Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [143,2,Mod(100,143)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(143, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("143.100");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 143 = 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 143.e (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: | \(1.14186074890\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.1714608.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 30x^{2} - 21x + 7 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 30x^{2} - 21x + 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} + 2\nu^{3} + 10\nu^{2} - 7\nu ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} - 18\nu^{3} + 22\nu^{2} - 28\nu + 7 ) / 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} - 5\nu^{4} + 25\nu^{3} - 29\nu^{2} + 56\nu - 14 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{3} - 3\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} + 3\beta_{4} - 4\beta_{3} + 2\beta_{2} - 6\beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( -4\beta_{5} - 5\beta_{4} - 8\beta_{3} + 5\beta_{2} + 9\beta _1 + 21 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/143\mathbb{Z}\right)^\times\).
| \(n\) | \(67\) | \(79\) |
| \(\chi(n)\) | \(\beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 100.1 |
|
−0.500000 | + | 0.866025i | −1.30084 | + | 2.25312i | 0.500000 | + | 0.866025i | 3.76873 | −1.30084 | − | 2.25312i | −0.0835276 | − | 0.144674i | −3.00000 | −1.88437 | − | 3.26382i | −1.88437 | + | 3.26382i | ||||||||||||||||||||||
| 100.2 | −0.500000 | + | 0.866025i | 0.169938 | − | 0.294342i | 0.500000 | + | 0.866025i | −2.88448 | 0.169938 | + | 0.294342i | 1.77230 | + | 3.06972i | −3.00000 | 1.44224 | + | 2.49804i | 1.44224 | − | 2.49804i | |||||||||||||||||||||||
| 100.3 | −0.500000 | + | 0.866025i | 1.13090 | − | 1.95878i | 0.500000 | + | 0.866025i | 2.11575 | 1.13090 | + | 1.95878i | −1.68878 | − | 2.92505i | −3.00000 | −1.05787 | − | 1.83229i | −1.05787 | + | 1.83229i | |||||||||||||||||||||||
| 133.1 | −0.500000 | − | 0.866025i | −1.30084 | − | 2.25312i | 0.500000 | − | 0.866025i | 3.76873 | −1.30084 | + | 2.25312i | −0.0835276 | + | 0.144674i | −3.00000 | −1.88437 | + | 3.26382i | −1.88437 | − | 3.26382i | |||||||||||||||||||||||
| 133.2 | −0.500000 | − | 0.866025i | 0.169938 | + | 0.294342i | 0.500000 | − | 0.866025i | −2.88448 | 0.169938 | − | 0.294342i | 1.77230 | − | 3.06972i | −3.00000 | 1.44224 | − | 2.49804i | 1.44224 | + | 2.49804i | |||||||||||||||||||||||
| 133.3 | −0.500000 | − | 0.866025i | 1.13090 | + | 1.95878i | 0.500000 | − | 0.866025i | 2.11575 | 1.13090 | − | 1.95878i | −1.68878 | + | 2.92505i | −3.00000 | −1.05787 | + | 1.83229i | −1.05787 | − | 1.83229i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 143.2.e.a | ✓ | 6 |
| 13.c | even | 3 | 1 | inner | 143.2.e.a | ✓ | 6 |
| 13.c | even | 3 | 1 | 1859.2.a.h | 3 | ||
| 13.e | even | 6 | 1 | 1859.2.a.e | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 143.2.e.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 143.2.e.a | ✓ | 6 | 13.c | even | 3 | 1 | inner |
| 1859.2.a.e | 3 | 13.e | even | 6 | 1 | ||
| 1859.2.a.h | 3 | 13.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(143, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{3} \)
|
| $3$ |
\( T^{6} + 6 T^{4} + \cdots + 4 \)
|
| $5$ |
\( (T^{3} - 3 T^{2} - 9 T + 23)^{2} \)
|
| $7$ |
\( T^{6} + 12 T^{4} + \cdots + 4 \)
|
| $11$ |
\( (T^{2} + T + 1)^{3} \)
|
| $13$ |
\( T^{6} - 9 T^{5} + \cdots + 2197 \)
|
| $17$ |
\( T^{6} - 3 T^{5} + \cdots + 2401 \)
|
| $19$ |
\( T^{6} + 6 T^{5} + \cdots + 4356 \)
|
| $23$ |
\( T^{6} + 24 T^{4} + \cdots + 256 \)
|
| $29$ |
\( T^{6} - 3 T^{5} + \cdots + 3249 \)
|
| $31$ |
\( (T^{3} - 6 T^{2} + \cdots + 536)^{2} \)
|
| $37$ |
\( T^{6} + 3 T^{5} + \cdots + 77841 \)
|
| $41$ |
\( T^{6} + 3 T^{5} + \cdots + 5329 \)
|
| $43$ |
\( T^{6} - 6 T^{5} + \cdots + 576 \)
|
| $47$ |
\( (T^{3} + 6 T^{2} + \cdots - 122)^{2} \)
|
| $53$ |
\( (T^{3} - 3 T^{2} - 45 T + 63)^{2} \)
|
| $59$ |
\( T^{6} - 18 T^{5} + \cdots + 33124 \)
|
| $61$ |
\( T^{6} + 9 T^{5} + \cdots + 35721 \)
|
| $67$ |
\( T^{6} + 12 T^{5} + \cdots + 40804 \)
|
| $71$ |
\( (T^{2} + 6 T + 36)^{3} \)
|
| $73$ |
\( (T^{3} - 9 T^{2} + \cdots + 2483)^{2} \)
|
| $79$ |
\( (T^{3} + 12 T^{2} + \cdots + 22)^{2} \)
|
| $83$ |
\( (T^{3} - 84 T + 294)^{2} \)
|
| $89$ |
\( T^{6} + 18 T^{5} + \cdots + 571536 \)
|
| $97$ |
\( T^{6} + 18 T^{5} + \cdots + 13456 \)
|
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