.i | 6 | 1.a | even | 1 | 1 | trivial | |
| 1568.3.d.i | 6 | 4.b | odd | 2 | 1 | inner | |
| 1568.3.d.l | 6 | 7.b | odd | 2 | 1 | ||
| 1568.3.d.l | 6 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1568, [\chi])\):
|
\( T_{3}^{6} + 27T_{3}^{4} + 75T_{3}^{2} + 1 \)
|
|
\( T_{5}^{3} + 9T_{5}^{2} + 3T_{5} - 29 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 27 T^{4} + \cdots + 1 \)
|
| $5$ |
\( (T^{3} + 9 T^{2} + 3 T - 29)^{2} \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + 363 T^{4} + \cdots + 452929 \)
|
| $13$ |
\( (T^{3} - 168 T - 784)^{2} \)
|
| $17$ |
\( (T^{3} - 3 T^{2} + \cdots - 1401)^{2} \)
|
| $19$ |
\( T^{6} + 1755 T^{4} + \cdots + 96059601 \)
|
| $23$ |
\( T^{6} + 2595 T^{4} + \cdots + 163763209 \)
|
| $29$ |
\( (T^{3} - 1176 T - 5488)^{2} \)
|
| $31$ |
\( T^{6} + 531 T^{4} + \cdots + 1885129 \)
|
| $37$ |
\( (T^{3} - 57 T^{2} + \cdots - 363)^{2} \)
|
| $41$ |
\( (T^{3} - 1848 T + 22736)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 2517630976 \)
|
| $47$ |
\( T^{6} + 4947 T^{4} + \cdots + 80802121 \)
|
| $53$ |
\( (T^{3} - 9 T^{2} + \cdots - 21531)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 60465334609 \)
|
| $61$ |
\( (T^{3} + 159 T^{2} + \cdots - 2323)^{2} \)
|
| $67$ |
\( T^{6} + 3387 T^{4} + \cdots + 4068289 \)
|
| $71$ |
\( T^{6} + \cdots + 37803802624 \)
|
| $73$ |
\( (T^{3} + 171 T^{2} + \cdots + 134121)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 83448765625 \)
|
| $83$ |
\( T^{6} + \cdots + 14201012224 \)
|
| $89$ |
\( (T^{3} + 75 T^{2} + \cdots - 357111)^{2} \)
|
| $97$ |
\( (T^{3} - 168 T - 784)^{2} \)
|
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