Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [128,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("128.65");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 128 = 2^{7} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(39.9852832620\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-85})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 84x^{2} + 1849 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{2} \) |
| 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
\( x^{4} - 84x^{2} + 1849 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} - 82\nu ) / 43 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -96\nu^{3} + 12192\nu ) / 43 \)
|
| \(\beta_{3}\) | \(=\) |
\( 48\nu^{2} - 2016 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 48\beta_1 ) / 192 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2016 ) / 48 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 41\beta_{2} + 6096\beta_1 ) / 192 \)
|
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 | − | 59.3970i | 0 | − | 442.538i | 0 | −1251.69 | 0 | −1341.00 | 0 | ||||||||||||||||||||||||||||
| 65.2 | 0 | − | 59.3970i | 0 | 442.538i | 0 | 1251.69 | 0 | −1341.00 | 0 | ||||||||||||||||||||||||||||||
| 65.3 | 0 | 59.3970i | 0 | − | 442.538i | 0 | 1251.69 | 0 | −1341.00 | 0 | ||||||||||||||||||||||||||||||
| 65.4 | 0 | 59.3970i | 0 | 442.538i | 0 | −1251.69 | 0 | −1341.00 | 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 | 128.8.b.d | ✓ | 4 |
| 4.b | odd | 2 | 1 | inner | 128.8.b.d | ✓ | 4 |
| 8.b | even | 2 | 1 | inner | 128.8.b.d | ✓ | 4 |
| 8.d | odd | 2 | 1 | inner | 128.8.b.d | ✓ | 4 |
| 16.e | even | 4 | 2 | 256.8.a.m | 4 | ||
| 16.f | odd | 4 | 2 | 256.8.a.m | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 128.8.b.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 128.8.b.d | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 128.8.b.d | ✓ | 4 | 8.b | even | 2 | 1 | inner |
| 128.8.b.d | ✓ | 4 | 8.d | odd | 2 | 1 | inner |
| 256.8.a.m | 4 | 16.e | even | 4 | 2 | ||
| 256.8.a.m | 4 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 3528 \)
acting on \(S_{8}^{\mathrm{new}}(128, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + 3528)^{2} \)
|
| $5$ |
\( (T^{2} + 195840)^{2} \)
|
| $7$ |
\( (T^{2} - 1566720)^{2} \)
|
| $11$ |
\( (T^{2} + 125000)^{2} \)
|
| $13$ |
\( (T^{2} + 44064000)^{2} \)
|
| $17$ |
\( (T + 36570)^{4} \)
|
| $19$ |
\( (T^{2} + 416449800)^{2} \)
|
| $23$ |
\( (T^{2} - 5090273280)^{2} \)
|
| $29$ |
\( (T^{2} + 43445341440)^{2} \)
|
| $31$ |
\( (T^{2} - 40108032000)^{2} \)
|
| $37$ |
\( (T^{2} + 1101600000)^{2} \)
|
| $41$ |
\( (T - 585642)^{4} \)
|
| $43$ |
\( (T^{2} + 3411049608)^{2} \)
|
| $47$ |
\( (T^{2} - 388552826880)^{2} \)
|
| $53$ |
\( (T^{2} + 256751136000)^{2} \)
|
| $59$ |
\( (T^{2} + 1918409544200)^{2} \)
|
| $61$ |
\( (T^{2} + 649413469440)^{2} \)
|
| $67$ |
\( (T^{2} + 17746665693192)^{2} \)
|
| $71$ |
\( (T^{2} - 33658198272000)^{2} \)
|
| $73$ |
\( (T + 2204210)^{4} \)
|
| $79$ |
\( (T^{2} - 28834071552000)^{2} \)
|
| $83$ |
\( (T^{2} + 8713158333128)^{2} \)
|
| $89$ |
\( (T - 3136734)^{4} \)
|
| $97$ |
\( (T + 5924170)^{4} \)
|
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