Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1792,3,Mod(1023,1792)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1792, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1792.1023");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1792 = 2^{8} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1792.d (of order \(2\), degree \(1\), not 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: | \(48.8284633734\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.1539727.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 3x^{3} - 8x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{9} \) |
| Twist minimal: | no (minimal twist has level 896) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 3x^{3} - 8x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + \nu^{2} - 2\nu + 2 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} - 3\nu^{2} + 4\nu + 6 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} + 3\nu^{2} + 4\nu - 8 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( -2\nu^{5} + 2\nu^{4} + 4\nu^{3} - 2\nu^{2} - 6\nu + 11 \)
|
| \(\beta_{5}\) | \(=\) |
\( 3\nu^{5} - \nu^{4} - 4\nu^{3} + 7\nu^{2} + 7\nu - 15 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta_{3} + 3\beta _1 + 5 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{4} + 2\beta_{3} - 2\beta _1 - 1 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - 2\beta_{3} + \beta_{2} - \beta _1 - 6 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - 3\beta_{3} - 4\beta_{2} - 13\beta _1 + 17 ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).
| \(n\) | \(1023\) | \(1025\) | \(1541\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1023.1 |
|
0 | − | 4.54500i | 0 | 0.632062 | 0 | − | 2.64575i | 0 | −11.6570 | 0 | ||||||||||||||||||||||||||||||||||
| 1023.2 | 0 | − | 2.98472i | 0 | 8.80536 | 0 | 2.64575i | 0 | 0.0914622 | 0 | ||||||||||||||||||||||||||||||||||||
| 1023.3 | 0 | − | 1.56028i | 0 | −1.43742 | 0 | 2.64575i | 0 | 6.56553 | 0 | ||||||||||||||||||||||||||||||||||||
| 1023.4 | 0 | 1.56028i | 0 | −1.43742 | 0 | − | 2.64575i | 0 | 6.56553 | 0 | ||||||||||||||||||||||||||||||||||||
| 1023.5 | 0 | 2.98472i | 0 | 8.80536 | 0 | − | 2.64575i | 0 | 0.0914622 | 0 | ||||||||||||||||||||||||||||||||||||
| 1023.6 | 0 | 4.54500i | 0 | 0.632062 | 0 | 2.64575i | 0 | −11.6570 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1792.3.d.e | 6 | |
| 4.b | odd | 2 | 1 | inner | 1792.3.d.e | 6 | |
| 8.b | even | 2 | 1 | 1792.3.d.d | 6 | ||
| 8.d | odd | 2 | 1 | 1792.3.d.d | 6 | ||
| 16.e | even | 4 | 2 | 896.3.g.d | ✓ | 12 | |
| 16.f | odd | 4 | 2 | 896.3.g.d | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 896.3.g.d | ✓ | 12 | 16.e | even | 4 | 2 | |
| 896.3.g.d | ✓ | 12 | 16.f | odd | 4 | 2 | |
| 1792.3.d.d | 6 | 8.b | even | 2 | 1 | ||
| 1792.3.d.d | 6 | 8.d | odd | 2 | 1 | ||
| 1792.3.d.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 1792.3.d.e | 6 | 4.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1792, [\chi])\):
|
\( T_{3}^{6} + 32T_{3}^{4} + 256T_{3}^{2} + 448 \)
|
|
\( T_{5}^{3} - 8T_{5}^{2} - 8T_{5} + 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 32 T^{4} + \cdots + 448 \)
|
| $5$ |
\( (T^{3} - 8 T^{2} - 8 T + 8)^{2} \)
|
| $7$ |
\( (T^{2} + 7)^{3} \)
|
| $11$ |
\( T^{6} + 480 T^{4} + \cdots + 458752 \)
|
| $13$ |
\( (T^{3} - 264 T - 1384)^{2} \)
|
| $17$ |
\( (T^{3} + 14 T^{2} + \cdots - 2648)^{2} \)
|
| $19$ |
\( T^{6} + 1408 T^{4} + \cdots + 5322688 \)
|
| $23$ |
\( T^{6} + 2816 T^{4} + \cdots + 80539648 \)
|
| $29$ |
\( (T^{3} - 24 T^{2} + \cdots + 3776)^{2} \)
|
| $31$ |
\( T^{6} + 1600 T^{4} + \cdots + 7340032 \)
|
| $37$ |
\( (T^{3} - 8 T^{2} + \cdots + 24896)^{2} \)
|
| $41$ |
\( (T^{3} + 38 T^{2} + \cdots - 34936)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 9578856448 \)
|
| $47$ |
\( T^{6} + 3392 T^{4} + \cdots + 1835008 \)
|
| $53$ |
\( (T^{3} + 88 T^{2} + \cdots - 57344)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 23919727552 \)
|
| $61$ |
\( (T^{3} + 72 T^{2} + \cdots + 50536)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 175066673152 \)
|
| $71$ |
\( T^{6} + \cdots + 27312259072 \)
|
| $73$ |
\( (T^{3} - 78 T^{2} + \cdots + 562264)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 5754585088 \)
|
| $83$ |
\( T^{6} + \cdots + 12352704448 \)
|
| $89$ |
\( (T^{3} - 110 T^{2} + \cdots + 446168)^{2} \)
|
| $97$ |
\( (T^{3} - 34 T^{2} + \cdots + 144104)^{2} \)
|
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