Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [546,2,Mod(545,546)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(546, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("546.545");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 546.e (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: | \(4.35983195036\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.10070523904.11 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 10x^{4} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 10x^{4} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - \nu^{2} ) / 18 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 7\nu ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 19\nu^{2} ) / 18 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} + 3\nu^{5} + 9\nu^{4} + \nu^{3} - \nu^{2} - 3\nu - 45 ) / 36 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 3\nu^{5} + 9\nu^{4} - \nu^{3} + \nu^{2} + 3\nu - 45 ) / 36 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - 10\nu^{3} ) / 27 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{7} + 2\beta_{6} - 2\beta_{5} + 2\beta_{3} + 2\beta_{2} + 2\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{6} + 2\beta_{5} + 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6\beta_{3} + 7\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{4} + 19\beta_{2} \)
|
| \(\nu^{7}\) | \(=\) |
\( -3\beta_{7} + 20\beta_{6} - 20\beta_{5} + 20\beta_{3} + 20\beta_{2} + 20\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).
| \(n\) | \(157\) | \(365\) | \(379\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 545.1 |
|
1.00000 | −1.68014 | − | 0.420861i | 1.00000 | − | 0.841723i | −1.68014 | − | 0.420861i | −0.595188 | + | 2.57794i | 1.00000 | 2.64575 | + | 1.41421i | − | 0.841723i | ||||||||||||||||||||||||||||||||
| 545.2 | 1.00000 | −1.68014 | + | 0.420861i | 1.00000 | 0.841723i | −1.68014 | + | 0.420861i | −0.595188 | − | 2.57794i | 1.00000 | 2.64575 | − | 1.41421i | 0.841723i | |||||||||||||||||||||||||||||||||||
| 545.3 | 1.00000 | −0.420861 | − | 1.68014i | 1.00000 | − | 3.36028i | −0.420861 | − | 1.68014i | −2.37608 | − | 1.16372i | 1.00000 | −2.64575 | + | 1.41421i | − | 3.36028i | |||||||||||||||||||||||||||||||||
| 545.4 | 1.00000 | −0.420861 | + | 1.68014i | 1.00000 | 3.36028i | −0.420861 | + | 1.68014i | −2.37608 | + | 1.16372i | 1.00000 | −2.64575 | − | 1.41421i | 3.36028i | |||||||||||||||||||||||||||||||||||
| 545.5 | 1.00000 | 0.420861 | − | 1.68014i | 1.00000 | − | 3.36028i | 0.420861 | − | 1.68014i | 2.37608 | + | 1.16372i | 1.00000 | −2.64575 | − | 1.41421i | − | 3.36028i | |||||||||||||||||||||||||||||||||
| 545.6 | 1.00000 | 0.420861 | + | 1.68014i | 1.00000 | 3.36028i | 0.420861 | + | 1.68014i | 2.37608 | − | 1.16372i | 1.00000 | −2.64575 | + | 1.41421i | 3.36028i | |||||||||||||||||||||||||||||||||||
| 545.7 | 1.00000 | 1.68014 | − | 0.420861i | 1.00000 | − | 0.841723i | 1.68014 | − | 0.420861i | 0.595188 | − | 2.57794i | 1.00000 | 2.64575 | − | 1.41421i | − | 0.841723i | |||||||||||||||||||||||||||||||||
| 545.8 | 1.00000 | 1.68014 | + | 0.420861i | 1.00000 | 0.841723i | 1.68014 | + | 0.420861i | 0.595188 | + | 2.57794i | 1.00000 | 2.64575 | + | 1.41421i | 0.841723i | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 39.d | odd | 2 | 1 | inner |
| 273.g | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 546.2.e.h | yes | 8 |
| 3.b | odd | 2 | 1 | 546.2.e.f | ✓ | 8 | |
| 7.b | odd | 2 | 1 | inner | 546.2.e.h | yes | 8 |
| 13.b | even | 2 | 1 | 546.2.e.f | ✓ | 8 | |
| 21.c | even | 2 | 1 | 546.2.e.f | ✓ | 8 | |
| 39.d | odd | 2 | 1 | inner | 546.2.e.h | yes | 8 |
| 91.b | odd | 2 | 1 | 546.2.e.f | ✓ | 8 | |
| 273.g | even | 2 | 1 | inner | 546.2.e.h | yes | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.2.e.f | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 546.2.e.f | ✓ | 8 | 13.b | even | 2 | 1 | |
| 546.2.e.f | ✓ | 8 | 21.c | even | 2 | 1 | |
| 546.2.e.f | ✓ | 8 | 91.b | odd | 2 | 1 | |
| 546.2.e.h | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 546.2.e.h | yes | 8 | 7.b | odd | 2 | 1 | inner |
| 546.2.e.h | yes | 8 | 39.d | odd | 2 | 1 | inner |
| 546.2.e.h | yes | 8 | 273.g | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\):
|
\( T_{5}^{4} + 12T_{5}^{2} + 8 \)
|
|
\( T_{17}^{4} - 80T_{17}^{2} + 1152 \)
|
|
\( T_{19}^{4} - 52T_{19}^{2} + 648 \)
|
|
\( T_{71}^{2} + 8T_{71} - 96 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( T^{8} - 10T^{4} + 81 \)
|
| $5$ |
\( (T^{4} + 12 T^{2} + 8)^{2} \)
|
| $7$ |
\( T^{8} + 4 T^{6} + \cdots + 2401 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + 310 T^{4} + 28561 \)
|
| $17$ |
\( (T^{4} - 80 T^{2} + 1152)^{2} \)
|
| $19$ |
\( (T^{4} - 52 T^{2} + 648)^{2} \)
|
| $23$ |
\( (T^{4} + 64 T^{2} + 16)^{2} \)
|
| $29$ |
\( (T^{4} + 32 T^{2} + 144)^{2} \)
|
| $31$ |
\( (T^{4} - 40 T^{2} + 288)^{2} \)
|
| $37$ |
\( (T^{4} + 32 T^{2} + 144)^{2} \)
|
| $41$ |
\( (T^{4} + 216 T^{2} + 11552)^{2} \)
|
| $43$ |
\( (T - 8)^{8} \)
|
| $47$ |
\( (T^{4} + 80 T^{2} + 1152)^{2} \)
|
| $53$ |
\( (T^{4} + 64 T^{2} + 16)^{2} \)
|
| $59$ |
\( (T^{4} + 20 T^{2} + 72)^{2} \)
|
| $61$ |
\( (T^{4} + 52 T^{2} + 648)^{2} \)
|
| $67$ |
\( (T^{4} + 176 T^{2} + 576)^{2} \)
|
| $71$ |
\( (T^{2} + 8 T - 96)^{4} \)
|
| $73$ |
\( (T^{4} - 168 T^{2} + 1568)^{2} \)
|
| $79$ |
\( (T^{2} + 12 T + 8)^{4} \)
|
| $83$ |
\( (T^{4} + 180 T^{2} + 6728)^{2} \)
|
| $89$ |
\( (T^{4} + 168 T^{2} + 1568)^{2} \)
|
| $97$ |
\( (T^{4} - 216 T^{2} + 11552)^{2} \)
|
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