Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8384,2,Mod(1,8384)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8384, 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("8384.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8384 = 2^{6} \cdot 131 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8384.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: | \(66.9465770546\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 15x^{6} + 15x^{5} + 68x^{4} - 66x^{3} - 87x^{2} + 68x + 20 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1048) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 15x^{6} + 15x^{5} + 68x^{4} - 66x^{3} - 87x^{2} + 68x + 20 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{7} - 5\nu^{6} - 33\nu^{5} + 41\nu^{4} + 90\nu^{3} - 50\nu^{2} - 37\nu - 14 ) / 26 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{7} - 5\nu^{6} - 59\nu^{5} + 67\nu^{4} + 298\nu^{3} - 258\nu^{2} - 245\nu + 194 ) / 52 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{7} - 9\nu^{6} + 81\nu^{5} + 79\nu^{4} - 358\nu^{3} - 142\nu^{2} + 287\nu + 110 ) / 52 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -5\nu^{7} - 9\nu^{6} + 81\nu^{5} + 131\nu^{4} - 410\nu^{3} - 506\nu^{2} + 599\nu + 318 ) / 52 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 11\nu^{7} - \nu^{6} - 147\nu^{5} + 3\nu^{4} + 590\nu^{3} + 42\nu^{2} - 673\nu - 138 ) / 52 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{5} - \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{3} + 7\beta_{2} + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{7} + \beta_{6} + 8\beta_{5} - 2\beta_{4} - 8\beta_{3} - \beta_{2} + 40\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10\beta_{7} + 10\beta_{6} - 3\beta_{5} - 3\beta_{4} - 11\beta_{3} + 45\beta_{2} + \beta _1 + 157 \)
|
| \(\nu^{7}\) | \(=\) |
\( 72\beta_{7} + 14\beta_{6} + 53\beta_{5} - 27\beta_{4} - 54\beta_{3} - 15\beta_{2} + 274\beta _1 + 5 \)
|
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 | −3.02048 | 0 | −4.20998 | 0 | −3.78458 | 0 | 6.12327 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.56521 | 0 | −3.11649 | 0 | 4.53276 | 0 | −0.550129 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.39286 | 0 | 3.29988 | 0 | 5.11596 | 0 | −1.05995 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.744671 | 0 | 3.49852 | 0 | −4.78597 | 0 | −2.44546 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.204857 | 0 | −0.918577 | 0 | −2.40192 | 0 | −2.95803 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.25388 | 0 | −4.10020 | 0 | 1.28477 | 0 | −1.42780 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.50553 | 0 | −1.24807 | 0 | −4.54028 | 0 | 3.27770 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 3.16866 | 0 | 1.79492 | 0 | −1.42074 | 0 | 7.04040 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(131\) | \( +1 \) |
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 | 8384.2.a.bq | 8 | |
| 4.b | odd | 2 | 1 | 8384.2.a.br | 8 | ||
| 8.b | even | 2 | 1 | 2096.2.a.q | 8 | ||
| 8.d | odd | 2 | 1 | 1048.2.a.f | ✓ | 8 | |
| 24.f | even | 2 | 1 | 9432.2.a.t | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1048.2.a.f | ✓ | 8 | 8.d | odd | 2 | 1 | |
| 2096.2.a.q | 8 | 8.b | even | 2 | 1 | ||
| 8384.2.a.bq | 8 | 1.a | even | 1 | 1 | trivial | |
| 8384.2.a.br | 8 | 4.b | odd | 2 | 1 | ||
| 9432.2.a.t | 8 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8384))\):
|
\( T_{3}^{8} - 16T_{3}^{6} - 5T_{3}^{5} + 69T_{3}^{4} + 49T_{3}^{3} - 67T_{3}^{2} - 64T_{3} - 10 \)
|
|
\( T_{5}^{8} + 5T_{5}^{7} - 24T_{5}^{6} - 131T_{5}^{5} + 146T_{5}^{4} + 1013T_{5}^{3} + 89T_{5}^{2} - 1976T_{5} - 1278 \)
|
|
\( T_{7}^{8} + 6T_{7}^{7} - 39T_{7}^{6} - 291T_{7}^{5} + 167T_{7}^{4} + 3698T_{7}^{3} + 4766T_{7}^{2} - 5177T_{7} - 8361 \)
|
|
\( T_{11}^{8} - 13T_{11}^{7} + 44T_{11}^{6} + 39T_{11}^{5} - 364T_{11}^{4} + 307T_{11}^{3} + 371T_{11}^{2} - 484T_{11} + 124 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 16 T^{6} + \cdots - 10 \)
|
| $5$ |
\( T^{8} + 5 T^{7} + \cdots - 1278 \)
|
| $7$ |
\( T^{8} + 6 T^{7} + \cdots - 8361 \)
|
| $11$ |
\( T^{8} - 13 T^{7} + \cdots + 124 \)
|
| $13$ |
\( T^{8} - 6 T^{7} + \cdots - 26918 \)
|
| $17$ |
\( T^{8} - 16 T^{7} + \cdots + 105984 \)
|
| $19$ |
\( T^{8} - 99 T^{6} + \cdots - 4416 \)
|
| $23$ |
\( T^{8} + 2 T^{7} + \cdots - 31104 \)
|
| $29$ |
\( T^{8} + 12 T^{7} + \cdots - 54464 \)
|
| $31$ |
\( T^{8} - 90 T^{6} + \cdots - 640 \)
|
| $37$ |
\( T^{8} + 20 T^{7} + \cdots - 944448 \)
|
| $41$ |
\( T^{8} + 2 T^{7} + \cdots - 150693 \)
|
| $43$ |
\( T^{8} + 24 T^{7} + \cdots + 49152 \)
|
| $47$ |
\( T^{8} + 8 T^{7} + \cdots + 2585600 \)
|
| $53$ |
\( T^{8} - 6 T^{7} + \cdots + 274408 \)
|
| $59$ |
\( T^{8} - 2 T^{7} + \cdots + 21058 \)
|
| $61$ |
\( T^{8} + 8 T^{7} + \cdots + 5764544 \)
|
| $67$ |
\( T^{8} - 2 T^{7} + \cdots + 61352960 \)
|
| $71$ |
\( T^{8} - 36 T^{7} + \cdots - 4631936 \)
|
| $73$ |
\( T^{8} + 4 T^{7} + \cdots - 203264 \)
|
| $79$ |
\( T^{8} - 10 T^{7} + \cdots - 1005568 \)
|
| $83$ |
\( T^{8} - 36 T^{7} + \cdots - 1771584 \)
|
| $89$ |
\( T^{8} - 18 T^{7} + \cdots + 31930212 \)
|
| $97$ |
\( T^{8} + 10 T^{7} + \cdots - 769024 \)
|
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