Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [39,4,Mod(4,39)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(39, base_ring=CyclotomicField(6))
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("39.4");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 39 = 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 39.j (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: | \(2.30107449022\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-17})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 17x^{2} + 289 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 17x^{2} + 289 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 17 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 17 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 17\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 17\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).
| \(n\) | \(14\) | \(28\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.1 |
|
−3.57071 | − | 2.06155i | 1.50000 | − | 2.59808i | 4.50000 | + | 7.79423i | − | 13.4424i | −10.7121 | + | 6.18466i | −27.2121 | + | 15.7109i | − | 4.12311i | −4.50000 | − | 7.79423i | −27.7121 | + | 47.9988i | ||||||||||||||
| 4.2 | 3.57071 | + | 2.06155i | 1.50000 | − | 2.59808i | 4.50000 | + | 7.79423i | 3.05006i | 10.7121 | − | 6.18466i | −5.78786 | + | 3.34162i | 4.12311i | −4.50000 | − | 7.79423i | −6.28786 | + | 10.8909i | |||||||||||||||||
| 10.1 | −3.57071 | + | 2.06155i | 1.50000 | + | 2.59808i | 4.50000 | − | 7.79423i | 13.4424i | −10.7121 | − | 6.18466i | −27.2121 | − | 15.7109i | 4.12311i | −4.50000 | + | 7.79423i | −27.7121 | − | 47.9988i | |||||||||||||||||
| 10.2 | 3.57071 | − | 2.06155i | 1.50000 | + | 2.59808i | 4.50000 | − | 7.79423i | − | 3.05006i | 10.7121 | + | 6.18466i | −5.78786 | − | 3.34162i | − | 4.12311i | −4.50000 | + | 7.79423i | −6.28786 | − | 10.8909i | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.e | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 39.4.j.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 117.4.q.d | 4 | ||
| 4.b | odd | 2 | 1 | 624.4.bv.c | 4 | ||
| 13.c | even | 3 | 1 | 507.4.b.e | 4 | ||
| 13.e | even | 6 | 1 | inner | 39.4.j.b | ✓ | 4 |
| 13.e | even | 6 | 1 | 507.4.b.e | 4 | ||
| 13.f | odd | 12 | 2 | 507.4.a.k | 4 | ||
| 39.h | odd | 6 | 1 | 117.4.q.d | 4 | ||
| 39.k | even | 12 | 2 | 1521.4.a.z | 4 | ||
| 52.i | odd | 6 | 1 | 624.4.bv.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 39.4.j.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 39.4.j.b | ✓ | 4 | 13.e | even | 6 | 1 | inner |
| 117.4.q.d | 4 | 3.b | odd | 2 | 1 | ||
| 117.4.q.d | 4 | 39.h | odd | 6 | 1 | ||
| 507.4.a.k | 4 | 13.f | odd | 12 | 2 | ||
| 507.4.b.e | 4 | 13.c | even | 3 | 1 | ||
| 507.4.b.e | 4 | 13.e | even | 6 | 1 | ||
| 624.4.bv.c | 4 | 4.b | odd | 2 | 1 | ||
| 624.4.bv.c | 4 | 52.i | odd | 6 | 1 | ||
| 1521.4.a.z | 4 | 39.k | even | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 17T_{2}^{2} + 289 \)
acting on \(S_{4}^{\mathrm{new}}(39, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 17T^{2} + 289 \)
|
| $3$ |
\( (T^{2} - 3 T + 9)^{2} \)
|
| $5$ |
\( T^{4} + 190T^{2} + 1681 \)
|
| $7$ |
\( T^{4} + 66 T^{3} + \cdots + 44100 \)
|
| $11$ |
\( T^{4} + 126 T^{3} + \cdots + 1705636 \)
|
| $13$ |
\( T^{4} - 40 T^{3} + \cdots + 4826809 \)
|
| $17$ |
\( T^{4} + 72 T^{3} + \cdots + 1550025 \)
|
| $19$ |
\( T^{4} - 222 T^{3} + \cdots + 7452900 \)
|
| $23$ |
\( T^{4} + 138 T^{3} + \cdots + 5116644 \)
|
| $29$ |
\( T^{4} + 6 T^{3} + \cdots + 608855625 \)
|
| $31$ |
\( T^{4} + 96480 T^{2} + 137733696 \)
|
| $37$ |
\( T^{4} - 492 T^{3} + \cdots + 203148009 \)
|
| $41$ |
\( T^{4} - 180 T^{3} + \cdots + 7198489 \)
|
| $43$ |
\( T^{4} + \cdots + 1914062500 \)
|
| $47$ |
\( T^{4} + \cdots + 4779509956 \)
|
| $53$ |
\( (T^{2} + 1134 T + 304965)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 150320594944 \)
|
| $61$ |
\( T^{4} + \cdots + 144110585161 \)
|
| $67$ |
\( T^{4} + 498 T^{3} + \cdots + 173448900 \)
|
| $71$ |
\( T^{4} + \cdots + 19312660900 \)
|
| $73$ |
\( (T^{2} + 151875)^{2} \)
|
| $79$ |
\( (T^{2} - 4 T - 809672)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 798145692100 \)
|
| $89$ |
\( T^{4} + \cdots + 980686167616 \)
|
| $97$ |
\( T^{4} + \cdots + 7495877379600 \)
|
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