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: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 56x^{2} - 116x + 105 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{15}\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 56x^{2} - 116x + 105 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -16\nu^{3} + 52\nu^{2} + 888\nu - 64 ) / 23 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 96\nu^{3} - 496\nu^{2} - 2752\nu + 5536 ) / 23 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 336\nu^{3} + 12\nu^{2} - 16440\nu - 29568 ) / 23 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 6\beta_{2} + 57\beta_1 ) / 768 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_{3} - 6\beta_{2} + 111\beta _1 + 10752 ) / 384 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 101\beta_{3} + 294\beta_{2} + 2781\beta _1 + 66816 ) / 768 \)
|
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 | −67.3033 | 0 | −50.5583 | 0 | 1686.22 | 0 | 2342.73 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −14.3917 | 0 | −167.560 | 0 | −1355.20 | 0 | −1979.88 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 31.1632 | 0 | 436.804 | 0 | 384.352 | 0 | −1215.86 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 90.5318 | 0 | −482.686 | 0 | 644.627 | 0 | 6009.00 | 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.g | yes | 4 |
| 4.b | odd | 2 | 1 | 128.8.a.e | ✓ | 4 | |
| 8.b | even | 2 | 1 | 128.8.a.f | yes | 4 | |
| 8.d | odd | 2 | 1 | 128.8.a.h | yes | 4 | |
| 16.e | even | 4 | 2 | 256.8.b.m | 8 | ||
| 16.f | odd | 4 | 2 | 256.8.b.n | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 128.8.a.e | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 128.8.a.f | yes | 4 | 8.b | even | 2 | 1 | |
| 128.8.a.g | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 128.8.a.h | yes | 4 | 8.d | odd | 2 | 1 | |
| 256.8.b.m | 8 | 16.e | even | 4 | 2 | ||
| 256.8.b.n | 8 | 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 \(S_{8}^{\mathrm{new}}(\Gamma_0(128))\):
|
\( T_{3}^{4} - 40T_{3}^{3} - 6152T_{3}^{2} + 112608T_{3} + 2732688 \)
|
|
\( T_{5}^{4} + 264T_{5}^{3} - 192360T_{5}^{2} - 45599200T_{5} - 1786134000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 40 T^{3} + \cdots + 2732688 \)
|
| $5$ |
\( T^{4} + \cdots - 1786134000 \)
|
| $7$ |
\( T^{4} + \cdots - 566184154880 \)
|
| $11$ |
\( T^{4} + \cdots + 57211935334544 \)
|
| $13$ |
\( T^{4} + \cdots - 200078912533488 \)
|
| $17$ |
\( T^{4} + \cdots + 45\!\cdots\!56 \)
|
| $19$ |
\( T^{4} + \cdots + 48\!\cdots\!88 \)
|
| $23$ |
\( T^{4} + \cdots - 59\!\cdots\!44 \)
|
| $29$ |
\( T^{4} + \cdots + 96\!\cdots\!28 \)
|
| $31$ |
\( T^{4} + \cdots - 99\!\cdots\!40 \)
|
| $37$ |
\( T^{4} + \cdots - 11\!\cdots\!20 \)
|
| $41$ |
\( T^{4} + \cdots - 54\!\cdots\!16 \)
|
| $43$ |
\( T^{4} + \cdots - 76\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots - 26\!\cdots\!92 \)
|
| $53$ |
\( T^{4} + \cdots + 39\!\cdots\!60 \)
|
| $59$ |
\( T^{4} + \cdots - 15\!\cdots\!36 \)
|
| $61$ |
\( T^{4} + \cdots + 29\!\cdots\!76 \)
|
| $67$ |
\( T^{4} + \cdots + 48\!\cdots\!20 \)
|
| $71$ |
\( T^{4} + \cdots + 74\!\cdots\!20 \)
|
| $73$ |
\( T^{4} + \cdots - 97\!\cdots\!84 \)
|
| $79$ |
\( T^{4} + \cdots + 34\!\cdots\!44 \)
|
| $83$ |
\( T^{4} + \cdots + 63\!\cdots\!44 \)
|
| $89$ |
\( T^{4} + \cdots - 49\!\cdots\!52 \)
|
| $97$ |
\( T^{4} + \cdots + 41\!\cdots\!36 \)
|
| show more | |
| show less |