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: | 8.0.303595776.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 8\nu^{4} + 16\nu^{2} + 45 ) / 24 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 4\nu^{5} + 2\nu^{3} - 9\nu ) / 54 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{6} + 16\nu^{4} + 80\nu^{2} + 153 ) / 72 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5\nu^{7} + 16\nu^{5} + 8\nu^{3} + 81\nu ) / 216 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -5\nu^{6} - 16\nu^{4} - 8\nu^{2} - 81 ) / 72 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} + 5\nu^{5} + 16\nu^{3} + 45\nu ) / 27 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{4} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} - 4\beta_{5} - \beta_{3} - 2\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -3\beta_{4} + 5\beta_{2} - 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{7} + 8\beta_{5} + 8\beta_{3} \)
|
| \(\nu^{6}\) | \(=\) |
\( -16\beta_{6} + 8\beta_{4} - 16\beta_{2} - 5 \)
|
| \(\nu^{7}\) | \(=\) |
\( 24\beta_{5} - 24\beta_{3} - 13\beta_1 \)
|
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 | −1.26217 | − | 1.18614i | 1.00000 | 3.37228i | −1.26217 | − | 1.18614i | −2.52434 | − | 0.792287i | 1.00000 | 0.186141 | + | 2.99422i | 3.37228i | ||||||||||||||||||||||||||||||||||
| 545.2 | 1.00000 | −1.26217 | + | 1.18614i | 1.00000 | − | 3.37228i | −1.26217 | + | 1.18614i | −2.52434 | + | 0.792287i | 1.00000 | 0.186141 | − | 2.99422i | − | 3.37228i | |||||||||||||||||||||||||||||||||
| 545.3 | 1.00000 | −0.396143 | − | 1.68614i | 1.00000 | 2.37228i | −0.396143 | − | 1.68614i | −0.792287 | + | 2.52434i | 1.00000 | −2.68614 | + | 1.33591i | 2.37228i | |||||||||||||||||||||||||||||||||||
| 545.4 | 1.00000 | −0.396143 | + | 1.68614i | 1.00000 | − | 2.37228i | −0.396143 | + | 1.68614i | −0.792287 | − | 2.52434i | 1.00000 | −2.68614 | − | 1.33591i | − | 2.37228i | |||||||||||||||||||||||||||||||||
| 545.5 | 1.00000 | 0.396143 | − | 1.68614i | 1.00000 | 2.37228i | 0.396143 | − | 1.68614i | 0.792287 | − | 2.52434i | 1.00000 | −2.68614 | − | 1.33591i | 2.37228i | |||||||||||||||||||||||||||||||||||
| 545.6 | 1.00000 | 0.396143 | + | 1.68614i | 1.00000 | − | 2.37228i | 0.396143 | + | 1.68614i | 0.792287 | + | 2.52434i | 1.00000 | −2.68614 | + | 1.33591i | − | 2.37228i | |||||||||||||||||||||||||||||||||
| 545.7 | 1.00000 | 1.26217 | − | 1.18614i | 1.00000 | 3.37228i | 1.26217 | − | 1.18614i | 2.52434 | + | 0.792287i | 1.00000 | 0.186141 | − | 2.99422i | 3.37228i | |||||||||||||||||||||||||||||||||||
| 545.8 | 1.00000 | 1.26217 | + | 1.18614i | 1.00000 | − | 3.37228i | 1.26217 | + | 1.18614i | 2.52434 | − | 0.792287i | 1.00000 | 0.186141 | + | 2.99422i | − | 3.37228i | |||||||||||||||||||||||||||||||||
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.g | yes | 8 |
| 3.b | odd | 2 | 1 | 546.2.e.e | ✓ | 8 | |
| 7.b | odd | 2 | 1 | inner | 546.2.e.g | yes | 8 |
| 13.b | even | 2 | 1 | 546.2.e.e | ✓ | 8 | |
| 21.c | even | 2 | 1 | 546.2.e.e | ✓ | 8 | |
| 39.d | odd | 2 | 1 | inner | 546.2.e.g | yes | 8 |
| 91.b | odd | 2 | 1 | 546.2.e.e | ✓ | 8 | |
| 273.g | even | 2 | 1 | inner | 546.2.e.g | yes | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.2.e.e | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 546.2.e.e | ✓ | 8 | 13.b | even | 2 | 1 | |
| 546.2.e.e | ✓ | 8 | 21.c | even | 2 | 1 | |
| 546.2.e.e | ✓ | 8 | 91.b | odd | 2 | 1 | |
| 546.2.e.g | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 546.2.e.g | yes | 8 | 7.b | odd | 2 | 1 | inner |
| 546.2.e.g | yes | 8 | 39.d | odd | 2 | 1 | inner |
| 546.2.e.g | 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 \(S_{2}^{\mathrm{new}}(546, [\chi])\):
|
\( T_{5}^{4} + 17T_{5}^{2} + 64 \)
|
|
\( T_{17}^{4} - 7T_{17}^{2} + 4 \)
|
|
\( T_{19}^{4} - 63T_{19}^{2} + 324 \)
|
|
\( T_{71} + 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( T^{8} + 5 T^{6} + \cdots + 81 \)
|
| $5$ |
\( (T^{4} + 17 T^{2} + 64)^{2} \)
|
| $7$ |
\( T^{8} - 34T^{4} + 2401 \)
|
| $11$ |
\( (T^{2} - 33)^{4} \)
|
| $13$ |
\( (T^{4} - 22 T^{2} + 169)^{2} \)
|
| $17$ |
\( (T^{4} - 7 T^{2} + 4)^{2} \)
|
| $19$ |
\( (T^{4} - 63 T^{2} + 324)^{2} \)
|
| $23$ |
\( (T^{4} + 46 T^{2} + 1)^{2} \)
|
| $29$ |
\( (T^{4} + 63 T^{2} + 324)^{2} \)
|
| $31$ |
\( (T^{4} - 51 T^{2} + 576)^{2} \)
|
| $37$ |
\( (T^{4} + 94 T^{2} + 1681)^{2} \)
|
| $41$ |
\( (T^{4} + 77 T^{2} + 484)^{2} \)
|
| $43$ |
\( (T^{2} + 11 T + 22)^{4} \)
|
| $47$ |
\( (T^{4} + 129 T^{2} + 2304)^{2} \)
|
| $53$ |
\( (T^{4} + 112 T^{2} + 1024)^{2} \)
|
| $59$ |
\( (T^{4} + 84 T^{2} + 576)^{2} \)
|
| $61$ |
\( (T^{4} + 74 T^{2} + 841)^{2} \)
|
| $67$ |
\( (T^{4} + 43 T^{2} + 256)^{2} \)
|
| $71$ |
\( (T + 2)^{8} \)
|
| $73$ |
\( (T^{4} - 118 T^{2} + 1369)^{2} \)
|
| $79$ |
\( (T^{2} + 13 T + 34)^{4} \)
|
| $83$ |
\( (T^{4} + 68 T^{2} + 1024)^{2} \)
|
| $89$ |
\( (T^{2} + 100)^{4} \)
|
| $97$ |
\( (T^{4} - 139 T^{2} + 4624)^{2} \)
|
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