Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1792,3,Mod(127,1792)]
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, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1792.127");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1792 = 2^{8} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1792.g (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: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{7})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 3x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 224) |
| 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
\( x^{4} - 3x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{3} + 5\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 3 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{2} + 5\beta_1 ) / 4 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 127.1 |
|
0 | −2.00000 | 0 | − | 7.29150i | 0 | 2.64575i | 0 | −5.00000 | 0 | |||||||||||||||||||||||||||||
| 127.2 | 0 | −2.00000 | 0 | − | 3.29150i | 0 | 2.64575i | 0 | −5.00000 | 0 | ||||||||||||||||||||||||||||||
| 127.3 | 0 | −2.00000 | 0 | 3.29150i | 0 | − | 2.64575i | 0 | −5.00000 | 0 | ||||||||||||||||||||||||||||||
| 127.4 | 0 | −2.00000 | 0 | 7.29150i | 0 | − | 2.64575i | 0 | −5.00000 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | 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.g.a | 4 | |
| 4.b | odd | 2 | 1 | 1792.3.g.c | 4 | ||
| 8.b | even | 2 | 1 | 1792.3.g.c | 4 | ||
| 8.d | odd | 2 | 1 | inner | 1792.3.g.a | 4 | |
| 16.e | even | 4 | 1 | 224.3.d.a | ✓ | 4 | |
| 16.e | even | 4 | 1 | 448.3.d.c | 4 | ||
| 16.f | odd | 4 | 1 | 224.3.d.a | ✓ | 4 | |
| 16.f | odd | 4 | 1 | 448.3.d.c | 4 | ||
| 48.i | odd | 4 | 1 | 2016.3.m.a | 4 | ||
| 48.k | even | 4 | 1 | 2016.3.m.a | 4 | ||
| 112.j | even | 4 | 1 | 1568.3.d.h | 4 | ||
| 112.l | odd | 4 | 1 | 1568.3.d.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 224.3.d.a | ✓ | 4 | 16.e | even | 4 | 1 | |
| 224.3.d.a | ✓ | 4 | 16.f | odd | 4 | 1 | |
| 448.3.d.c | 4 | 16.e | even | 4 | 1 | ||
| 448.3.d.c | 4 | 16.f | odd | 4 | 1 | ||
| 1568.3.d.h | 4 | 112.j | even | 4 | 1 | ||
| 1568.3.d.h | 4 | 112.l | odd | 4 | 1 | ||
| 1792.3.g.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 1792.3.g.a | 4 | 8.d | odd | 2 | 1 | inner | |
| 1792.3.g.c | 4 | 4.b | odd | 2 | 1 | ||
| 1792.3.g.c | 4 | 8.b | even | 2 | 1 | ||
| 2016.3.m.a | 4 | 48.i | odd | 4 | 1 | ||
| 2016.3.m.a | 4 | 48.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} + 2 \)
acting on \(S_{3}^{\mathrm{new}}(1792, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 2)^{4} \)
|
| $5$ |
\( T^{4} + 64T^{2} + 576 \)
|
| $7$ |
\( (T^{2} + 7)^{2} \)
|
| $11$ |
\( (T^{2} + 8 T - 96)^{2} \)
|
| $13$ |
\( T^{4} + 512 T^{2} + 61504 \)
|
| $17$ |
\( (T^{2} + 28 T + 84)^{2} \)
|
| $19$ |
\( (T^{2} - 20 T - 348)^{2} \)
|
| $23$ |
\( T^{4} + 512T^{2} + 1024 \)
|
| $29$ |
\( T^{4} + 2216 T^{2} + 824464 \)
|
| $31$ |
\( (T^{2} + 16)^{2} \)
|
| $37$ |
\( T^{4} + 2536 T^{2} + 1089936 \)
|
| $41$ |
\( (T^{2} - 60 T + 788)^{2} \)
|
| $43$ |
\( (T^{2} - 24 T - 2656)^{2} \)
|
| $47$ |
\( T^{4} + 3488 T^{2} + 719104 \)
|
| $53$ |
\( T^{4} + 10696 T^{2} + 19820304 \)
|
| $59$ |
\( (T^{2} + 36 T - 1468)^{2} \)
|
| $61$ |
\( T^{4} + 512 T^{2} + 61504 \)
|
| $67$ |
\( (T^{2} + 48 T - 2224)^{2} \)
|
| $71$ |
\( T^{4} + 11776 T^{2} + 5308416 \)
|
| $73$ |
\( (T^{2} - 76 T - 5724)^{2} \)
|
| $79$ |
\( (T^{2} + 11200)^{2} \)
|
| $83$ |
\( (T^{2} + 124 T - 188)^{2} \)
|
| $89$ |
\( (T^{2} + 68 T - 636)^{2} \)
|
| $97$ |
\( (T^{2} + 300 T + 22388)^{2} \)
|
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