Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [546,2,Mod(373,546)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(546, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("546.373");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 546.k (of order \(3\), 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: | \(4.35983195036\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.447703281.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 2x^{6} + 2x^{5} + 3x^{4} + 4x^{3} - 8x^{2} - 8x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 2x^{6} + 2x^{5} + 3x^{4} + 4x^{3} - 8x^{2} - 8x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 2\nu^{5} + 2\nu^{4} + 3\nu^{3} + 4\nu^{2} - 8\nu - 8 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 2\nu^{3} - \nu^{2} + 6\nu + 4 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - \nu^{6} + 2\nu^{4} + \nu^{3} - 6\nu^{2} - 4\nu ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} + 6\nu^{5} + 2\nu^{4} - 3\nu^{3} - 12\nu^{2} + 4\nu + 32 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{5} + \nu^{4} - \nu^{3} - 5\nu^{2} - \nu + 8 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2\nu^{7} + \nu^{6} - 5\nu^{5} - 4\nu^{4} + 4\nu^{3} + 17\nu^{2} + 2\nu - 24 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 2\beta_{5} + \beta_{4} - \beta_{3} + \beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} - \beta_{5} + 2\beta_{3} + 3\beta_{2} + \beta _1 + 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} + \beta_{5} - 2\beta_{4} - 2\beta_{3} + \beta_{2} + 2\beta _1 - 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{7} + 2\beta_{5} + 3\beta_{4} + 4\beta_{3} - 5\beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( -\beta_{7} - 4\beta_{6} + 4\beta_{5} - \beta_{3} - 2\beta _1 - 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).
| \(n\) | \(157\) | \(365\) | \(379\) |
| \(\chi(n)\) | \(-1 - \beta_{3}\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 373.1 |
|
−0.500000 | − | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | −1.97513 | + | 3.42102i | 0.500000 | + | 0.866025i | −1.48662 | − | 2.18860i | 1.00000 | 1.00000 | 3.95025 | ||||||||||||||||||||||||||||||||
| 373.2 | −0.500000 | − | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | −0.228205 | + | 0.395262i | 0.500000 | + | 0.866025i | 2.45374 | + | 0.989520i | 1.00000 | 1.00000 | 0.456409 | |||||||||||||||||||||||||||||||||
| 373.3 | −0.500000 | − | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | 1.14553 | − | 1.98411i | 0.500000 | + | 0.866025i | −1.12588 | + | 2.39424i | 1.00000 | 1.00000 | −2.29105 | |||||||||||||||||||||||||||||||||
| 373.4 | −0.500000 | − | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | 2.05781 | − | 3.56422i | 0.500000 | + | 0.866025i | 1.65876 | − | 2.06119i | 1.00000 | 1.00000 | −4.11561 | |||||||||||||||||||||||||||||||||
| 445.1 | −0.500000 | + | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | −1.97513 | − | 3.42102i | 0.500000 | − | 0.866025i | −1.48662 | + | 2.18860i | 1.00000 | 1.00000 | 3.95025 | |||||||||||||||||||||||||||||||||
| 445.2 | −0.500000 | + | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | −0.228205 | − | 0.395262i | 0.500000 | − | 0.866025i | 2.45374 | − | 0.989520i | 1.00000 | 1.00000 | 0.456409 | |||||||||||||||||||||||||||||||||
| 445.3 | −0.500000 | + | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | 1.14553 | + | 1.98411i | 0.500000 | − | 0.866025i | −1.12588 | − | 2.39424i | 1.00000 | 1.00000 | −2.29105 | |||||||||||||||||||||||||||||||||
| 445.4 | −0.500000 | + | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | 2.05781 | + | 3.56422i | 0.500000 | − | 0.866025i | 1.65876 | + | 2.06119i | 1.00000 | 1.00000 | −4.11561 | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 91.g | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 546.2.k.b | yes | 8 |
| 3.b | odd | 2 | 1 | 1638.2.p.i | 8 | ||
| 7.c | even | 3 | 1 | 546.2.j.d | ✓ | 8 | |
| 13.c | even | 3 | 1 | 546.2.j.d | ✓ | 8 | |
| 21.h | odd | 6 | 1 | 1638.2.m.g | 8 | ||
| 39.i | odd | 6 | 1 | 1638.2.m.g | 8 | ||
| 91.g | even | 3 | 1 | inner | 546.2.k.b | yes | 8 |
| 273.bm | odd | 6 | 1 | 1638.2.p.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.2.j.d | ✓ | 8 | 7.c | even | 3 | 1 | |
| 546.2.j.d | ✓ | 8 | 13.c | even | 3 | 1 | |
| 546.2.k.b | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 546.2.k.b | yes | 8 | 91.g | even | 3 | 1 | inner |
| 1638.2.m.g | 8 | 21.h | odd | 6 | 1 | ||
| 1638.2.m.g | 8 | 39.i | odd | 6 | 1 | ||
| 1638.2.p.i | 8 | 3.b | odd | 2 | 1 | ||
| 1638.2.p.i | 8 | 273.bm | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 2T_{5}^{7} + 21T_{5}^{6} - 26T_{5}^{5} + 332T_{5}^{4} - 442T_{5}^{3} + 1189T_{5}^{2} + 510T_{5} + 289 \)
acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{4} \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( T^{8} - 2 T^{7} + \cdots + 289 \)
|
| $7$ |
\( T^{8} - 3 T^{7} + \cdots + 2401 \)
|
| $11$ |
\( (T^{4} - 6 T^{3} - 10 T^{2} + \cdots - 67)^{2} \)
|
| $13$ |
\( T^{8} + 11 T^{7} + \cdots + 28561 \)
|
| $17$ |
\( T^{8} - 4 T^{7} + \cdots + 1 \)
|
| $19$ |
\( (T^{4} + 6 T^{3} + 5 T^{2} + \cdots - 13)^{2} \)
|
| $23$ |
\( T^{8} + 10 T^{7} + \cdots + 27889 \)
|
| $29$ |
\( T^{8} - 2 T^{7} + \cdots + 4489 \)
|
| $31$ |
\( T^{8} - 6 T^{7} + \cdots + 210681 \)
|
| $37$ |
\( T^{8} + 28 T^{7} + \cdots + 44521 \)
|
| $41$ |
\( T^{8} + 92 T^{6} + \cdots + 1912689 \)
|
| $43$ |
\( T^{8} + 6 T^{7} + \cdots + 4489 \)
|
| $47$ |
\( T^{8} - T^{7} + \cdots + 2601 \)
|
| $53$ |
\( T^{8} - 7 T^{7} + \cdots + 6561 \)
|
| $59$ |
\( T^{8} - 2 T^{7} + \cdots + 96721 \)
|
| $61$ |
\( (T^{4} + 24 T^{3} + \cdots - 4176)^{2} \)
|
| $67$ |
\( (T^{4} - 15 T^{3} + \cdots + 167)^{2} \)
|
| $71$ |
\( T^{8} - 6 T^{7} + \cdots + 674041 \)
|
| $73$ |
\( T^{8} - T^{7} + 8 T^{6} + \cdots + 1 \)
|
| $79$ |
\( T^{8} + 12 T^{7} + \cdots + 6985449 \)
|
| $83$ |
\( (T^{4} + 16 T^{3} + \cdots + 1919)^{2} \)
|
| $89$ |
\( T^{8} - 25 T^{7} + \cdots + 729 \)
|
| $97$ |
\( T^{8} + T^{7} + \cdots + 45873529 \)
|
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