Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [52,5,Mod(3,52)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(52, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 2]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("52.3");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 52 = 2^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 52.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: | \(5.37523808036\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{6}]$ |
$q$-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
| \(\beta_{1}\) | \(=\) |
\( 24\zeta_{12} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{12}^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( 24\zeta_{12}^{3} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_1 ) / 24 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( ( \beta_{3} ) / 24 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/52\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(41\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
2.00000 | + | 3.46410i | 0 | −8.00000 | + | 13.8564i | −34.5692 | 0 | 0 | −64.0000 | −40.5000 | + | 70.1481i | −69.1384 | − | 119.751i | ||||||||||||||||||||||
| 3.2 | 2.00000 | + | 3.46410i | 0 | −8.00000 | + | 13.8564i | 48.5692 | 0 | 0 | −64.0000 | −40.5000 | + | 70.1481i | 97.1384 | + | 168.249i | |||||||||||||||||||||||
| 35.1 | 2.00000 | − | 3.46410i | 0 | −8.00000 | − | 13.8564i | −34.5692 | 0 | 0 | −64.0000 | −40.5000 | − | 70.1481i | −69.1384 | + | 119.751i | |||||||||||||||||||||||
| 35.2 | 2.00000 | − | 3.46410i | 0 | −8.00000 | − | 13.8564i | 48.5692 | 0 | 0 | −64.0000 | −40.5000 | − | 70.1481i | 97.1384 | − | 168.249i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
| 13.c | even | 3 | 1 | inner |
| 52.j | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 52.5.j.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | CM | 52.5.j.a | ✓ | 4 |
| 13.c | even | 3 | 1 | inner | 52.5.j.a | ✓ | 4 |
| 52.j | odd | 6 | 1 | inner | 52.5.j.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 52.5.j.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 52.5.j.a | ✓ | 4 | 4.b | odd | 2 | 1 | CM |
| 52.5.j.a | ✓ | 4 | 13.c | even | 3 | 1 | inner |
| 52.5.j.a | ✓ | 4 | 52.j | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} \)
acting on \(S_{5}^{\mathrm{new}}(52, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 4 T + 16)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} - 14 T - 1679)^{2} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} - 238 T^{3} + \cdots + 815730721 \)
|
| $17$ |
\( T^{4} + \cdots + 21573440641 \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( T^{4} + \cdots + 4473728384161 \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} + \cdots + 899157201121 \)
|
| $41$ |
\( T^{4} + \cdots + 565746169921 \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( (T^{2} + 2482 T - 17511119)^{2} \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( T^{4} + \cdots + 47282690290081 \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( (T^{2} + 1442 T - 83115359)^{2} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( (T^{2} - 9758 T + 95218564)^{2} \)
|
| $97$ |
\( (T^{2} - 1918 T + 3678724)^{2} \)
|
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