Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [128,8,Mod(1,128)]
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, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("128.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 128 = 2^{7} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 128.a (trivial) |
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: | yes |
| Analytic conductor: | \(39.9852832620\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.56328.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 50x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 50x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( 8\nu - 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 32\nu^{2} - 40\nu - 1065 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 3 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} + 5\beta _1 + 1080 ) / 32 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −51.2165 | 0 | 145.477 | 0 | 192.521 | 0 | 436.129 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 2.95798 | 0 | −362.486 | 0 | −715.055 | 0 | −2178.25 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 62.2585 | 0 | 203.009 | 0 | 534.535 | 0 | 1689.12 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 128.8.a.c | yes | 3 |
| 4.b | odd | 2 | 1 | 128.8.a.a | ✓ | 3 | |
| 8.b | even | 2 | 1 | 128.8.a.b | yes | 3 | |
| 8.d | odd | 2 | 1 | 128.8.a.d | yes | 3 | |
| 16.e | even | 4 | 2 | 256.8.b.k | 6 | ||
| 16.f | odd | 4 | 2 | 256.8.b.l | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 128.8.a.a | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 128.8.a.b | yes | 3 | 8.b | even | 2 | 1 | |
| 128.8.a.c | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 128.8.a.d | yes | 3 | 8.d | odd | 2 | 1 | |
| 256.8.b.k | 6 | 16.e | even | 4 | 2 | ||
| 256.8.b.l | 6 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(128))\):
|
\( T_{3}^{3} - 14T_{3}^{2} - 3156T_{3} + 9432 \)
|
|
\( T_{5}^{3} + 14T_{5}^{2} - 96788T_{5} + 10705320 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 14 T^{2} + \cdots + 9432 \)
|
| $5$ |
\( T^{3} + 14 T^{2} + \cdots + 10705320 \)
|
| $7$ |
\( T^{3} - 12 T^{2} + \cdots + 73585600 \)
|
| $11$ |
\( T^{3} + \cdots + 58585322728 \)
|
| $13$ |
\( T^{3} + \cdots + 172883400168 \)
|
| $17$ |
\( T^{3} + \cdots - 5603612104824 \)
|
| $19$ |
\( T^{3} + \cdots + 18044266112856 \)
|
| $23$ |
\( T^{3} + \cdots - 36954823141312 \)
|
| $29$ |
\( T^{3} + \cdots - 634620069297912 \)
|
| $31$ |
\( T^{3} + \cdots - 17\!\cdots\!52 \)
|
| $37$ |
\( T^{3} + \cdots + 19\!\cdots\!44 \)
|
| $41$ |
\( T^{3} + \cdots - 31\!\cdots\!36 \)
|
| $43$ |
\( T^{3} + \cdots + 15\!\cdots\!00 \)
|
| $47$ |
\( T^{3} + \cdots + 41\!\cdots\!92 \)
|
| $53$ |
\( T^{3} + \cdots + 13\!\cdots\!80 \)
|
| $59$ |
\( T^{3} + \cdots - 21\!\cdots\!36 \)
|
| $61$ |
\( T^{3} + \cdots + 21\!\cdots\!28 \)
|
| $67$ |
\( T^{3} + \cdots - 13\!\cdots\!92 \)
|
| $71$ |
\( T^{3} + \cdots - 35\!\cdots\!40 \)
|
| $73$ |
\( T^{3} + \cdots - 48\!\cdots\!92 \)
|
| $79$ |
\( T^{3} + \cdots + 13\!\cdots\!32 \)
|
| $83$ |
\( T^{3} + \cdots - 99\!\cdots\!92 \)
|
| $89$ |
\( T^{3} + \cdots - 95\!\cdots\!24 \)
|
| $97$ |
\( T^{3} + \cdots - 83\!\cdots\!64 \)
|
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