Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [64,8,Mod(33,64)]
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([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("64.33");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 64.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: | \(19.9926416310\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.2705346343547136.10 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 301x^{6} + 68101x^{4} - 6772500x^{2} + 506250000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{42} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 301x^{6} + 68101x^{4} - 6772500x^{2} + 506250000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 130784\nu^{6} - 39770984\nu^{4} + 5841976184\nu^{2} - 438311070000 ) / 383068125 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 32\nu^{6} + 111254416 ) / 68101 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 30784\nu^{6} - 8815984\nu^{4} + 1539231184\nu^{2} - 109304820000 ) / 34824375 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 408457\nu^{7} - 92413057\nu^{5} + 17090422657\nu^{3} - 686981250000\nu ) / 114920437500 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 104732\nu^{7} - 349494332\nu^{5} + 56165073932\nu^{3} - 8572919940000\nu ) / 28730109375 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -1204\nu^{7} + 272404\nu^{5} - 41403604\nu^{3} + 2025000000\nu ) / 190265625 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 256468\nu^{7} + 267773132\nu^{5} - 31566992732\nu^{3} + 4287965940000\nu ) / 28730109375 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} + 7\beta_{6} - \beta_{5} + 16\beta_{4} ) / 128 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 73\beta_{3} - 8\beta_{2} - 178\beta _1 + 38528 ) / 512 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1357\beta_{6} + 2416\beta_{4} ) / 64 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 10723\beta_{3} + 2408\beta_{2} - 31078\beta _1 - 5836928 ) / 512 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 206183\beta_{7} + 2007656\beta_{6} + 115883\beta_{5} + 2937728\beta_{4} ) / 1024 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 68101\beta_{2} - 111254416 ) / 32 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 33193583\beta_{7} - 360044456\beta_{6} + 12763283\beta_{5} - 449376128\beta_{4} ) / 1024 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 33.1 |
|
0 | − | 69.0306i | 0 | − | 232.201i | 0 | −1504.06 | 0 | −2578.22 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 33.2 | 0 | − | 69.0306i | 0 | 232.201i | 0 | 1504.06 | 0 | −2578.22 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 33.3 | 0 | − | 29.0306i | 0 | − | 107.493i | 0 | 534.108 | 0 | 1344.22 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 33.4 | 0 | − | 29.0306i | 0 | 107.493i | 0 | −534.108 | 0 | 1344.22 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 33.5 | 0 | 29.0306i | 0 | − | 107.493i | 0 | −534.108 | 0 | 1344.22 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 33.6 | 0 | 29.0306i | 0 | 107.493i | 0 | 534.108 | 0 | 1344.22 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 33.7 | 0 | 69.0306i | 0 | − | 232.201i | 0 | 1504.06 | 0 | −2578.22 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 33.8 | 0 | 69.0306i | 0 | 232.201i | 0 | −1504.06 | 0 | −2578.22 | 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 | 64.8.b.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | 576.8.d.f | 8 | ||
| 4.b | odd | 2 | 1 | inner | 64.8.b.c | ✓ | 8 |
| 8.b | even | 2 | 1 | inner | 64.8.b.c | ✓ | 8 |
| 8.d | odd | 2 | 1 | inner | 64.8.b.c | ✓ | 8 |
| 12.b | even | 2 | 1 | 576.8.d.f | 8 | ||
| 16.e | even | 4 | 1 | 256.8.a.j | 4 | ||
| 16.e | even | 4 | 1 | 256.8.a.p | 4 | ||
| 16.f | odd | 4 | 1 | 256.8.a.j | 4 | ||
| 16.f | odd | 4 | 1 | 256.8.a.p | 4 | ||
| 24.f | even | 2 | 1 | 576.8.d.f | 8 | ||
| 24.h | odd | 2 | 1 | 576.8.d.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 64.8.b.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 64.8.b.c | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 64.8.b.c | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 64.8.b.c | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
| 256.8.a.j | 4 | 16.e | even | 4 | 1 | ||
| 256.8.a.j | 4 | 16.f | odd | 4 | 1 | ||
| 256.8.a.p | 4 | 16.e | even | 4 | 1 | ||
| 256.8.a.p | 4 | 16.f | odd | 4 | 1 | ||
| 576.8.d.f | 8 | 3.b | odd | 2 | 1 | ||
| 576.8.d.f | 8 | 12.b | even | 2 | 1 | ||
| 576.8.d.f | 8 | 24.f | even | 2 | 1 | ||
| 576.8.d.f | 8 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 5608T_{3}^{2} + 4016016 \)
acting on \(S_{8}^{\mathrm{new}}(64, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} + 5608 T^{2} + 4016016)^{2} \)
|
| $5$ |
\( (T^{4} + 65472 T^{2} + 623001600)^{2} \)
|
| $7$ |
\( (T^{4} - 2547456 T^{2} + 645335875584)^{2} \)
|
| $11$ |
\( (T^{4} + \cdots + 77526545767056)^{2} \)
|
| $13$ |
\( (T^{4} + \cdots + 55\!\cdots\!04)^{2} \)
|
| $17$ |
\( (T^{2} - 22740 T + 61426404)^{4} \)
|
| $19$ |
\( (T^{4} + \cdots + 11\!\cdots\!96)^{2} \)
|
| $23$ |
\( (T^{4} + \cdots + 38\!\cdots\!00)^{2} \)
|
| $29$ |
\( (T^{4} + \cdots + 91\!\cdots\!64)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots + 29\!\cdots\!64)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots + 33\!\cdots\!04)^{2} \)
|
| $41$ |
\( (T^{2} + 209388 T - 538628183964)^{4} \)
|
| $43$ |
\( (T^{4} + \cdots + 10\!\cdots\!76)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots + 34\!\cdots\!04)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots + 99\!\cdots\!00)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots + 60\!\cdots\!36)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots + 42\!\cdots\!04)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots + 45\!\cdots\!76)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 13\!\cdots\!84)^{2} \)
|
| $73$ |
\( (T^{2} + \cdots + 2806911930244)^{4} \)
|
| $79$ |
\( (T^{4} + \cdots + 23\!\cdots\!84)^{2} \)
|
| $83$ |
\( (T^{4} + \cdots + 74\!\cdots\!00)^{2} \)
|
| $89$ |
\( (T^{2} + \cdots - 32517358628796)^{4} \)
|
| $97$ |
\( (T^{2} + \cdots - 30423027470684)^{4} \)
|
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