Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [130,2,Mod(47,130)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(130, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("130.47");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 130 = 2 \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 130.j (of order \(4\), 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: | \(1.03805522628\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{11})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 5x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{4} - 5x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{2} + 2\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/130\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(41\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 47.1 |
|
− | 1.00000i | −1.15831 | + | 1.15831i | −1.00000 | 1.65831 | + | 1.50000i | 1.15831 | + | 1.15831i | 3.00000 | 1.00000i | 0.316625i | 1.50000 | − | 1.65831i | |||||||||||||||||||||
| 47.2 | − | 1.00000i | 2.15831 | − | 2.15831i | −1.00000 | −1.65831 | + | 1.50000i | −2.15831 | − | 2.15831i | 3.00000 | 1.00000i | − | 6.31662i | 1.50000 | + | 1.65831i | |||||||||||||||||||||
| 83.1 | 1.00000i | −1.15831 | − | 1.15831i | −1.00000 | 1.65831 | − | 1.50000i | 1.15831 | − | 1.15831i | 3.00000 | − | 1.00000i | − | 0.316625i | 1.50000 | + | 1.65831i | |||||||||||||||||||||
| 83.2 | 1.00000i | 2.15831 | + | 2.15831i | −1.00000 | −1.65831 | − | 1.50000i | −2.15831 | + | 2.15831i | 3.00000 | − | 1.00000i | 6.31662i | 1.50000 | − | 1.65831i | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 130.2.j.d | yes | 4 |
| 3.b | odd | 2 | 1 | 1170.2.w.e | 4 | ||
| 4.b | odd | 2 | 1 | 1040.2.cd.i | 4 | ||
| 5.b | even | 2 | 1 | 650.2.j.f | 4 | ||
| 5.c | odd | 4 | 1 | 130.2.g.d | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 650.2.g.g | 4 | ||
| 13.d | odd | 4 | 1 | 130.2.g.d | ✓ | 4 | |
| 15.e | even | 4 | 1 | 1170.2.m.e | 4 | ||
| 20.e | even | 4 | 1 | 1040.2.bg.k | 4 | ||
| 39.f | even | 4 | 1 | 1170.2.m.e | 4 | ||
| 52.f | even | 4 | 1 | 1040.2.bg.k | 4 | ||
| 65.f | even | 4 | 1 | inner | 130.2.j.d | yes | 4 |
| 65.g | odd | 4 | 1 | 650.2.g.g | 4 | ||
| 65.k | even | 4 | 1 | 650.2.j.f | 4 | ||
| 195.u | odd | 4 | 1 | 1170.2.w.e | 4 | ||
| 260.l | odd | 4 | 1 | 1040.2.cd.i | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 130.2.g.d | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 130.2.g.d | ✓ | 4 | 13.d | odd | 4 | 1 | |
| 130.2.j.d | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 130.2.j.d | yes | 4 | 65.f | even | 4 | 1 | inner |
| 650.2.g.g | 4 | 5.c | odd | 4 | 1 | ||
| 650.2.g.g | 4 | 65.g | odd | 4 | 1 | ||
| 650.2.j.f | 4 | 5.b | even | 2 | 1 | ||
| 650.2.j.f | 4 | 65.k | even | 4 | 1 | ||
| 1040.2.bg.k | 4 | 20.e | even | 4 | 1 | ||
| 1040.2.bg.k | 4 | 52.f | even | 4 | 1 | ||
| 1040.2.cd.i | 4 | 4.b | odd | 2 | 1 | ||
| 1040.2.cd.i | 4 | 260.l | odd | 4 | 1 | ||
| 1170.2.m.e | 4 | 15.e | even | 4 | 1 | ||
| 1170.2.m.e | 4 | 39.f | even | 4 | 1 | ||
| 1170.2.w.e | 4 | 3.b | odd | 2 | 1 | ||
| 1170.2.w.e | 4 | 195.u | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 2T_{3}^{3} + 2T_{3}^{2} + 10T_{3} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(130, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{2} \)
|
| $3$ |
\( T^{4} - 2 T^{3} + \cdots + 25 \)
|
| $5$ |
\( T^{4} - T^{2} + 25 \)
|
| $7$ |
\( (T - 3)^{4} \)
|
| $11$ |
\( T^{4} + 484 \)
|
| $13$ |
\( (T^{2} + 6 T + 13)^{2} \)
|
| $17$ |
\( T^{4} + 6 T^{3} + \cdots + 1 \)
|
| $19$ |
\( (T^{2} - 4 T + 8)^{2} \)
|
| $23$ |
\( T^{4} + 484 \)
|
| $29$ |
\( T^{4} + 40T^{2} + 4 \)
|
| $31$ |
\( (T^{2} - 2 T + 2)^{2} \)
|
| $37$ |
\( (T - 3)^{4} \)
|
| $41$ |
\( T^{4} + 12 T^{3} + \cdots + 16 \)
|
| $43$ |
\( T^{4} + 10 T^{3} + \cdots + 1369 \)
|
| $47$ |
\( (T^{2} + 12 T + 25)^{2} \)
|
| $53$ |
\( T^{4} - 12 T^{3} + \cdots + 4900 \)
|
| $59$ |
\( T^{4} + 12 T^{3} + \cdots + 16 \)
|
| $61$ |
\( (T - 2)^{4} \)
|
| $67$ |
\( T^{4} + 248T^{2} + 5476 \)
|
| $71$ |
\( T^{4} + 18 T^{3} + \cdots + 1225 \)
|
| $73$ |
\( (T^{2} + 16)^{2} \)
|
| $79$ |
\( T^{4} + 216T^{2} + 8100 \)
|
| $83$ |
\( (T^{2} - 6 T - 2)^{2} \)
|
| $89$ |
\( T^{4} + 12 T^{3} + \cdots + 16 \)
|
| $97$ |
\( T^{4} + 200T^{2} + 9604 \)
|
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