Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [592,3,Mod(47,592)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(592, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("592.47");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 592 = 2^{4} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 592.z (of order \(6\), degree \(2\), 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: | \(16.1308316501\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{6}]$ |
$q$-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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\zeta_{12}^{3} + 4\zeta_{12} \)
|
| \(\beta_{3}\) | \(=\) |
\( -4\zeta_{12}^{3} + 8\zeta_{12} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 12 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} ) / 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).
| \(n\) | \(113\) | \(149\) | \(223\) |
| \(\chi(n)\) | \(-\beta_{1}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 47.1 |
|
0 | 0 | 0 | −4.96410 | − | 8.59808i | 0 | 0 | 0 | −4.50000 | + | 7.79423i | 0 | ||||||||||||||||||||||||||
| 47.2 | 0 | 0 | 0 | 1.96410 | + | 3.40192i | 0 | 0 | 0 | −4.50000 | + | 7.79423i | 0 | |||||||||||||||||||||||||||
| 63.1 | 0 | 0 | 0 | −4.96410 | + | 8.59808i | 0 | 0 | 0 | −4.50000 | − | 7.79423i | 0 | |||||||||||||||||||||||||||
| 63.2 | 0 | 0 | 0 | 1.96410 | − | 3.40192i | 0 | 0 | 0 | −4.50000 | − | 7.79423i | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
| 37.c | even | 3 | 1 | inner |
| 148.i | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 592.3.z.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | CM | 592.3.z.a | ✓ | 4 |
| 37.c | even | 3 | 1 | inner | 592.3.z.a | ✓ | 4 |
| 148.i | odd | 6 | 1 | inner | 592.3.z.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 592.3.z.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 592.3.z.a | ✓ | 4 | 4.b | odd | 2 | 1 | CM |
| 592.3.z.a | ✓ | 4 | 37.c | even | 3 | 1 | inner |
| 592.3.z.a | ✓ | 4 | 148.i | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} \)
acting on \(S_{3}^{\mathrm{new}}(592, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 6 T^{3} + \cdots + 1521 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( (T^{2} + 10 T + 100)^{2} \)
|
| $17$ |
\( T^{4} + 30 T^{3} + \cdots + 1089 \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( (T^{2} + 42 T - 759)^{2} \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} - 70 T^{3} + \cdots + 1874161 \)
|
| $41$ |
\( T^{4} - 18 T^{3} + \cdots + 22268961 \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( (T^{2} + 90 T + 8100)^{2} \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( T^{4} + 22 T^{3} + \cdots + 114041041 \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( (T + 110)^{4} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} + 78 T^{3} + \cdots + 312547041 \)
|
| $97$ |
\( (T^{2} + 130 T - 11327)^{2} \)
|
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