Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [48,8,Mod(47,48)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(48, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48.47");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 48.c (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: | \(14.9944812232\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{30}, \sqrt{-123})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} + 3x^{2} - 2x + 3691 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{4} \) |
| Twist minimal: | yes |
| 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
\( x^{4} - 2x^{3} + 3x^{2} - 2x + 3691 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 8\nu^{3} - 12\nu^{2} + 18\nu - 7 ) / 27 \)
|
| \(\beta_{2}\) | \(=\) |
\( 6\nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{2} - 4\nu + 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} + 2\beta_{2} - 6 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{3} - \beta_{2} + 27\beta _1 - 8 ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(31\) | \(37\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 47.1 |
|
0 | −32.8634 | − | 33.2716i | 0 | 242.981i | 0 | − | 199.630i | 0 | −27.0000 | + | 2186.83i | 0 | |||||||||||||||||||||||||
| 47.2 | 0 | −32.8634 | + | 33.2716i | 0 | − | 242.981i | 0 | 199.630i | 0 | −27.0000 | − | 2186.83i | 0 | ||||||||||||||||||||||||||
| 47.3 | 0 | 32.8634 | − | 33.2716i | 0 | − | 242.981i | 0 | − | 199.630i | 0 | −27.0000 | − | 2186.83i | 0 | |||||||||||||||||||||||||
| 47.4 | 0 | 32.8634 | + | 33.2716i | 0 | 242.981i | 0 | 199.630i | 0 | −27.0000 | + | 2186.83i | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 48.8.c.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 48.8.c.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | inner | 48.8.c.b | ✓ | 4 |
| 8.b | even | 2 | 1 | 192.8.c.c | 4 | ||
| 8.d | odd | 2 | 1 | 192.8.c.c | 4 | ||
| 12.b | even | 2 | 1 | inner | 48.8.c.b | ✓ | 4 |
| 24.f | even | 2 | 1 | 192.8.c.c | 4 | ||
| 24.h | odd | 2 | 1 | 192.8.c.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 48.8.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 48.8.c.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 48.8.c.b | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 48.8.c.b | ✓ | 4 | 12.b | even | 2 | 1 | inner |
| 192.8.c.c | 4 | 8.b | even | 2 | 1 | ||
| 192.8.c.c | 4 | 8.d | odd | 2 | 1 | ||
| 192.8.c.c | 4 | 24.f | even | 2 | 1 | ||
| 192.8.c.c | 4 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 59040 \)
acting on \(S_{8}^{\mathrm{new}}(48, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 54 T^{2} + 4782969 \)
|
| $5$ |
\( (T^{2} + 59040)^{2} \)
|
| $7$ |
\( (T^{2} + 39852)^{2} \)
|
| $11$ |
\( (T^{2} - 8748000)^{2} \)
|
| $13$ |
\( (T - 310)^{4} \)
|
| $17$ |
\( (T^{2} + 590400000)^{2} \)
|
| $19$ |
\( (T^{2} + 2200826700)^{2} \)
|
| $23$ |
\( (T^{2} - 11086865280)^{2} \)
|
| $29$ |
\( (T^{2} + 30521377440)^{2} \)
|
| $31$ |
\( (T^{2} + 37111178700)^{2} \)
|
| $37$ |
\( (T - 188030)^{4} \)
|
| $41$ |
\( (T^{2} + 182466898560)^{2} \)
|
| $43$ |
\( (T^{2} + 328943548908)^{2} \)
|
| $47$ |
\( (T^{2} - 288496442880)^{2} \)
|
| $53$ |
\( (T^{2} + 91513476000)^{2} \)
|
| $59$ |
\( (T^{2} - 4986578700000)^{2} \)
|
| $61$ |
\( (T - 2183462)^{4} \)
|
| $67$ |
\( (T^{2} + 2989414927692)^{2} \)
|
| $71$ |
\( (T^{2} - 13932449712000)^{2} \)
|
| $73$ |
\( (T - 3790330)^{4} \)
|
| $79$ |
\( (T^{2} + 60134038764300)^{2} \)
|
| $83$ |
\( (T^{2} - 769575206880)^{2} \)
|
| $89$ |
\( (T^{2} + 18167892946560)^{2} \)
|
| $97$ |
\( (T - 4439890)^{4} \)
|
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