Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [117,4,Mod(64,117)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(117, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("117.64");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 117 = 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 117.b (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: | \(6.90322347067\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.5054412.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 29x^{2} + 48 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 39) |
| 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
\( x^{4} + 29x^{2} + 48 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 25\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 15 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 15 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} - 25\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).
| \(n\) | \(28\) | \(92\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 64.1 |
|
− | 5.21898i | 0 | −19.2377 | 5.83936i | 0 | 31.3139i | 58.6495i | 0 | 30.4755 | |||||||||||||||||||||||||||||
| 64.2 | − | 1.32750i | 0 | 6.23774 | − | 15.4241i | 0 | 7.96501i | − | 18.9006i | 0 | −20.4755 | ||||||||||||||||||||||||||||
| 64.3 | 1.32750i | 0 | 6.23774 | 15.4241i | 0 | − | 7.96501i | 18.9006i | 0 | −20.4755 | ||||||||||||||||||||||||||||||
| 64.4 | 5.21898i | 0 | −19.2377 | − | 5.83936i | 0 | − | 31.3139i | − | 58.6495i | 0 | 30.4755 | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 117.4.b.d | 4 | |
| 3.b | odd | 2 | 1 | 39.4.b.a | ✓ | 4 | |
| 12.b | even | 2 | 1 | 624.4.c.e | 4 | ||
| 13.b | even | 2 | 1 | inner | 117.4.b.d | 4 | |
| 13.d | odd | 4 | 2 | 1521.4.a.x | 4 | ||
| 39.d | odd | 2 | 1 | 39.4.b.a | ✓ | 4 | |
| 39.f | even | 4 | 2 | 507.4.a.j | 4 | ||
| 156.h | even | 2 | 1 | 624.4.c.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 39.4.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 39.4.b.a | ✓ | 4 | 39.d | odd | 2 | 1 | |
| 117.4.b.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 117.4.b.d | 4 | 13.b | even | 2 | 1 | inner | |
| 507.4.a.j | 4 | 39.f | even | 4 | 2 | ||
| 624.4.c.e | 4 | 12.b | even | 2 | 1 | ||
| 624.4.c.e | 4 | 156.h | even | 2 | 1 | ||
| 1521.4.a.x | 4 | 13.d | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 29T_{2}^{2} + 48 \)
acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 29T^{2} + 48 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 272T^{2} + 8112 \)
|
| $7$ |
\( T^{4} + 1044 T^{2} + 62208 \)
|
| $11$ |
\( T^{4} + 428 T^{2} + 43200 \)
|
| $13$ |
\( T^{4} + 72 T^{3} + \cdots + 4826809 \)
|
| $17$ |
\( (T + 54)^{4} \)
|
| $19$ |
\( T^{4} + 11556 T^{2} + 31492800 \)
|
| $23$ |
\( (T^{2} + 60 T - 22464)^{2} \)
|
| $29$ |
\( (T^{2} - 24 T - 23220)^{2} \)
|
| $31$ |
\( T^{4} + 17028 T^{2} + 47044800 \)
|
| $37$ |
\( T^{4} + \cdots + 2293235712 \)
|
| $41$ |
\( T^{4} + 45392 T^{2} + 128314800 \)
|
| $43$ |
\( (T^{2} - 532 T + 47392)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 62387841792 \)
|
| $53$ |
\( (T^{2} - 432 T - 163620)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 2436066048 \)
|
| $61$ |
\( (T^{2} + 1140 T + 314516)^{2} \)
|
| $67$ |
\( T^{4} + 727668 T^{2} + 143327232 \)
|
| $71$ |
\( T^{4} + \cdots + 3298756800 \)
|
| $73$ |
\( T^{4} + \cdots + 68976790272 \)
|
| $79$ |
\( (T^{2} - 144 T - 160960)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 106682723328 \)
|
| $89$ |
\( T^{4} + 60464 T^{2} + 249304368 \)
|
| $97$ |
\( T^{4} + \cdots + 3383532000000 \)
|
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