Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [133,2,Mod(75,133)]
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([5, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("133.75");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 133 = 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 133.o (of order \(6\), 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 \(\Q(\zeta_{6})\) |
| Coefficient field: | 6.0.6967728.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 8\nu^{4} + 64\nu^{3} - 50\nu^{2} + 7\nu - 56 ) / 393 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 56\nu^{5} - 55\nu^{4} + 440\nu^{3} + 344\nu^{2} + 2750\nu - 385 ) / 393 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 70\nu^{5} - 36\nu^{4} + 550\nu^{3} + 561\nu^{2} + 3634\nu + 534 ) / 393 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 77\nu^{5} - 92\nu^{4} + 605\nu^{3} + 211\nu^{2} + 3683\nu - 1430 ) / 393 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{5} - \beta_{4} + 4\beta_{3} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 7\beta_{2} - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{5} + 16\beta_{4} - 31\beta_{3} - 6\beta _1 - 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( 28\beta_{5} + 14\beta_{4} - 48\beta_{3} - 55\beta_{2} - 55\beta _1 + 28 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/133\mathbb{Z}\right)^\times\).
| \(n\) | \(78\) | \(115\) |
| \(\chi(n)\) | \(-1\) | \(1 + \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 75.1 |
|
1.50000 | + | 0.866025i | −1.63654 | − | 2.83456i | 0.500000 | + | 0.866025i | −0.583430 | − | 0.336844i | − | 5.66913i | 2.21997 | − | 1.43936i | − | 1.73205i | −3.85650 | + | 6.67966i | −0.583430 | − | 1.01053i | ||||||||||||||||||||
| 75.2 | 1.50000 | + | 0.866025i | −0.429782 | − | 0.744405i | 0.500000 | + | 0.866025i | 1.99014 | + | 1.14901i | − | 1.48881i | −1.56036 | + | 2.13665i | − | 1.73205i | 1.13057 | − | 1.95821i | 1.99014 | + | 3.44702i | |||||||||||||||||||||
| 75.3 | 1.50000 | + | 0.866025i | 1.06632 | + | 1.84692i | 0.500000 | + | 0.866025i | −2.90671 | − | 1.67819i | 3.69384i | 1.84039 | + | 1.90078i | − | 1.73205i | −0.774071 | + | 1.34073i | −2.90671 | − | 5.03457i | ||||||||||||||||||||||
| 94.1 | 1.50000 | − | 0.866025i | −1.63654 | + | 2.83456i | 0.500000 | − | 0.866025i | −0.583430 | + | 0.336844i | 5.66913i | 2.21997 | + | 1.43936i | 1.73205i | −3.85650 | − | 6.67966i | −0.583430 | + | 1.01053i | |||||||||||||||||||||||
| 94.2 | 1.50000 | − | 0.866025i | −0.429782 | + | 0.744405i | 0.500000 | − | 0.866025i | 1.99014 | − | 1.14901i | 1.48881i | −1.56036 | − | 2.13665i | 1.73205i | 1.13057 | + | 1.95821i | 1.99014 | − | 3.44702i | |||||||||||||||||||||||
| 94.3 | 1.50000 | − | 0.866025i | 1.06632 | − | 1.84692i | 0.500000 | − | 0.866025i | −2.90671 | + | 1.67819i | − | 3.69384i | 1.84039 | − | 1.90078i | 1.73205i | −0.774071 | − | 1.34073i | −2.90671 | + | 5.03457i | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 133.o | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 133.2.o.e | yes | 6 |
| 7.b | odd | 2 | 1 | 931.2.o.f | 6 | ||
| 7.c | even | 3 | 1 | 931.2.c.e | 6 | ||
| 7.c | even | 3 | 1 | 931.2.o.e | 6 | ||
| 7.d | odd | 6 | 1 | 133.2.o.d | ✓ | 6 | |
| 7.d | odd | 6 | 1 | 931.2.c.d | 6 | ||
| 19.b | odd | 2 | 1 | 133.2.o.d | ✓ | 6 | |
| 133.c | even | 2 | 1 | 931.2.o.e | 6 | ||
| 133.o | even | 6 | 1 | inner | 133.2.o.e | yes | 6 |
| 133.o | even | 6 | 1 | 931.2.c.e | 6 | ||
| 133.r | odd | 6 | 1 | 931.2.c.d | 6 | ||
| 133.r | odd | 6 | 1 | 931.2.o.f | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 133.2.o.d | ✓ | 6 | 7.d | odd | 6 | 1 | |
| 133.2.o.d | ✓ | 6 | 19.b | odd | 2 | 1 | |
| 133.2.o.e | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 133.2.o.e | yes | 6 | 133.o | even | 6 | 1 | inner |
| 931.2.c.d | 6 | 7.d | odd | 6 | 1 | ||
| 931.2.c.d | 6 | 133.r | odd | 6 | 1 | ||
| 931.2.c.e | 6 | 7.c | even | 3 | 1 | ||
| 931.2.c.e | 6 | 133.o | even | 6 | 1 | ||
| 931.2.o.e | 6 | 7.c | even | 3 | 1 | ||
| 931.2.o.e | 6 | 133.c | even | 2 | 1 | ||
| 931.2.o.f | 6 | 7.b | odd | 2 | 1 | ||
| 931.2.o.f | 6 | 133.r | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 3T_{2} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(133, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 3 T + 3)^{3} \)
|
| $3$ |
\( T^{6} + 2 T^{5} + \cdots + 36 \)
|
| $5$ |
\( T^{6} + 3 T^{5} + \cdots + 27 \)
|
| $7$ |
\( T^{6} - 5 T^{5} + \cdots + 343 \)
|
| $11$ |
\( T^{6} - 5 T^{5} + \cdots + 81 \)
|
| $13$ |
\( (T^{3} + 4 T^{2} - 12 T + 6)^{2} \)
|
| $17$ |
\( T^{6} + 12 T^{5} + \cdots + 48 \)
|
| $19$ |
\( T^{6} + 11 T^{5} + \cdots + 6859 \)
|
| $23$ |
\( T^{6} + T^{5} + \cdots + 441 \)
|
| $29$ |
\( T^{6} + 108 T^{4} + \cdots + 38988 \)
|
| $31$ |
\( T^{6} - 16 T^{5} + \cdots + 12996 \)
|
| $37$ |
\( T^{6} - 12 T^{5} + \cdots + 972 \)
|
| $41$ |
\( (T^{3} + 6 T^{2} + \cdots - 216)^{2} \)
|
| $43$ |
\( (T^{3} - 5 T^{2} + \cdots + 167)^{2} \)
|
| $47$ |
\( T^{6} + 27 T^{5} + \cdots + 6627 \)
|
| $53$ |
\( T^{6} + 6 T^{5} + \cdots + 428652 \)
|
| $59$ |
\( (T^{2} - 6 T + 36)^{3} \)
|
| $61$ |
\( T^{6} - 15 T^{5} + \cdots + 3 \)
|
| $67$ |
\( T^{6} + 6 T^{5} + \cdots + 1728 \)
|
| $71$ |
\( T^{6} + 108 T^{4} + \cdots + 108 \)
|
| $73$ |
\( T^{6} + 3 T^{5} + \cdots + 11907 \)
|
| $79$ |
\( T^{6} - 198 T^{4} + \cdots + 38988 \)
|
| $83$ |
\( T^{6} + 341 T^{4} + \cdots + 1232643 \)
|
| $89$ |
\( T^{6} + 48 T^{4} + \cdots + 15876 \)
|
| $97$ |
\( (T^{3} - 4 T^{2} + \cdots + 288)^{2} \)
|
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