Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [567,2,Mod(190,567)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(567, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("567.190");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 567 = 3^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 567.f (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: | \(4.52751779461\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.1156923.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - \nu + 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} + 8\nu^{2} - 7\nu + 6 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{5} - 5\nu^{4} + 22\nu^{3} - 28\nu^{2} + 43\nu - 16 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{5} - 13\nu^{4} + 54\nu^{3} - 71\nu^{2} + 99\nu - 40 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -6\nu^{5} + 15\nu^{4} - 64\nu^{3} + 82\nu^{2} - 121\nu + 50 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{5} - 2\beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{5} - 2\beta_{4} - \beta_{3} - \beta_{2} + 4\beta _1 - 7 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{5} + 2\beta_{4} + 4\beta_{3} + \beta_{2} - 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 20\beta_{5} + 14\beta_{4} + 25\beta_{3} + 13\beta_{2} - 25\beta _1 + 16 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -34\beta_{5} - 16\beta_{4} - 59\beta_{3} + 7\beta_{2} - 28\beta _1 + 121 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).
| \(n\) | \(325\) | \(407\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 190.1 |
|
−1.33454 | − | 2.31149i | 0 | −2.56199 | + | 4.43750i | 0.727452 | − | 1.25998i | 0 | −0.500000 | − | 0.866025i | 8.33816 | 0 | −3.88325 | ||||||||||||||||||||||||||||
| 190.2 | 0.261988 | + | 0.453777i | 0 | 0.862724 | − | 1.49428i | −1.10074 | + | 1.90653i | 0 | −0.500000 | − | 0.866025i | 1.95205 | 0 | −1.15352 | |||||||||||||||||||||||||||||
| 190.3 | 1.07255 | + | 1.85771i | 0 | −1.30073 | + | 2.25294i | 1.87328 | − | 3.24462i | 0 | −0.500000 | − | 0.866025i | −1.29021 | 0 | 8.03677 | |||||||||||||||||||||||||||||
| 379.1 | −1.33454 | + | 2.31149i | 0 | −2.56199 | − | 4.43750i | 0.727452 | + | 1.25998i | 0 | −0.500000 | + | 0.866025i | 8.33816 | 0 | −3.88325 | |||||||||||||||||||||||||||||
| 379.2 | 0.261988 | − | 0.453777i | 0 | 0.862724 | + | 1.49428i | −1.10074 | − | 1.90653i | 0 | −0.500000 | + | 0.866025i | 1.95205 | 0 | −1.15352 | |||||||||||||||||||||||||||||
| 379.3 | 1.07255 | − | 1.85771i | 0 | −1.30073 | − | 2.25294i | 1.87328 | + | 3.24462i | 0 | −0.500000 | + | 0.866025i | −1.29021 | 0 | 8.03677 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 567.2.f.m | 6 | |
| 3.b | odd | 2 | 1 | 567.2.f.l | 6 | ||
| 9.c | even | 3 | 1 | 567.2.a.e | ✓ | 3 | |
| 9.c | even | 3 | 1 | inner | 567.2.f.m | 6 | |
| 9.d | odd | 6 | 1 | 567.2.a.f | yes | 3 | |
| 9.d | odd | 6 | 1 | 567.2.f.l | 6 | ||
| 36.f | odd | 6 | 1 | 9072.2.a.bu | 3 | ||
| 36.h | even | 6 | 1 | 9072.2.a.cb | 3 | ||
| 63.l | odd | 6 | 1 | 3969.2.a.o | 3 | ||
| 63.o | even | 6 | 1 | 3969.2.a.n | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 567.2.a.e | ✓ | 3 | 9.c | even | 3 | 1 | |
| 567.2.a.f | yes | 3 | 9.d | odd | 6 | 1 | |
| 567.2.f.l | 6 | 3.b | odd | 2 | 1 | ||
| 567.2.f.l | 6 | 9.d | odd | 6 | 1 | ||
| 567.2.f.m | 6 | 1.a | even | 1 | 1 | trivial | |
| 567.2.f.m | 6 | 9.c | even | 3 | 1 | inner | |
| 3969.2.a.n | 3 | 63.o | even | 6 | 1 | ||
| 3969.2.a.o | 3 | 63.l | odd | 6 | 1 | ||
| 9072.2.a.bu | 3 | 36.f | odd | 6 | 1 | ||
| 9072.2.a.cb | 3 | 36.h | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\):
|
\( T_{2}^{6} + 6T_{2}^{4} - 6T_{2}^{3} + 36T_{2}^{2} - 18T_{2} + 9 \)
|
|
\( T_{5}^{6} - 3T_{5}^{5} + 15T_{5}^{4} - 6T_{5}^{3} + 72T_{5}^{2} - 72T_{5} + 144 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 6 T^{4} + \cdots + 9 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 3 T^{5} + \cdots + 144 \)
|
| $7$ |
\( (T^{2} + T + 1)^{3} \)
|
| $11$ |
\( T^{6} - 6 T^{5} + \cdots + 36 \)
|
| $13$ |
\( T^{6} + 3 T^{5} + \cdots + 12544 \)
|
| $17$ |
\( (T^{3} + 3 T^{2} - 24 T + 12)^{2} \)
|
| $19$ |
\( (T^{3} - 48 T - 56)^{2} \)
|
| $23$ |
\( T^{6} + 6 T^{5} + \cdots + 9216 \)
|
| $29$ |
\( T^{6} - 3 T^{5} + \cdots + 1296 \)
|
| $31$ |
\( (T^{2} + 2 T + 4)^{3} \)
|
| $37$ |
\( (T - 5)^{6} \)
|
| $41$ |
\( T^{6} + 12 T^{5} + \cdots + 5184 \)
|
| $43$ |
\( T^{6} + 12 T^{5} + \cdots + 425104 \)
|
| $47$ |
\( T^{6} + 24 T^{4} + \cdots + 576 \)
|
| $53$ |
\( (T^{3} + 12 T^{2} + 21 T - 6)^{2} \)
|
| $59$ |
\( T^{6} + 18 T^{5} + \cdots + 304704 \)
|
| $61$ |
\( T^{6} - 3 T^{5} + \cdots + 35344 \)
|
| $67$ |
\( T^{6} + 6 T^{5} + \cdots + 68644 \)
|
| $71$ |
\( (T^{3} - 81 T + 108)^{2} \)
|
| $73$ |
\( (T^{3} - 9 T^{2} - 12 T + 16)^{2} \)
|
| $79$ |
\( T^{6} + 6 T^{5} + \cdots + 68644 \)
|
| $83$ |
\( T^{6} - 12 T^{5} + \cdots + 5184 \)
|
| $89$ |
\( (T^{3} + 15 T^{2} + \cdots - 48)^{2} \)
|
| $97$ |
\( T^{6} + 12 T^{5} + \cdots + 33856 \)
|
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