Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [768,6,Mod(1,768)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(768, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("768.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 768 = 2^{8} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 768.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: | \(123.174773616\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{61}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 15 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 384) |
| 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 \(\beta = 4\sqrt{61}\). We also show the integral \(q\)-expansion of the trace form.
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 | 9.00000 | 0 | −51.2410 | 0 | −164.205 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 9.00000 | 0 | 11.2410 | 0 | 148.205 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
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 | 768.6.a.r | 2 | |
| 4.b | odd | 2 | 1 | 768.6.a.m | 2 | ||
| 8.b | even | 2 | 1 | 768.6.a.q | 2 | ||
| 8.d | odd | 2 | 1 | 768.6.a.v | 2 | ||
| 16.e | even | 4 | 2 | 384.6.d.h | yes | 4 | |
| 16.f | odd | 4 | 2 | 384.6.d.g | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.6.d.g | ✓ | 4 | 16.f | odd | 4 | 2 | |
| 384.6.d.h | yes | 4 | 16.e | even | 4 | 2 | |
| 768.6.a.m | 2 | 4.b | odd | 2 | 1 | ||
| 768.6.a.q | 2 | 8.b | even | 2 | 1 | ||
| 768.6.a.r | 2 | 1.a | even | 1 | 1 | trivial | |
| 768.6.a.v | 2 | 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(768))\):
|
\( T_{5}^{2} + 40T_{5} - 576 \)
|
|
\( T_{7}^{2} + 16T_{7} - 24336 \)
|
|
\( T_{11}^{2} - 344T_{11} + 13968 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( (T - 9)^{2} \)
|
| $5$ |
\( T^{2} + 40T - 576 \)
|
| $7$ |
\( T^{2} + 16T - 24336 \)
|
| $11$ |
\( T^{2} - 344T + 13968 \)
|
| $13$ |
\( T^{2} + 216T - 179632 \)
|
| $17$ |
\( T^{2} - 924 T - 2425660 \)
|
| $19$ |
\( T^{2} + 536T - 693360 \)
|
| $23$ |
\( T^{2} + 3584 T + 2738880 \)
|
| $29$ |
\( T^{2} + 8648 T + 14823232 \)
|
| $31$ |
\( T^{2} - 3536 T - 19127952 \)
|
| $37$ |
\( T^{2} - 3112 T - 23829360 \)
|
| $41$ |
\( T^{2} + 2516 T - 164087580 \)
|
| $43$ |
\( T^{2} + 3688 T - 248470128 \)
|
| $47$ |
\( T^{2} - 4576 T - 81437760 \)
|
| $53$ |
\( T^{2} + 968T - 828608 \)
|
| $59$ |
\( T^{2} + 27816 T - 382235760 \)
|
| $61$ |
\( T^{2} + 28440 T - 238537584 \)
|
| $67$ |
\( T^{2} + \cdots + 1430533008 \)
|
| $71$ |
\( T^{2} + \cdots + 1179861696 \)
|
| $73$ |
\( T^{2} + \cdots - 2681559900 \)
|
| $79$ |
\( T^{2} + \cdots + 1974396528 \)
|
| $83$ |
\( T^{2} + \cdots - 3636999792 \)
|
| $89$ |
\( T^{2} + \cdots + 2033493540 \)
|
| $97$ |
\( T^{2} + \cdots - 5288174460 \)
|
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