Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [64,5,Mod(31,64)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(64, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("64.31");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 64.d (of order \(2\), degree \(1\), 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: | \(6.61567763737\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{51})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 25x^{2} + 169 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | yes |
| 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
\( x^{4} - 25x^{2} + 169 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} + 76\nu ) / 13 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{3} - 96\nu ) / 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( 8\nu^{2} - 100 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 4\beta_1 ) / 16 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 100 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 19\beta_{2} + 24\beta_1 ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(63\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0 | −14.2829 | 0 | − | 28.5657i | 0 | 56.0000i | 0 | 123.000 | 0 | |||||||||||||||||||||||||||||
| 31.2 | 0 | −14.2829 | 0 | 28.5657i | 0 | − | 56.0000i | 0 | 123.000 | 0 | ||||||||||||||||||||||||||||||
| 31.3 | 0 | 14.2829 | 0 | − | 28.5657i | 0 | − | 56.0000i | 0 | 123.000 | 0 | |||||||||||||||||||||||||||||
| 31.4 | 0 | 14.2829 | 0 | 28.5657i | 0 | 56.0000i | 0 | 123.000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 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 | 64.5.d.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 576.5.b.a | 4 | ||
| 4.b | odd | 2 | 1 | inner | 64.5.d.b | ✓ | 4 |
| 8.b | even | 2 | 1 | inner | 64.5.d.b | ✓ | 4 |
| 8.d | odd | 2 | 1 | inner | 64.5.d.b | ✓ | 4 |
| 12.b | even | 2 | 1 | 576.5.b.a | 4 | ||
| 16.e | even | 4 | 2 | 256.5.c.h | 4 | ||
| 16.f | odd | 4 | 2 | 256.5.c.h | 4 | ||
| 24.f | even | 2 | 1 | 576.5.b.a | 4 | ||
| 24.h | odd | 2 | 1 | 576.5.b.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 64.5.d.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 64.5.d.b | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 64.5.d.b | ✓ | 4 | 8.b | even | 2 | 1 | inner |
| 64.5.d.b | ✓ | 4 | 8.d | odd | 2 | 1 | inner |
| 256.5.c.h | 4 | 16.e | even | 4 | 2 | ||
| 256.5.c.h | 4 | 16.f | odd | 4 | 2 | ||
| 576.5.b.a | 4 | 3.b | odd | 2 | 1 | ||
| 576.5.b.a | 4 | 12.b | even | 2 | 1 | ||
| 576.5.b.a | 4 | 24.f | even | 2 | 1 | ||
| 576.5.b.a | 4 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 204 \)
acting on \(S_{5}^{\mathrm{new}}(64, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} - 204)^{2} \)
|
| $5$ |
\( (T^{2} + 816)^{2} \)
|
| $7$ |
\( (T^{2} + 3136)^{2} \)
|
| $11$ |
\( (T^{2} - 9996)^{2} \)
|
| $13$ |
\( (T^{2} + 66096)^{2} \)
|
| $17$ |
\( (T - 378)^{4} \)
|
| $19$ |
\( (T^{2} - 16524)^{2} \)
|
| $23$ |
\( (T^{2} + 46656)^{2} \)
|
| $29$ |
\( (T^{2} + 359856)^{2} \)
|
| $31$ |
\( (T^{2} + 50176)^{2} \)
|
| $37$ |
\( (T^{2} + 3238704)^{2} \)
|
| $41$ |
\( (T + 1134)^{4} \)
|
| $43$ |
\( (T^{2} - 809676)^{2} \)
|
| $47$ |
\( (T^{2} + 9144576)^{2} \)
|
| $53$ |
\( (T^{2} + 999600)^{2} \)
|
| $59$ |
\( (T^{2} - 45900)^{2} \)
|
| $61$ |
\( (T^{2} + 594864)^{2} \)
|
| $67$ |
\( (T^{2} - 65583756)^{2} \)
|
| $71$ |
\( (T^{2} + 20575296)^{2} \)
|
| $73$ |
\( (T - 490)^{4} \)
|
| $79$ |
\( (T^{2} + 7840000)^{2} \)
|
| $83$ |
\( (T^{2} - 67917516)^{2} \)
|
| $89$ |
\( (T - 9450)^{4} \)
|
| $97$ |
\( (T + 16198)^{4} \)
|
| show more | |
| show less |