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: | 4.4.7232.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 5x^{2} + 4x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 5x^{2} + 4x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - \nu ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 5\nu + 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 4\nu^{2} + \nu - 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{3} - \beta_{2} + 9\beta _1 + 17 ) / 2 \)
|
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 |
|
−2.06644 | 1.00000 | 2.27016 | −1.00000 | −2.06644 | 0 | −0.558268 | 1.00000 | 2.06644 | ||||||||||||||||||||||||||||||
| 1.2 | −0.222191 | 1.00000 | −1.95063 | −1.00000 | −0.222191 | 0 | 0.877796 | 1.00000 | 0.222191 | |||||||||||||||||||||||||||||||
| 1.3 | 1.65222 | 1.00000 | 0.729840 | −1.00000 | 1.65222 | 0 | −2.09859 | 1.00000 | −1.65222 | |||||||||||||||||||||||||||||||
| 1.4 | 2.63640 | 1.00000 | 4.95063 | −1.00000 | 2.63640 | 0 | 7.77906 | 1.00000 | −2.63640 | |||||||||||||||||||||||||||||||
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.bq | 4 | |
| 7.b | odd | 2 | 1 | 1155.2.a.v | ✓ | 4 | |
| 21.c | even | 2 | 1 | 3465.2.a.bj | 4 | ||
| 35.c | odd | 2 | 1 | 5775.2.a.by | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1155.2.a.v | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 3465.2.a.bj | 4 | 21.c | even | 2 | 1 | ||
| 5775.2.a.by | 4 | 35.c | odd | 2 | 1 | ||
| 8085.2.a.bq | 4 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8085))\):
|
\( T_{2}^{4} - 2T_{2}^{3} - 5T_{2}^{2} + 8T_{2} + 2 \)
|
|
\( T_{13}^{4} + 4T_{13}^{3} - 14T_{13}^{2} - 24T_{13} - 8 \)
|
|
\( T_{17}^{4} + 6T_{17}^{3} - 11T_{17}^{2} - 4T_{17} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2 T^{3} + \cdots + 2 \)
|
| $3$ |
\( (T - 1)^{4} \)
|
| $5$ |
\( (T + 1)^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T - 1)^{4} \)
|
| $13$ |
\( T^{4} + 4 T^{3} + \cdots - 8 \)
|
| $17$ |
\( T^{4} + 6 T^{3} + \cdots + 4 \)
|
| $19$ |
\( T^{4} - 6 T^{3} + \cdots - 16 \)
|
| $23$ |
\( T^{4} - 10 T^{3} + \cdots - 128 \)
|
| $29$ |
\( T^{4} - 6 T^{3} + \cdots + 284 \)
|
| $31$ |
\( T^{4} - 8 T^{3} + \cdots - 256 \)
|
| $37$ |
\( T^{4} - 12 T^{3} + \cdots + 824 \)
|
| $41$ |
\( T^{4} - 106 T^{2} + \cdots + 968 \)
|
| $43$ |
\( T^{4} - 6 T^{3} + \cdots + 6256 \)
|
| $47$ |
\( T^{4} - 146 T^{2} + \cdots + 2176 \)
|
| $53$ |
\( T^{4} - 14 T^{3} + \cdots + 68 \)
|
| $59$ |
\( T^{4} + 2 T^{3} + \cdots + 4336 \)
|
| $61$ |
\( T^{4} + 2 T^{3} + \cdots + 196 \)
|
| $67$ |
\( T^{4} + 12 T^{3} + \cdots - 6592 \)
|
| $71$ |
\( T^{4} - 4 T^{3} + \cdots + 2176 \)
|
| $73$ |
\( T^{4} + 20 T^{3} + \cdots - 32 \)
|
| $79$ |
\( T^{4} + 4 T^{3} + \cdots + 1088 \)
|
| $83$ |
\( T^{4} + 14 T^{3} + \cdots + 1136 \)
|
| $89$ |
\( T^{4} - 6 T^{3} + \cdots + 188 \)
|
| $97$ |
\( T^{4} - 14 T^{3} + \cdots - 4 \)
|
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