Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [390,4,Mod(61,390)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(390, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("390.61");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 390.i (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: | \(23.0107449022\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{10})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 10x^{2} + 100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| 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
\( x^{4} + 10x^{2} + 100 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 10\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 10\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).
| \(n\) | \(131\) | \(157\) | \(301\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 61.1 |
|
1.00000 | − | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | − | 3.46410i | −5.00000 | 3.00000 | + | 5.19615i | −9.24342 | − | 16.0101i | −8.00000 | −4.50000 | − | 7.79423i | −5.00000 | + | 8.66025i | ||||||||||||||||
| 61.2 | 1.00000 | − | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | − | 3.46410i | −5.00000 | 3.00000 | + | 5.19615i | 0.243416 | + | 0.421610i | −8.00000 | −4.50000 | − | 7.79423i | −5.00000 | + | 8.66025i | |||||||||||||||||
| 211.1 | 1.00000 | + | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | −5.00000 | 3.00000 | − | 5.19615i | −9.24342 | + | 16.0101i | −8.00000 | −4.50000 | + | 7.79423i | −5.00000 | − | 8.66025i | |||||||||||||||||
| 211.2 | 1.00000 | + | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | −5.00000 | 3.00000 | − | 5.19615i | 0.243416 | − | 0.421610i | −8.00000 | −4.50000 | + | 7.79423i | −5.00000 | − | 8.66025i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 390.4.i.c | ✓ | 4 |
| 13.c | even | 3 | 1 | inner | 390.4.i.c | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 390.4.i.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 390.4.i.c | ✓ | 4 | 13.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 18T_{7}^{3} + 333T_{7}^{2} - 162T_{7} + 81 \)
acting on \(S_{4}^{\mathrm{new}}(390, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 4)^{2} \)
|
| $3$ |
\( (T^{2} + 3 T + 9)^{2} \)
|
| $5$ |
\( (T + 5)^{4} \)
|
| $7$ |
\( T^{4} + 18 T^{3} + \cdots + 81 \)
|
| $11$ |
\( (T^{2} + 6 T + 36)^{2} \)
|
| $13$ |
\( T^{4} - 78 T^{3} + \cdots + 4826809 \)
|
| $17$ |
\( T^{4} - 112 T^{3} + \cdots + 60516 \)
|
| $19$ |
\( T^{4} - 44 T^{3} + \cdots + 4309776 \)
|
| $23$ |
\( T^{4} - 36 T^{3} + \cdots + 99856 \)
|
| $29$ |
\( T^{4} + \cdots + 1051445476 \)
|
| $31$ |
\( (T^{2} - 2 T - 54759)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 3190442256 \)
|
| $41$ |
\( T^{4} + 284 T^{3} + \cdots + 24542116 \)
|
| $43$ |
\( T^{4} - 370 T^{3} + \cdots + 421686225 \)
|
| $47$ |
\( (T^{2} - 524 T + 26394)^{2} \)
|
| $53$ |
\( (T^{2} - 528 T + 69336)^{2} \)
|
| $59$ |
\( T^{4} + 320 T^{3} + \cdots + 178222500 \)
|
| $61$ |
\( T^{4} - 614 T^{3} + \cdots + 193515921 \)
|
| $67$ |
\( T^{4} + \cdots + 142298700625 \)
|
| $71$ |
\( T^{4} + \cdots + 4230982116 \)
|
| $73$ |
\( (T^{2} - 242 T - 83369)^{2} \)
|
| $79$ |
\( (T^{2} - 718 T - 224559)^{2} \)
|
| $83$ |
\( (T^{2} + 16 T - 399936)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 24113741796 \)
|
| $97$ |
\( T^{4} + \cdots + 306161915761 \)
|
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