Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [129,2,Mod(49,129)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(129, 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("129.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 129 = 3 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 129.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: | \(1.03007018607\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.64827.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 3\beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/129\mathbb{Z}\right)^\times\).
| \(n\) | \(44\) | \(46\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
−1.24698 | −0.500000 | + | 0.866025i | −0.445042 | 1.57942 | − | 2.73563i | 0.623490 | − | 1.07992i | 0.777479 | + | 1.34663i | 3.04892 | −0.500000 | − | 0.866025i | −1.96950 | + | 3.41127i | ||||||||||||||||||||||||
| 49.2 | 0.445042 | −0.500000 | + | 0.866025i | −1.80194 | −2.14795 | + | 3.72036i | −0.222521 | + | 0.385418i | 0.0990311 | + | 0.171527i | −1.69202 | −0.500000 | − | 0.866025i | −0.955927 | + | 1.65571i | |||||||||||||||||||||||||
| 49.3 | 1.80194 | −0.500000 | + | 0.866025i | 1.24698 | 1.06853 | − | 1.85075i | −0.900969 | + | 1.56052i | 1.62349 | + | 2.81197i | −1.35690 | −0.500000 | − | 0.866025i | 1.92543 | − | 3.33494i | |||||||||||||||||||||||||
| 79.1 | −1.24698 | −0.500000 | − | 0.866025i | −0.445042 | 1.57942 | + | 2.73563i | 0.623490 | + | 1.07992i | 0.777479 | − | 1.34663i | 3.04892 | −0.500000 | + | 0.866025i | −1.96950 | − | 3.41127i | |||||||||||||||||||||||||
| 79.2 | 0.445042 | −0.500000 | − | 0.866025i | −1.80194 | −2.14795 | − | 3.72036i | −0.222521 | − | 0.385418i | 0.0990311 | − | 0.171527i | −1.69202 | −0.500000 | + | 0.866025i | −0.955927 | − | 1.65571i | |||||||||||||||||||||||||
| 79.3 | 1.80194 | −0.500000 | − | 0.866025i | 1.24698 | 1.06853 | + | 1.85075i | −0.900969 | − | 1.56052i | 1.62349 | − | 2.81197i | −1.35690 | −0.500000 | + | 0.866025i | 1.92543 | + | 3.33494i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 43.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 129.2.e.c | ✓ | 6 |
| 3.b | odd | 2 | 1 | 387.2.h.e | 6 | ||
| 4.b | odd | 2 | 1 | 2064.2.q.n | 6 | ||
| 43.c | even | 3 | 1 | inner | 129.2.e.c | ✓ | 6 |
| 43.c | even | 3 | 1 | 5547.2.a.n | 3 | ||
| 43.d | odd | 6 | 1 | 5547.2.a.j | 3 | ||
| 129.f | odd | 6 | 1 | 387.2.h.e | 6 | ||
| 172.g | odd | 6 | 1 | 2064.2.q.n | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 129.2.e.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 129.2.e.c | ✓ | 6 | 43.c | even | 3 | 1 | inner |
| 387.2.h.e | 6 | 3.b | odd | 2 | 1 | ||
| 387.2.h.e | 6 | 129.f | odd | 6 | 1 | ||
| 2064.2.q.n | 6 | 4.b | odd | 2 | 1 | ||
| 2064.2.q.n | 6 | 172.g | odd | 6 | 1 | ||
| 5547.2.a.j | 3 | 43.d | odd | 6 | 1 | ||
| 5547.2.a.n | 3 | 43.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - T_{2}^{2} - 2T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(129, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} - T^{2} - 2 T + 1)^{2} \)
|
| $3$ |
\( (T^{2} + T + 1)^{3} \)
|
| $5$ |
\( T^{6} - T^{5} + \cdots + 841 \)
|
| $7$ |
\( T^{6} - 5 T^{5} + \cdots + 1 \)
|
| $11$ |
\( (T^{3} - 10 T^{2} + \cdots + 41)^{2} \)
|
| $13$ |
\( T^{6} + 3 T^{5} + \cdots + 729 \)
|
| $17$ |
\( T^{6} + 4 T^{5} + \cdots + 1 \)
|
| $19$ |
\( T^{6} + 5 T^{5} + \cdots + 9409 \)
|
| $23$ |
\( T^{6} + 2 T^{5} + \cdots + 5041 \)
|
| $29$ |
\( T^{6} - 4 T^{5} + \cdots + 1 \)
|
| $31$ |
\( T^{6} + T^{5} + \cdots + 841 \)
|
| $37$ |
\( T^{6} + 9 T^{5} + \cdots + 28561 \)
|
| $41$ |
\( (T^{3} - 6 T^{2} + \cdots + 559)^{2} \)
|
| $43$ |
\( T^{6} - 18 T^{5} + \cdots + 79507 \)
|
| $47$ |
\( (T^{3} - 4 T^{2} - 67 T + 29)^{2} \)
|
| $53$ |
\( T^{6} - 5 T^{5} + \cdots + 169 \)
|
| $59$ |
\( (T^{3} + 7 T^{2} + \cdots - 301)^{2} \)
|
| $61$ |
\( T^{6} - T^{5} + \cdots + 169 \)
|
| $67$ |
\( T^{6} + 13 T^{5} + \cdots + 169 \)
|
| $71$ |
\( T^{6} - 8 T^{5} + \cdots + 64 \)
|
| $73$ |
\( T^{6} - 7 T^{5} + \cdots + 41209 \)
|
| $79$ |
\( T^{6} + 12 T^{5} + \cdots + 2283121 \)
|
| $83$ |
\( T^{6} - T^{5} + \cdots + 841 \)
|
| $89$ |
\( T^{6} + 10 T^{5} + \cdots + 64 \)
|
| $97$ |
\( (T^{3} - 16 T^{2} + \cdots + 2059)^{2} \)
|
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