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: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
|
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 1 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} - 1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/133\mathbb{Z}\right)^\times\).
| \(n\) | \(78\) | \(115\) |
| \(\chi(n)\) | \(\beta_{3}\) | \(1\) |
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 |
|
−0.809017 | − | 1.40126i | −1.11803 | − | 1.93649i | −0.309017 | + | 0.535233i | −0.500000 | − | 0.866025i | −1.80902 | + | 3.13331i | 1.00000 | −2.23607 | −1.00000 | + | 1.73205i | −0.809017 | + | 1.40126i | ||||||||||||||||
| 64.2 | 0.309017 | + | 0.535233i | 1.11803 | + | 1.93649i | 0.809017 | − | 1.40126i | −0.500000 | − | 0.866025i | −0.690983 | + | 1.19682i | 1.00000 | 2.23607 | −1.00000 | + | 1.73205i | 0.309017 | − | 0.535233i | |||||||||||||||||
| 106.1 | −0.809017 | + | 1.40126i | −1.11803 | + | 1.93649i | −0.309017 | − | 0.535233i | −0.500000 | + | 0.866025i | −1.80902 | − | 3.13331i | 1.00000 | −2.23607 | −1.00000 | − | 1.73205i | −0.809017 | − | 1.40126i | |||||||||||||||||
| 106.2 | 0.309017 | − | 0.535233i | 1.11803 | − | 1.93649i | 0.809017 | + | 1.40126i | −0.500000 | + | 0.866025i | −0.690983 | − | 1.19682i | 1.00000 | 2.23607 | −1.00000 | − | 1.73205i | 0.309017 | + | 0.535233i | |||||||||||||||||
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.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1197.2.k.c | 4 | ||
| 7.b | odd | 2 | 1 | 931.2.e.a | 4 | ||
| 7.c | even | 3 | 1 | 931.2.g.a | 4 | ||
| 7.c | even | 3 | 1 | 931.2.h.b | 4 | ||
| 7.d | odd | 6 | 1 | 931.2.g.b | 4 | ||
| 7.d | odd | 6 | 1 | 931.2.h.a | 4 | ||
| 19.c | even | 3 | 1 | inner | 133.2.e.a | ✓ | 4 |
| 19.c | even | 3 | 1 | 2527.2.a.c | 2 | ||
| 19.d | odd | 6 | 1 | 2527.2.a.b | 2 | ||
| 57.h | odd | 6 | 1 | 1197.2.k.c | 4 | ||
| 133.g | even | 3 | 1 | 931.2.h.b | 4 | ||
| 133.h | even | 3 | 1 | 931.2.g.a | 4 | ||
| 133.k | odd | 6 | 1 | 931.2.h.a | 4 | ||
| 133.m | odd | 6 | 1 | 931.2.e.a | 4 | ||
| 133.t | odd | 6 | 1 | 931.2.g.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 133.2.e.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 133.2.e.a | ✓ | 4 | 19.c | even | 3 | 1 | inner |
| 931.2.e.a | 4 | 7.b | odd | 2 | 1 | ||
| 931.2.e.a | 4 | 133.m | odd | 6 | 1 | ||
| 931.2.g.a | 4 | 7.c | even | 3 | 1 | ||
| 931.2.g.a | 4 | 133.h | even | 3 | 1 | ||
| 931.2.g.b | 4 | 7.d | odd | 6 | 1 | ||
| 931.2.g.b | 4 | 133.t | odd | 6 | 1 | ||
| 931.2.h.a | 4 | 7.d | odd | 6 | 1 | ||
| 931.2.h.a | 4 | 133.k | odd | 6 | 1 | ||
| 931.2.h.b | 4 | 7.c | even | 3 | 1 | ||
| 931.2.h.b | 4 | 133.g | even | 3 | 1 | ||
| 1197.2.k.c | 4 | 3.b | odd | 2 | 1 | ||
| 1197.2.k.c | 4 | 57.h | odd | 6 | 1 | ||
| 2527.2.a.b | 2 | 19.d | odd | 6 | 1 | ||
| 2527.2.a.c | 2 | 19.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + T_{2}^{3} + 2T_{2}^{2} - T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(133, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + T^{3} + 2 T^{2} + \cdots + 1 \)
|
| $3$ |
\( T^{4} + 5T^{2} + 25 \)
|
| $5$ |
\( (T^{2} + T + 1)^{2} \)
|
| $7$ |
\( (T - 1)^{4} \)
|
| $11$ |
\( (T^{2} + 4 T - 16)^{2} \)
|
| $13$ |
\( (T^{2} + 5 T + 25)^{2} \)
|
| $17$ |
\( (T^{2} + 5 T + 25)^{2} \)
|
| $19$ |
\( (T^{2} - 4 T + 19)^{2} \)
|
| $23$ |
\( T^{4} - 8 T^{3} + \cdots + 121 \)
|
| $29$ |
\( T^{4} + 2 T^{3} + \cdots + 361 \)
|
| $31$ |
\( (T - 4)^{4} \)
|
| $37$ |
\( (T^{2} + 8 T - 4)^{2} \)
|
| $41$ |
\( T^{4} + 6 T^{3} + \cdots + 121 \)
|
| $43$ |
\( T^{4} - 16 T^{3} + \cdots + 3481 \)
|
| $47$ |
\( T^{4} + 5T^{2} + 25 \)
|
| $53$ |
\( (T^{2} + 3 T + 9)^{2} \)
|
| $59$ |
\( T^{4} - 4 T^{3} + \cdots + 1 \)
|
| $61$ |
\( T^{4} + 6 T^{3} + \cdots + 121 \)
|
| $67$ |
\( T^{4} - 16 T^{3} + \cdots + 3481 \)
|
| $71$ |
\( T^{4} - 20 T^{3} + \cdots + 3025 \)
|
| $73$ |
\( T^{4} + 2 T^{3} + \cdots + 6241 \)
|
| $79$ |
\( T^{4} + 12 T^{3} + \cdots + 961 \)
|
| $83$ |
\( (T - 8)^{4} \)
|
| $89$ |
\( T^{4} + 10 T^{3} + \cdots + 3025 \)
|
| $97$ |
\( T^{4} - 6 T^{3} + \cdots + 5041 \)
|
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