Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8128,2,Mod(1,8128)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8128, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8128.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8128 = 2^{6} \cdot 127 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8128.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: | \(64.9024067629\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 8x^{5} + 15x^{4} + 17x^{3} - 28x^{2} - 11x + 15 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 127) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 8x^{5} + 15x^{4} + 17x^{3} - 28x^{2} - 11x + 15 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} - \nu + 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 8\nu^{4} + 6\nu^{3} + 16\nu^{2} - 5\nu - 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 6\nu^{4} + 12\nu^{3} + 4\nu^{2} - 11\nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 13\nu^{2} + 10\nu - 14 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 5\nu^{4} + 19\nu^{3} - \nu^{2} - 20\nu + 6 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 4\nu^{4} + 19\nu^{3} - 9\nu^{2} - 19\nu + 17 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + \beta_{4} - \beta_{3} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - \beta_{3} + \beta _1 + 7 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{6} + \beta_{5} + 3\beta_{4} - 7\beta_{3} + 2\beta_{2} + \beta _1 + 7 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{6} + 6\beta_{5} + 7\beta_{4} - 7\beta_{3} + 8\beta _1 + 33 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10\beta_{6} + 3\beta_{5} + 7\beta_{4} - 20\beta_{3} + 7\beta_{2} + 5\beta _1 + 21 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 9\beta_{6} + 32\beta_{5} + 41\beta_{4} - 43\beta_{3} + 4\beta_{2} + 52\beta _1 + 173 ) / 2 \)
|
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 | −2.64231 | 0 | −3.73266 | 0 | −1.84347 | 0 | 3.98183 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.48317 | 0 | −1.52027 | 0 | −1.05834 | 0 | 3.16612 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.182004 | 0 | 0.919173 | 0 | 4.13370 | 0 | −2.96687 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.12872 | 0 | −2.74338 | 0 | 0.135055 | 0 | −1.72599 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.44873 | 0 | −0.395790 | 0 | −3.35479 | 0 | −0.901182 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.74369 | 0 | −3.18668 | 0 | 3.85075 | 0 | 4.52784 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.98635 | 0 | 2.65960 | 0 | 1.13710 | 0 | 5.91827 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(127\) | \(1\) |
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 | 8128.2.a.bj | 7 | |
| 4.b | odd | 2 | 1 | 8128.2.a.bi | 7 | ||
| 8.b | even | 2 | 1 | 2032.2.a.p | 7 | ||
| 8.d | odd | 2 | 1 | 127.2.a.b | ✓ | 7 | |
| 24.f | even | 2 | 1 | 1143.2.a.i | 7 | ||
| 40.e | odd | 2 | 1 | 3175.2.a.j | 7 | ||
| 56.e | even | 2 | 1 | 6223.2.a.h | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 127.2.a.b | ✓ | 7 | 8.d | odd | 2 | 1 | |
| 1143.2.a.i | 7 | 24.f | even | 2 | 1 | ||
| 2032.2.a.p | 7 | 8.b | even | 2 | 1 | ||
| 3175.2.a.j | 7 | 40.e | odd | 2 | 1 | ||
| 6223.2.a.h | 7 | 56.e | even | 2 | 1 | ||
| 8128.2.a.bi | 7 | 4.b | odd | 2 | 1 | ||
| 8128.2.a.bj | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8128))\):
|
\( T_{3}^{7} - 3T_{3}^{6} - 12T_{3}^{5} + 39T_{3}^{4} + 26T_{3}^{3} - 128T_{3}^{2} + 64T_{3} + 16 \)
|
|
\( T_{5}^{7} + 8T_{5}^{6} + 11T_{5}^{5} - 53T_{5}^{4} - 146T_{5}^{3} - 32T_{5}^{2} + 128T_{5} + 48 \)
|
|
\( T_{7}^{7} - 3T_{7}^{6} - 20T_{7}^{5} + 41T_{7}^{4} + 114T_{7}^{3} - 64T_{7}^{2} - 112T_{7} + 16 \)
|
|
\( T_{11}^{7} - 28T_{11}^{5} - 17T_{11}^{4} + 88T_{11}^{3} - 37T_{11}^{2} - 5T_{11} + 3 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - 3 T^{6} + \cdots + 16 \)
|
| $5$ |
\( T^{7} + 8 T^{6} + \cdots + 48 \)
|
| $7$ |
\( T^{7} - 3 T^{6} + \cdots + 16 \)
|
| $11$ |
\( T^{7} - 28 T^{5} + \cdots + 3 \)
|
| $13$ |
\( T^{7} - T^{6} + \cdots - 5383 \)
|
| $17$ |
\( T^{7} - 24 T^{6} + \cdots + 38235 \)
|
| $19$ |
\( T^{7} + 5 T^{6} + \cdots + 853 \)
|
| $23$ |
\( T^{7} - T^{6} + \cdots - 8016 \)
|
| $29$ |
\( T^{7} - 7 T^{6} + \cdots + 5520 \)
|
| $31$ |
\( T^{7} - 8 T^{6} + \cdots + 2845 \)
|
| $37$ |
\( T^{7} - 6 T^{6} + \cdots + 920 \)
|
| $41$ |
\( T^{7} - 14 T^{6} + \cdots + 4032 \)
|
| $43$ |
\( T^{7} + T^{6} + \cdots + 10096 \)
|
| $47$ |
\( T^{7} + 25 T^{6} + \cdots + 1046391 \)
|
| $53$ |
\( T^{7} + 29 T^{6} + \cdots + 755376 \)
|
| $59$ |
\( T^{7} + 12 T^{6} + \cdots - 339120 \)
|
| $61$ |
\( T^{7} + 7 T^{6} + \cdots - 3625 \)
|
| $67$ |
\( T^{7} + 25 T^{6} + \cdots - 64784 \)
|
| $71$ |
\( T^{7} + 7 T^{6} + \cdots + 84633 \)
|
| $73$ |
\( T^{7} - 13 T^{6} + \cdots + 17401 \)
|
| $79$ |
\( T^{7} - 23 T^{6} + \cdots - 1841711 \)
|
| $83$ |
\( T^{7} - 26 T^{6} + \cdots + 16464 \)
|
| $89$ |
\( T^{7} - 13 T^{6} + \cdots + 432 \)
|
| $97$ |
\( T^{7} + 5 T^{6} + \cdots - 12656 \)
|
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