Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3328,2,Mod(1,3328)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3328, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3328.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3328 = 2^{8} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3328.a (trivial) |
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: | yes |
| Analytic conductor: | \(26.5742137927\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.13968.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 7x^{2} + 8x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 832) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{4} - 2x^{3} - 7x^{2} + 8x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 7\nu - 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 5\nu + 6 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} + 7\beta _1 + 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −2.27743 | 0 | 2.27743 | 0 | −0.121816 | 0 | 2.18667 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.386509 | 0 | 0.386509 | 0 | 4.17452 | 0 | −2.85061 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.38651 | 0 | −1.38651 | 0 | −2.44247 | 0 | −1.07759 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.27743 | 0 | −3.27743 | 0 | −1.61023 | 0 | 7.74153 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3328.2.a.bm | 4 | |
| 4.b | odd | 2 | 1 | 3328.2.a.bi | 4 | ||
| 8.b | even | 2 | 1 | 3328.2.a.bj | 4 | ||
| 8.d | odd | 2 | 1 | 3328.2.a.bn | 4 | ||
| 16.e | even | 4 | 2 | 832.2.b.c | ✓ | 8 | |
| 16.f | odd | 4 | 2 | 832.2.b.d | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 832.2.b.c | ✓ | 8 | 16.e | even | 4 | 2 | |
| 832.2.b.d | yes | 8 | 16.f | odd | 4 | 2 | |
| 3328.2.a.bi | 4 | 4.b | odd | 2 | 1 | ||
| 3328.2.a.bj | 4 | 8.b | even | 2 | 1 | ||
| 3328.2.a.bm | 4 | 1.a | even | 1 | 1 | trivial | |
| 3328.2.a.bn | 4 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3328))\):
|
\( T_{3}^{4} - 2T_{3}^{3} - 7T_{3}^{2} + 8T_{3} + 4 \)
|
|
\( T_{5}^{4} + 2T_{5}^{3} - 7T_{5}^{2} - 8T_{5} + 4 \)
|
|
\( T_{7}^{4} - 13T_{7}^{2} - 18T_{7} - 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 2 T^{3} + \cdots + 4 \)
|
| $5$ |
\( T^{4} + 2 T^{3} + \cdots + 4 \)
|
| $7$ |
\( T^{4} - 13 T^{2} + \cdots - 2 \)
|
| $11$ |
\( T^{4} + 2 T^{3} + \cdots + 16 \)
|
| $13$ |
\( (T - 1)^{4} \)
|
| $17$ |
\( T^{4} + 2 T^{3} + \cdots - 104 \)
|
| $19$ |
\( T^{4} + 6 T^{3} + \cdots + 64 \)
|
| $23$ |
\( T^{4} + 12 T^{3} + \cdots - 128 \)
|
| $29$ |
\( T^{4} - 16 T^{3} + \cdots - 1184 \)
|
| $31$ |
\( T^{4} + 10 T^{3} + \cdots - 1472 \)
|
| $37$ |
\( T^{4} + 14 T^{3} + \cdots - 764 \)
|
| $41$ |
\( T^{4} + 20 T^{3} + \cdots - 368 \)
|
| $43$ |
\( T^{4} + 14 T^{3} + \cdots + 436 \)
|
| $47$ |
\( T^{4} + 20 T^{3} + \cdots - 194 \)
|
| $53$ |
\( T^{4} - 4 T^{3} + \cdots - 128 \)
|
| $59$ |
\( T^{4} + 2 T^{3} + \cdots + 184 \)
|
| $61$ |
\( T^{4} - 108 T^{2} + \cdots + 576 \)
|
| $67$ |
\( T^{4} - 6 T^{3} + \cdots - 8 \)
|
| $71$ |
\( T^{4} + 28 T^{3} + \cdots + 1366 \)
|
| $73$ |
\( T^{4} - 8 T^{3} + \cdots + 5584 \)
|
| $79$ |
\( T^{4} + 16 T^{3} + \cdots - 32 \)
|
| $83$ |
\( T^{4} + 22 T^{3} + \cdots - 872 \)
|
| $89$ |
\( T^{4} - 8 T^{3} + \cdots + 64 \)
|
| $97$ |
\( T^{4} + 20 T^{3} + \cdots - 32 \)
|
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