Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7280,2,Mod(1,7280)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7280, 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("7280.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7280 = 2^{4} \cdot 5 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7280.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: | \(58.1310926715\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.2112217.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 10x^{3} - x^{2} + 10x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 3640) |
| 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
\( x^{5} - 10x^{3} - x^{2} + 10x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 9\nu^{2} + 8\nu + 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 10\nu^{2} + 8\nu + 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\nu^{4} - \nu^{3} - 19\nu^{2} + 8\nu + 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} - \beta_{2} + 8\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - 10\beta_{3} + 9\beta_{2} + 32 \)
|
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 | −2.94326 | 0 | 1.00000 | 0 | −1.00000 | 0 | 5.66276 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.769609 | 0 | 1.00000 | 0 | −1.00000 | 0 | −2.40770 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.492015 | 0 | 1.00000 | 0 | −1.00000 | 0 | −2.75792 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.19075 | 0 | 1.00000 | 0 | −1.00000 | 0 | −1.58211 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.01413 | 0 | 1.00000 | 0 | −1.00000 | 0 | 6.08498 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
| \(13\) | \(1\) |
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 | 7280.2.a.cd | 5 | |
| 4.b | odd | 2 | 1 | 3640.2.a.z | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3640.2.a.z | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 7280.2.a.cd | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7280))\):
|
\( T_{3}^{5} - 10T_{3}^{3} - T_{3}^{2} + 10T_{3} + 4 \)
|
|
\( T_{11}^{5} + 7T_{11}^{4} - 8T_{11}^{3} - 68T_{11}^{2} + 80T_{11} + 16 \)
|
|
\( T_{17}^{5} - 3T_{17}^{4} - 17T_{17}^{3} + 50T_{17}^{2} + 48T_{17} - 128 \)
|
|
\( T_{19}^{5} + 3T_{19}^{4} - 67T_{19}^{3} - 212T_{19}^{2} + 1032T_{19} + 3352 \)
|
|
\( T_{23}^{5} + 3T_{23}^{4} - 30T_{23}^{3} - 76T_{23}^{2} + 208T_{23} + 448 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 10 T^{3} + \cdots + 4 \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( (T + 1)^{5} \)
|
| $11$ |
\( T^{5} + 7 T^{4} + \cdots + 16 \)
|
| $13$ |
\( (T + 1)^{5} \)
|
| $17$ |
\( T^{5} - 3 T^{4} + \cdots - 128 \)
|
| $19$ |
\( T^{5} + 3 T^{4} + \cdots + 3352 \)
|
| $23$ |
\( T^{5} + 3 T^{4} + \cdots + 448 \)
|
| $29$ |
\( T^{5} - 7 T^{4} + \cdots - 952 \)
|
| $31$ |
\( T^{5} + 10 T^{4} + \cdots - 28 \)
|
| $37$ |
\( T^{5} - 10 T^{4} + \cdots - 124 \)
|
| $41$ |
\( T^{5} - 74 T^{3} + \cdots + 256 \)
|
| $43$ |
\( T^{5} + 4 T^{4} + \cdots - 3808 \)
|
| $47$ |
\( T^{5} - 3 T^{4} + \cdots - 256 \)
|
| $53$ |
\( T^{5} - 26 T^{4} + \cdots + 3584 \)
|
| $59$ |
\( T^{5} + 3 T^{4} + \cdots - 224 \)
|
| $61$ |
\( T^{5} - 13 T^{4} + \cdots - 1984 \)
|
| $67$ |
\( T^{5} + 12 T^{4} + \cdots + 57808 \)
|
| $71$ |
\( T^{5} + 8 T^{4} + \cdots + 22016 \)
|
| $73$ |
\( T^{5} - T^{4} + \cdots - 4336 \)
|
| $79$ |
\( T^{5} - 6 T^{4} + \cdots - 256 \)
|
| $83$ |
\( T^{5} - 8 T^{4} + \cdots + 128 \)
|
| $89$ |
\( T^{5} - 3 T^{4} + \cdots - 22400 \)
|
| $97$ |
\( T^{5} + 15 T^{4} + \cdots + 16576 \)
|
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