Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8048,2,Mod(1,8048)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8048, 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("8048.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8048 = 2^{4} \cdot 503 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8048.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: | \(64.2636035467\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.205225.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1006) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + 2\nu^{2} + 3\nu - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + \nu + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 6\beta_{3} + 7\beta_{2} + 9\beta _1 + 14 \)
|
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 | −1.54180 | 0 | −2.22418 | 0 | 4.85942 | 0 | −0.622849 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.170728 | 0 | −0.411173 | 0 | 3.93028 | 0 | −2.97085 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.581067 | 0 | −1.45190 | 0 | 1.38596 | 0 | −2.66236 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.35347 | 0 | 3.94839 | 0 | 3.24145 | 0 | 2.53883 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.77799 | 0 | −2.86114 | 0 | −4.41712 | 0 | 4.71723 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(503\) | \(1\) |
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 | 8048.2.a.o | 5 | |
| 4.b | odd | 2 | 1 | 1006.2.a.h | ✓ | 5 | |
| 12.b | even | 2 | 1 | 9054.2.a.ba | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1006.2.a.h | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 8048.2.a.o | 5 | 1.a | even | 1 | 1 | trivial | |
| 9054.2.a.ba | 5 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):
|
\( T_{3}^{5} - 4T_{3}^{4} + 11T_{3}^{2} - 4T_{3} - 1 \)
|
|
\( T_{5}^{5} + 3T_{5}^{4} - 11T_{5}^{3} - 50T_{5}^{2} - 55T_{5} - 15 \)
|
|
\( T_{7}^{5} - 9T_{7}^{4} + 5T_{7}^{3} + 156T_{7}^{2} - 479T_{7} + 379 \)
|
|
\( T_{13}^{5} + 5T_{13}^{4} - 27T_{13}^{3} - 85T_{13}^{2} + 10T_{13} + 15 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 4 T^{4} + \cdots - 1 \)
|
| $5$ |
\( T^{5} + 3 T^{4} + \cdots - 15 \)
|
| $7$ |
\( T^{5} - 9 T^{4} + \cdots + 379 \)
|
| $11$ |
\( T^{5} - 15 T^{4} + \cdots - 9 \)
|
| $13$ |
\( T^{5} + 5 T^{4} + \cdots + 15 \)
|
| $17$ |
\( T^{5} + 6 T^{4} + \cdots - 219 \)
|
| $19$ |
\( T^{5} + 10 T^{4} + \cdots + 333 \)
|
| $23$ |
\( T^{5} - 12 T^{4} + \cdots + 1497 \)
|
| $29$ |
\( T^{5} + 16 T^{4} + \cdots - 4119 \)
|
| $31$ |
\( T^{5} - 33 T^{4} + \cdots - 9797 \)
|
| $37$ |
\( T^{5} - 2 T^{4} + \cdots - 1895 \)
|
| $41$ |
\( T^{5} + 12 T^{4} + \cdots + 687 \)
|
| $43$ |
\( T^{5} - 17 T^{4} + \cdots + 7067 \)
|
| $47$ |
\( T^{5} - 7 T^{4} + \cdots - 66081 \)
|
| $53$ |
\( T^{5} - 88 T^{3} + \cdots + 3347 \)
|
| $59$ |
\( T^{5} + 18 T^{4} + \cdots - 1563 \)
|
| $61$ |
\( T^{5} - T^{4} + \cdots - 8863 \)
|
| $67$ |
\( T^{5} + 6 T^{4} + \cdots - 19989 \)
|
| $71$ |
\( T^{5} - 26 T^{4} + \cdots + 51891 \)
|
| $73$ |
\( T^{5} - 9 T^{4} + \cdots - 1 \)
|
| $79$ |
\( T^{5} - 21 T^{4} + \cdots + 729 \)
|
| $83$ |
\( T^{5} - 312 T^{3} + \cdots - 729 \)
|
| $89$ |
\( T^{5} - 15 T^{4} + \cdots - 45027 \)
|
| $97$ |
\( T^{5} - 26 T^{4} + \cdots - 30449 \)
|
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