Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [592,2,Mod(417,592)]
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([0, 0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("592.417");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 592 = 2^{4} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 592.i (of order \(3\), degree \(2\), not 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: | \(4.72714379966\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.591408.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 4x^{4} + x^{3} + 10x^{2} - 3x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 296) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{6} - x^{5} + 4x^{4} + x^{3} + 10x^{2} - 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3\nu^{5} - 12\nu^{4} + 11\nu^{3} - 30\nu^{2} + 9\nu - 73 ) / 37 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} + 20\nu^{4} - 43\nu^{3} + 50\nu^{2} - 15\nu + 97 ) / 37 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 12\nu^{5} - 11\nu^{4} + 44\nu^{3} + 28\nu^{2} + 110\nu + 4 ) / 37 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 21\nu^{5} - 10\nu^{4} + 77\nu^{3} + 49\nu^{2} + 285\nu + 7 ) / 37 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -24\nu^{5} + 22\nu^{4} - 88\nu^{3} - 19\nu^{2} - 220\nu + 66 ) / 37 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{2} - 5\beta _1 - 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{5} - \beta_{4} - 7\beta_{3} - 5\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -23\beta_{5} - 11\beta_{4} - 16\beta_{3} + 11\beta_{2} + 16 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).
| \(n\) | \(113\) | \(149\) | \(223\) |
| \(\chi(n)\) | \(-\beta_{3}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 417.1 |
|
0 | −0.854638 | − | 1.48028i | 0 | 1.35464 | + | 2.34630i | 0 | 0.315449 | + | 0.546373i | 0 | 0.0391889 | − | 0.0678771i | 0 | ||||||||||||||||||||||||||||
| 417.2 | 0 | 0.403032 | + | 0.698071i | 0 | 0.0969683 | + | 0.167954i | 0 | −2.07816 | − | 3.59948i | 0 | 1.17513 | − | 2.03539i | 0 | |||||||||||||||||||||||||||||
| 417.3 | 0 | 1.45161 | + | 2.51426i | 0 | −0.951606 | − | 1.64823i | 0 | 0.762714 | + | 1.32106i | 0 | −2.71432 | + | 4.70134i | 0 | |||||||||||||||||||||||||||||
| 433.1 | 0 | −0.854638 | + | 1.48028i | 0 | 1.35464 | − | 2.34630i | 0 | 0.315449 | − | 0.546373i | 0 | 0.0391889 | + | 0.0678771i | 0 | |||||||||||||||||||||||||||||
| 433.2 | 0 | 0.403032 | − | 0.698071i | 0 | 0.0969683 | − | 0.167954i | 0 | −2.07816 | + | 3.59948i | 0 | 1.17513 | + | 2.03539i | 0 | |||||||||||||||||||||||||||||
| 433.3 | 0 | 1.45161 | − | 2.51426i | 0 | −0.951606 | + | 1.64823i | 0 | 0.762714 | − | 1.32106i | 0 | −2.71432 | − | 4.70134i | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 592.2.i.g | 6 | |
| 4.b | odd | 2 | 1 | 296.2.i.b | ✓ | 6 | |
| 12.b | even | 2 | 1 | 2664.2.r.i | 6 | ||
| 37.c | even | 3 | 1 | inner | 592.2.i.g | 6 | |
| 148.i | odd | 6 | 1 | 296.2.i.b | ✓ | 6 | |
| 444.t | even | 6 | 1 | 2664.2.r.i | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 296.2.i.b | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 296.2.i.b | ✓ | 6 | 148.i | odd | 6 | 1 | |
| 592.2.i.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 592.2.i.g | 6 | 37.c | even | 3 | 1 | inner | |
| 2664.2.r.i | 6 | 12.b | even | 2 | 1 | ||
| 2664.2.r.i | 6 | 444.t | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 2T_{3}^{5} + 8T_{3}^{4} + 24T_{3}^{2} - 16T_{3} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(592, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots + 16 \)
|
| $5$ |
\( T^{6} - T^{5} + 6 T^{4} + \cdots + 1 \)
|
| $7$ |
\( T^{6} + 2 T^{5} + \cdots + 16 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( (T^{2} - 2 T + 4)^{3} \)
|
| $17$ |
\( T^{6} + 5 T^{5} + \cdots + 18769 \)
|
| $19$ |
\( T^{6} - 8 T^{5} + \cdots + 25600 \)
|
| $23$ |
\( (T^{3} + 6 T^{2} + \cdots - 100)^{2} \)
|
| $29$ |
\( (T^{3} - 3 T^{2} - 25 T - 25)^{2} \)
|
| $31$ |
\( (T^{3} - 16 T^{2} + \cdots - 32)^{2} \)
|
| $37$ |
\( T^{6} - 15 T^{5} + \cdots + 50653 \)
|
| $41$ |
\( T^{6} + 5 T^{5} + \cdots + 289 \)
|
| $43$ |
\( (T^{3} - 10 T^{2} + \cdots + 604)^{2} \)
|
| $47$ |
\( (T^{3} - 4 T^{2} - 16 T + 32)^{2} \)
|
| $53$ |
\( T^{6} - 6 T^{5} + \cdots + 64 \)
|
| $59$ |
\( T^{6} + 8 T^{5} + \cdots + 25600 \)
|
| $61$ |
\( T^{6} + 27 T^{5} + \cdots + 180625 \)
|
| $67$ |
\( T^{6} - 4 T^{5} + \cdots + 4096 \)
|
| $71$ |
\( T^{6} + 2 T^{5} + \cdots + 67600 \)
|
| $73$ |
\( (T^{3} + 6 T^{2} + \cdots - 712)^{2} \)
|
| $79$ |
\( T^{6} - 8 T^{5} + \cdots + 295936 \)
|
| $83$ |
\( T^{6} + 6 T^{5} + \cdots + 1607824 \)
|
| $89$ |
\( T^{6} + 13 T^{5} + \cdots + 2512225 \)
|
| $97$ |
\( (T^{3} - 9 T^{2} + \cdots + 877)^{2} \)
|
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