Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [132,3,Mod(67,132)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(132, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("132.67");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 132 = 2^{2} \cdot 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 132.g (of order \(2\), degree \(1\), 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: | \(3.59673948956\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-11})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 2\nu^{2} - 2\nu - 6 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{3} + 2\nu^{2} + 10\nu + 3 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + 4 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} - \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 5\beta _1 + 5 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/132\mathbb{Z}\right)^\times\).
| \(n\) | \(13\) | \(67\) | \(89\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
−1.00000 | − | 1.73205i | − | 1.73205i | −2.00000 | + | 3.46410i | −4.74456 | −3.00000 | + | 1.73205i | 11.6819i | 8.00000 | −3.00000 | 4.74456 | + | 8.21782i | |||||||||||||||||||||
| 67.2 | −1.00000 | − | 1.73205i | − | 1.73205i | −2.00000 | + | 3.46410i | 6.74456 | −3.00000 | + | 1.73205i | − | 8.21782i | 8.00000 | −3.00000 | −6.74456 | − | 11.6819i | |||||||||||||||||||||
| 67.3 | −1.00000 | + | 1.73205i | 1.73205i | −2.00000 | − | 3.46410i | −4.74456 | −3.00000 | − | 1.73205i | − | 11.6819i | 8.00000 | −3.00000 | 4.74456 | − | 8.21782i | ||||||||||||||||||||||
| 67.4 | −1.00000 | + | 1.73205i | 1.73205i | −2.00000 | − | 3.46410i | 6.74456 | −3.00000 | − | 1.73205i | 8.21782i | 8.00000 | −3.00000 | −6.74456 | + | 11.6819i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 132.3.g.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 396.3.g.a | 4 | ||
| 4.b | odd | 2 | 1 | inner | 132.3.g.a | ✓ | 4 |
| 8.b | even | 2 | 1 | 2112.3.l.a | 4 | ||
| 8.d | odd | 2 | 1 | 2112.3.l.a | 4 | ||
| 12.b | even | 2 | 1 | 396.3.g.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 132.3.g.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 132.3.g.a | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 396.3.g.a | 4 | 3.b | odd | 2 | 1 | ||
| 396.3.g.a | 4 | 12.b | even | 2 | 1 | ||
| 2112.3.l.a | 4 | 8.b | even | 2 | 1 | ||
| 2112.3.l.a | 4 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 2T_{5} - 32 \)
acting on \(S_{3}^{\mathrm{new}}(132, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 4)^{2} \)
|
| $3$ |
\( (T^{2} + 3)^{2} \)
|
| $5$ |
\( (T^{2} - 2 T - 32)^{2} \)
|
| $7$ |
\( T^{4} + 204T^{2} + 9216 \)
|
| $11$ |
\( (T^{2} + 11)^{2} \)
|
| $13$ |
\( (T^{2} - 10 T - 8)^{2} \)
|
| $17$ |
\( (T^{2} + 40 T + 268)^{2} \)
|
| $19$ |
\( T^{4} + 816 T^{2} + 147456 \)
|
| $23$ |
\( T^{4} + 924 T^{2} + 69696 \)
|
| $29$ |
\( (T^{2} - 56 T + 652)^{2} \)
|
| $31$ |
\( T^{4} + 3552 T^{2} + 1937664 \)
|
| $37$ |
\( (T^{2} + 20 T - 428)^{2} \)
|
| $41$ |
\( (T^{2} + 4 T - 4748)^{2} \)
|
| $43$ |
\( T^{4} + 3552 T^{2} + 1937664 \)
|
| $47$ |
\( T^{4} + 2268 T^{2} + 419904 \)
|
| $53$ |
\( (T^{2} - 26 T - 128)^{2} \)
|
| $59$ |
\( T^{4} + 7344 T^{2} + 11943936 \)
|
| $61$ |
\( (T^{2} + 2 T - 2672)^{2} \)
|
| $67$ |
\( (T^{2} + 3888)^{2} \)
|
| $71$ |
\( T^{4} + 7548 T^{2} + 3968064 \)
|
| $73$ |
\( (T^{2} - 124 T - 908)^{2} \)
|
| $79$ |
\( T^{4} + 21804 T^{2} + 35426304 \)
|
| $83$ |
\( (T^{2} + 5808)^{2} \)
|
| $89$ |
\( (T^{2} + 292 T + 20788)^{2} \)
|
| $97$ |
\( (T^{2} - 304 T + 21916)^{2} \)
|
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