Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8880,2,Mod(1,8880)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8880, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("8880.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8880.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: | \(70.9071569949\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 29x^{3} + 6x^{2} + 176x + 128 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 4440) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 29x^{3} + 6x^{2} + 176x + 128 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 3\nu - 12 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 29\nu^{2} + 22\nu + 128 ) / 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{4} + 14\nu^{3} + 55\nu^{2} - 154\nu - 240 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + 3\beta _1 + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + 6\beta_{3} + 8\beta_{2} + 23\beta _1 + 30 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} + 28\beta_{3} + 74\beta_{2} + 111\beta _1 + 280 \)
|
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 | −1.00000 | 0 | −1.00000 | 0 | −4.52112 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | −1.00000 | 0 | −3.46338 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | −1.00000 | 0 | −1.84151 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | −1.00000 | 0 | 2.02084 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.00000 | 0 | −1.00000 | 0 | 4.80516 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(37\) | \(-1\) |
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 | 8880.2.a.cj | 5 | |
| 4.b | odd | 2 | 1 | 4440.2.a.y | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4440.2.a.y | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 8880.2.a.cj | 5 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8880))\):
|
\( T_{7}^{5} + 3T_{7}^{4} - 27T_{7}^{3} - 83T_{7}^{2} + 98T_{7} + 280 \)
|
|
\( T_{11}^{5} - 2T_{11}^{4} - 30T_{11}^{3} + 89T_{11}^{2} + 38T_{11} - 200 \)
|
|
\( T_{13}^{5} + 7T_{13}^{4} - 25T_{13}^{3} - 155T_{13}^{2} + 152T_{13} + 52 \)
|
|
\( T_{23}^{5} + 6T_{23}^{4} - 93T_{23}^{3} - 446T_{23}^{2} + 1856T_{23} + 6016 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( (T + 1)^{5} \)
|
| $5$ |
\( (T + 1)^{5} \)
|
| $7$ |
\( T^{5} + 3 T^{4} + \cdots + 280 \)
|
| $11$ |
\( T^{5} - 2 T^{4} + \cdots - 200 \)
|
| $13$ |
\( T^{5} + 7 T^{4} + \cdots + 52 \)
|
| $17$ |
\( T^{5} - 12 T^{4} + \cdots - 3020 \)
|
| $19$ |
\( T^{5} + T^{4} + \cdots + 40 \)
|
| $23$ |
\( T^{5} + 6 T^{4} + \cdots + 6016 \)
|
| $29$ |
\( T^{5} + T^{4} + \cdots + 86 \)
|
| $31$ |
\( T^{5} + 3 T^{4} + \cdots + 280 \)
|
| $37$ |
\( (T - 1)^{5} \)
|
| $41$ |
\( T^{5} - 11 T^{4} + \cdots + 7264 \)
|
| $43$ |
\( T^{5} + 9 T^{4} + \cdots - 4 \)
|
| $47$ |
\( T^{5} + 11 T^{4} + \cdots + 5488 \)
|
| $53$ |
\( T^{5} + 10 T^{4} + \cdots - 32 \)
|
| $59$ |
\( T^{5} + 8 T^{4} + \cdots - 56 \)
|
| $61$ |
\( T^{5} - T^{4} + \cdots + 720 \)
|
| $67$ |
\( T^{5} + 31 T^{4} + \cdots + 2176 \)
|
| $71$ |
\( T^{5} + 11 T^{4} + \cdots + 256 \)
|
| $73$ |
\( T^{5} - 3 T^{4} + \cdots - 14144 \)
|
| $79$ |
\( T^{5} - T^{4} + \cdots - 57472 \)
|
| $83$ |
\( T^{5} + 2 T^{4} + \cdots + 21368 \)
|
| $89$ |
\( T^{5} - 27 T^{4} + \cdots + 604 \)
|
| $97$ |
\( T^{5} - 33 T^{4} + \cdots + 688 \)
|
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