Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [133,2,Mod(64,133)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(133, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("133.64");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 133 = 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 133.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: | \(1.06201034688\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.309123.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - \nu + 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/133\mathbb{Z}\right)^\times\).
| \(n\) | \(78\) | \(115\) |
| \(\chi(n)\) | \(-1 + \beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 64.1 |
|
−1.34981 | − | 2.33795i | 0.849814 | + | 1.47192i | −2.64400 | + | 4.57954i | 0.555632 | + | 0.962383i | 2.29418 | − | 3.97364i | 1.00000 | 8.87636 | 0.0556321 | − | 0.0963576i | 1.50000 | − | 2.59808i | ||||||||||||||||||||||
| 64.2 | −0.380438 | − | 0.658939i | −0.119562 | − | 0.207087i | 0.710533 | − | 1.23068i | 1.97141 | + | 3.41458i | −0.0909717 | + | 0.157568i | 1.00000 | −2.60301 | 1.47141 | − | 2.54856i | 1.50000 | − | 2.59808i | |||||||||||||||||||||||
| 64.3 | 0.730252 | + | 1.26483i | −1.23025 | − | 2.13086i | −0.0665372 | + | 0.115246i | −1.02704 | − | 1.77889i | 1.79679 | − | 3.11213i | 1.00000 | 2.72665 | −1.52704 | + | 2.64491i | 1.50000 | − | 2.59808i | |||||||||||||||||||||||
| 106.1 | −1.34981 | + | 2.33795i | 0.849814 | − | 1.47192i | −2.64400 | − | 4.57954i | 0.555632 | − | 0.962383i | 2.29418 | + | 3.97364i | 1.00000 | 8.87636 | 0.0556321 | + | 0.0963576i | 1.50000 | + | 2.59808i | |||||||||||||||||||||||
| 106.2 | −0.380438 | + | 0.658939i | −0.119562 | + | 0.207087i | 0.710533 | + | 1.23068i | 1.97141 | − | 3.41458i | −0.0909717 | − | 0.157568i | 1.00000 | −2.60301 | 1.47141 | + | 2.54856i | 1.50000 | + | 2.59808i | |||||||||||||||||||||||
| 106.3 | 0.730252 | − | 1.26483i | −1.23025 | + | 2.13086i | −0.0665372 | − | 0.115246i | −1.02704 | + | 1.77889i | 1.79679 | + | 3.11213i | 1.00000 | 2.72665 | −1.52704 | − | 2.64491i | 1.50000 | + | 2.59808i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 133.2.e.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1197.2.k.d | 6 | ||
| 7.b | odd | 2 | 1 | 931.2.e.b | 6 | ||
| 7.c | even | 3 | 1 | 931.2.g.d | 6 | ||
| 7.c | even | 3 | 1 | 931.2.h.c | 6 | ||
| 7.d | odd | 6 | 1 | 931.2.g.c | 6 | ||
| 7.d | odd | 6 | 1 | 931.2.h.d | 6 | ||
| 19.c | even | 3 | 1 | inner | 133.2.e.b | ✓ | 6 |
| 19.c | even | 3 | 1 | 2527.2.a.h | 3 | ||
| 19.d | odd | 6 | 1 | 2527.2.a.g | 3 | ||
| 57.h | odd | 6 | 1 | 1197.2.k.d | 6 | ||
| 133.g | even | 3 | 1 | 931.2.h.c | 6 | ||
| 133.h | even | 3 | 1 | 931.2.g.d | 6 | ||
| 133.k | odd | 6 | 1 | 931.2.h.d | 6 | ||
| 133.m | odd | 6 | 1 | 931.2.e.b | 6 | ||
| 133.t | odd | 6 | 1 | 931.2.g.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 133.2.e.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 133.2.e.b | ✓ | 6 | 19.c | even | 3 | 1 | inner |
| 931.2.e.b | 6 | 7.b | odd | 2 | 1 | ||
| 931.2.e.b | 6 | 133.m | odd | 6 | 1 | ||
| 931.2.g.c | 6 | 7.d | odd | 6 | 1 | ||
| 931.2.g.c | 6 | 133.t | odd | 6 | 1 | ||
| 931.2.g.d | 6 | 7.c | even | 3 | 1 | ||
| 931.2.g.d | 6 | 133.h | even | 3 | 1 | ||
| 931.2.h.c | 6 | 7.c | even | 3 | 1 | ||
| 931.2.h.c | 6 | 133.g | even | 3 | 1 | ||
| 931.2.h.d | 6 | 7.d | odd | 6 | 1 | ||
| 931.2.h.d | 6 | 133.k | odd | 6 | 1 | ||
| 1197.2.k.d | 6 | 3.b | odd | 2 | 1 | ||
| 1197.2.k.d | 6 | 57.h | odd | 6 | 1 | ||
| 2527.2.a.g | 3 | 19.d | odd | 6 | 1 | ||
| 2527.2.a.h | 3 | 19.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 2T_{2}^{5} + 7T_{2}^{4} + 15T_{2}^{2} + 9T_{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(133, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 2 T^{5} + \cdots + 9 \)
|
| $3$ |
\( T^{6} + T^{5} + 5 T^{4} + \cdots + 1 \)
|
| $5$ |
\( T^{6} - 3 T^{5} + \cdots + 81 \)
|
| $7$ |
\( (T - 1)^{6} \)
|
| $11$ |
\( (T^{3} - T^{2} - 12 T - 9)^{2} \)
|
| $13$ |
\( T^{6} - 2 T^{5} + \cdots + 441 \)
|
| $17$ |
\( T^{6} - 7 T^{5} + \cdots + 9 \)
|
| $19$ |
\( (T^{2} + 7 T + 19)^{3} \)
|
| $23$ |
\( T^{6} + 6 T^{5} + \cdots + 9801 \)
|
| $29$ |
\( T^{6} + 33 T^{4} + \cdots + 81 \)
|
| $31$ |
\( (T^{3} + 5 T^{2} + \cdots - 489)^{2} \)
|
| $37$ |
\( (T^{3} - 11 T^{2} + \cdots + 89)^{2} \)
|
| $41$ |
\( T^{6} + 11 T^{5} + \cdots + 1089 \)
|
| $43$ |
\( T^{6} + 12 T^{5} + \cdots + 90601 \)
|
| $47$ |
\( T^{6} + 10 T^{5} + \cdots + 28224 \)
|
| $53$ |
\( T^{6} + 15 T^{5} + \cdots + 3969 \)
|
| $59$ |
\( T^{6} + 8 T^{5} + \cdots + 40401 \)
|
| $61$ |
\( T^{6} - T^{5} + \cdots + 14641 \)
|
| $67$ |
\( T^{6} + 3 T^{5} + \cdots + 37249 \)
|
| $71$ |
\( T^{6} + 28 T^{5} + \cdots + 576081 \)
|
| $73$ |
\( T^{6} - 13 T^{5} + \cdots + 502681 \)
|
| $79$ |
\( T^{6} + 31 T^{5} + \cdots + 239121 \)
|
| $83$ |
\( (T^{3} + 6 T^{2} + \cdots - 1593)^{2} \)
|
| $89$ |
\( T^{6} - 8 T^{5} + \cdots + 3415104 \)
|
| $97$ |
\( T^{6} - 10 T^{5} + \cdots + 66049 \)
|
| show more | |
| show less |