Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [128,10,Mod(65,128)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(128, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("128.65");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 128 = 2^{7} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 128.b (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: | \(65.9245870290\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{130})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 4225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{11} \) |
| 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
\( x^{4} + 4225 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 4\nu^{2} ) / 65 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{3} + 520\nu ) / 65 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -32\nu^{3} + 2080\nu ) / 65 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 4\beta_{2} ) / 64 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 65\beta_1 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -65\beta_{3} + 260\beta_{2} ) / 64 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(127\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | − | 273.642i | 0 | − | 1020.00i | 0 | −9121.40 | 0 | −55197.0 | 0 | ||||||||||||||||||||||||||||
| 65.2 | 0 | − | 273.642i | 0 | 1020.00i | 0 | 9121.40 | 0 | −55197.0 | 0 | ||||||||||||||||||||||||||||||
| 65.3 | 0 | 273.642i | 0 | − | 1020.00i | 0 | 9121.40 | 0 | −55197.0 | 0 | ||||||||||||||||||||||||||||||
| 65.4 | 0 | 273.642i | 0 | 1020.00i | 0 | −9121.40 | 0 | −55197.0 | 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 | 128.10.b.c | ✓ | 4 |
| 4.b | odd | 2 | 1 | inner | 128.10.b.c | ✓ | 4 |
| 8.b | even | 2 | 1 | inner | 128.10.b.c | ✓ | 4 |
| 8.d | odd | 2 | 1 | inner | 128.10.b.c | ✓ | 4 |
| 16.e | even | 4 | 1 | 256.10.a.g | 2 | ||
| 16.e | even | 4 | 1 | 256.10.a.i | 2 | ||
| 16.f | odd | 4 | 1 | 256.10.a.g | 2 | ||
| 16.f | odd | 4 | 1 | 256.10.a.i | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 128.10.b.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 128.10.b.c | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 128.10.b.c | ✓ | 4 | 8.b | even | 2 | 1 | inner |
| 128.10.b.c | ✓ | 4 | 8.d | odd | 2 | 1 | inner |
| 256.10.a.g | 2 | 16.e | even | 4 | 1 | ||
| 256.10.a.g | 2 | 16.f | odd | 4 | 1 | ||
| 256.10.a.i | 2 | 16.e | even | 4 | 1 | ||
| 256.10.a.i | 2 | 16.f | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 74880 \)
acting on \(S_{10}^{\mathrm{new}}(128, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + 74880)^{2} \)
|
| $5$ |
\( (T^{2} + 1040400)^{2} \)
|
| $7$ |
\( (T^{2} - 83200000)^{2} \)
|
| $11$ |
\( (T^{2} + 666340480)^{2} \)
|
| $13$ |
\( (T^{2} + 23791211536)^{2} \)
|
| $17$ |
\( (T + 47362)^{4} \)
|
| $19$ |
\( (T^{2} + 2310705280)^{2} \)
|
| $23$ |
\( (T^{2} - 2776548935680)^{2} \)
|
| $29$ |
\( (T^{2} + 15658830351376)^{2} \)
|
| $31$ |
\( (T^{2} - 15967251988480)^{2} \)
|
| $37$ |
\( (T^{2} + 223514818969744)^{2} \)
|
| $41$ |
\( (T - 23024970)^{4} \)
|
| $43$ |
\( (T^{2} + 7524485937280)^{2} \)
|
| $47$ |
\( (T^{2} - 536401247150080)^{2} \)
|
| $53$ |
\( (T^{2} + 424109705717776)^{2} \)
|
| $59$ |
\( (T^{2} + 17\!\cdots\!80)^{2} \)
|
| $61$ |
\( (T^{2} + 10\!\cdots\!84)^{2} \)
|
| $67$ |
\( (T^{2} + 12\!\cdots\!20)^{2} \)
|
| $71$ |
\( (T^{2} - 37\!\cdots\!80)^{2} \)
|
| $73$ |
\( (T - 302241606)^{4} \)
|
| $79$ |
\( (T^{2} - 47\!\cdots\!00)^{2} \)
|
| $83$ |
\( (T^{2} + 23\!\cdots\!20)^{2} \)
|
| $89$ |
\( (T + 175887690)^{4} \)
|
| $97$ |
\( (T + 1168949138)^{4} \)
|
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