Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [578,4,Mod(1,578)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(578, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("578.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 578 = 2 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 578.a (trivial) |
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: | yes |
| Analytic conductor: | \(34.1031039833\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{13}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.00000 | −8.90833 | 4.00000 | −1.30278 | 17.8167 | −32.9361 | −8.00000 | 52.3583 | 2.60555 | ||||||||||||||||||||||||
| 1.2 | −2.00000 | 1.90833 | 4.00000 | 2.30278 | −3.81665 | 13.9361 | −8.00000 | −23.3583 | −4.60555 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(17\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 578.4.a.e | ✓ | 2 |
| 17.b | even | 2 | 1 | 578.4.a.g | yes | 2 | |
| 17.c | even | 4 | 2 | 578.4.b.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 578.4.a.e | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 578.4.a.g | yes | 2 | 17.b | even | 2 | 1 | |
| 578.4.b.f | 4 | 17.c | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(578))\):
|
\( T_{3}^{2} + 7T_{3} - 17 \)
|
|
\( T_{5}^{2} - T_{5} - 3 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{2} \)
|
| $3$ |
\( T^{2} + 7T - 17 \)
|
| $5$ |
\( T^{2} - T - 3 \)
|
| $7$ |
\( T^{2} + 19T - 459 \)
|
| $11$ |
\( T^{2} + 28T - 441 \)
|
| $13$ |
\( T^{2} - 45T - 1525 \)
|
| $17$ |
\( T^{2} \)
|
| $19$ |
\( T^{2} + 99T + 1901 \)
|
| $23$ |
\( T^{2} - 111T + 3051 \)
|
| $29$ |
\( T^{2} + 91T - 66261 \)
|
| $31$ |
\( T^{2} - 58T - 3371 \)
|
| $37$ |
\( T^{2} - 438T + 47324 \)
|
| $41$ |
\( T^{2} - 364T + 22932 \)
|
| $43$ |
\( T^{2} - 350T + 28077 \)
|
| $47$ |
\( T^{2} - 384T + 36747 \)
|
| $53$ |
\( T^{2} + 933T + 207063 \)
|
| $59$ |
\( T^{2} + 720T - 117972 \)
|
| $61$ |
\( T^{2} + 188T - 641 \)
|
| $67$ |
\( T^{2} + 22T - 567732 \)
|
| $71$ |
\( T^{2} - 1332 T + 413604 \)
|
| $73$ |
\( T^{2} + 1284 T + 322607 \)
|
| $79$ |
\( T^{2} + 324T - 626044 \)
|
| $83$ |
\( T^{2} - 7T - 1633701 \)
|
| $89$ |
\( T^{2} + 1272 T + 366588 \)
|
| $97$ |
\( T^{2} + 2631 T + 1601837 \)
|
| show more | |
| show less |