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: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.34196.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 8x^{2} - 2x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} - 8x^{2} - 2x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 8\nu - 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 2\nu^{2} + 6\nu - 6 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} + 8\beta _1 + 2 \)
|
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 |
|
−2.16748 | −1.00000 | 2.69797 | −1.00000 | 2.16748 | 0 | −1.51284 | 1.00000 | 2.16748 | ||||||||||||||||||||||||||||||
| 1.2 | −1.33099 | −1.00000 | −0.228467 | −1.00000 | 1.33099 | 0 | 2.96607 | 1.00000 | 1.33099 | |||||||||||||||||||||||||||||||
| 1.3 | 1.79515 | −1.00000 | 1.22257 | −1.00000 | −1.79515 | 0 | −1.39560 | 1.00000 | −1.79515 | |||||||||||||||||||||||||||||||
| 1.4 | 2.70332 | −1.00000 | 5.30793 | −1.00000 | −2.70332 | 0 | 8.94237 | 1.00000 | −2.70332 | |||||||||||||||||||||||||||||||
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.bo | ✓ | 4 |
| 7.b | odd | 2 | 1 | 8085.2.a.bp | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8085.2.a.bo | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 8085.2.a.bp | yes | 4 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8085))\):
|
\( T_{2}^{4} - T_{2}^{3} - 8T_{2}^{2} + 4T_{2} + 14 \)
|
|
\( T_{13}^{4} + 8T_{13}^{3} + 16T_{13}^{2} - 2T_{13} - 16 \)
|
|
\( T_{17}^{4} + 5T_{17}^{3} + T_{17}^{2} - 9T_{17} - 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} + \cdots + 14 \)
|
| $3$ |
\( (T + 1)^{4} \)
|
| $5$ |
\( (T + 1)^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T + 1)^{4} \)
|
| $13$ |
\( T^{4} + 8 T^{3} + \cdots - 16 \)
|
| $17$ |
\( T^{4} + 5 T^{3} + \cdots - 2 \)
|
| $19$ |
\( T^{4} + 3 T^{3} + \cdots - 4 \)
|
| $23$ |
\( T^{4} - 7 T^{3} + \cdots - 608 \)
|
| $29$ |
\( T^{4} - 5 T^{3} + \cdots - 16 \)
|
| $31$ |
\( T^{4} + 14 T^{3} + \cdots - 1792 \)
|
| $37$ |
\( T^{4} - 92 T^{2} + \cdots + 1876 \)
|
| $41$ |
\( T^{4} - 10 T^{3} + \cdots - 20 \)
|
| $43$ |
\( T^{4} - 11 T^{3} + \cdots + 8 \)
|
| $47$ |
\( T^{4} - 2 T^{3} + \cdots + 152 \)
|
| $53$ |
\( T^{4} + 3 T^{3} + \cdots + 1852 \)
|
| $59$ |
\( T^{4} - 19 T^{3} + \cdots - 278 \)
|
| $61$ |
\( T^{4} + T^{3} + \cdots + 578 \)
|
| $67$ |
\( T^{4} - 8 T^{3} + \cdots + 272 \)
|
| $71$ |
\( T^{4} - 4 T^{3} + \cdots - 560 \)
|
| $73$ |
\( T^{4} - 10 T^{3} + \cdots + 1504 \)
|
| $79$ |
\( T^{4} - 10 T^{3} + \cdots + 100 \)
|
| $83$ |
\( T^{4} - 5 T^{3} + \cdots + 610 \)
|
| $89$ |
\( T^{4} + 3 T^{3} + \cdots + 8446 \)
|
| $97$ |
\( T^{4} + 3 T^{3} + \cdots + 8974 \)
|
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