Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8085,2,Mod(1,8085)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8085, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("8085.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8085.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: | \(64.5590500342\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{8} - 13x^{6} + 53x^{4} - 72x^{2} - 6x + 24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1155) |
| 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
\( x^{8} - 13x^{6} + 53x^{4} - 72x^{2} - 6x + 24 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 6\nu^{2} + 7\nu - 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 11\nu^{5} + 33\nu^{3} - 2\nu^{2} - 20\nu + 2 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{6} + 11\nu^{5} - 20\nu^{4} - 35\nu^{3} + 54\nu^{2} + 28\nu - 28 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 7\beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 7\beta_{3} + \beta_{2} + 28\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 10\beta_{4} + \beta_{3} + 44\beta_{2} + \beta _1 + 75 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2\beta_{6} + 11\beta_{5} + 11\beta_{4} + 44\beta_{3} + 13\beta_{2} + 163\beta _1 + 4 \)
|
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 |
|
−2.56369 | −1.00000 | 4.57249 | 1.00000 | 2.56369 | 0 | −6.59506 | 1.00000 | −2.56369 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.93459 | −1.00000 | 1.74263 | 1.00000 | 1.93459 | 0 | 0.497910 | 1.00000 | −1.93459 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.53073 | −1.00000 | 0.343144 | 1.00000 | 1.53073 | 0 | 2.53620 | 1.00000 | −1.53073 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.614552 | −1.00000 | −1.62233 | 1.00000 | 0.614552 | 0 | 2.22611 | 1.00000 | −0.614552 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.856504 | −1.00000 | −1.26640 | 1.00000 | −0.856504 | 0 | −2.79769 | 1.00000 | 0.856504 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.08995 | −1.00000 | −0.812006 | 1.00000 | −1.08995 | 0 | −3.06495 | 1.00000 | 1.08995 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.27410 | −1.00000 | 3.17154 | 1.00000 | −2.27410 | 0 | 2.66421 | 1.00000 | 2.27410 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.42300 | −1.00000 | 3.87093 | 1.00000 | −2.42300 | 0 | 4.53326 | 1.00000 | 2.42300 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
| \(11\) | \(1\) |
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 | 8085.2.a.ce | 8 | |
| 7.b | odd | 2 | 1 | 8085.2.a.cf | 8 | ||
| 7.d | odd | 6 | 2 | 1155.2.q.j | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1155.2.q.j | ✓ | 16 | 7.d | odd | 6 | 2 | |
| 8085.2.a.ce | 8 | 1.a | even | 1 | 1 | trivial | |
| 8085.2.a.cf | 8 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8085))\):
|
\( T_{2}^{8} - 13T_{2}^{6} + 53T_{2}^{4} - 72T_{2}^{2} + 6T_{2} + 24 \)
|
|
\( T_{13}^{8} + 4T_{13}^{7} - 70T_{13}^{6} - 236T_{13}^{5} + 1320T_{13}^{4} + 2436T_{13}^{3} - 8810T_{13}^{2} + 3428T_{13} + 7 \)
|
|
\( T_{17}^{8} + 4T_{17}^{7} - 76T_{17}^{6} - 208T_{17}^{5} + 1984T_{17}^{4} + 2880T_{17}^{3} - 18624T_{17}^{2} - 6912T_{17} + 29952 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 13 T^{6} + \cdots + 24 \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( (T - 1)^{8} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T + 1)^{8} \)
|
| $13$ |
\( T^{8} + 4 T^{7} + \cdots + 7 \)
|
| $17$ |
\( T^{8} + 4 T^{7} + \cdots + 29952 \)
|
| $19$ |
\( T^{8} + 9 T^{7} + \cdots + 177920 \)
|
| $23$ |
\( T^{8} + 5 T^{7} + \cdots - 12000 \)
|
| $29$ |
\( T^{8} + 5 T^{7} + \cdots - 2048736 \)
|
| $31$ |
\( T^{8} + 5 T^{7} + \cdots - 1780 \)
|
| $37$ |
\( T^{8} - 7 T^{7} + \cdots + 2395424 \)
|
| $41$ |
\( T^{8} + 9 T^{7} + \cdots + 777600 \)
|
| $43$ |
\( T^{8} - 14 T^{7} + \cdots + 84133 \)
|
| $47$ |
\( T^{8} - 5 T^{7} + \cdots + 3206688 \)
|
| $53$ |
\( T^{8} + T^{7} + \cdots - 943872 \)
|
| $59$ |
\( T^{8} + 16 T^{7} + \cdots - 32496 \)
|
| $61$ |
\( T^{8} + 26 T^{7} + \cdots - 24064 \)
|
| $67$ |
\( T^{8} - 3 T^{7} + \cdots + 821344 \)
|
| $71$ |
\( T^{8} + 30 T^{7} + \cdots + 2088384 \)
|
| $73$ |
\( T^{8} + 15 T^{7} + \cdots - 72020 \)
|
| $79$ |
\( T^{8} - 11 T^{7} + \cdots - 578560 \)
|
| $83$ |
\( T^{8} + 12 T^{7} + \cdots - 35281392 \)
|
| $89$ |
\( T^{8} - 230 T^{6} + \cdots - 51324 \)
|
| $97$ |
\( T^{8} + 44 T^{7} + \cdots + 7517120 \)
|
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