Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [500,2,Mod(249,500)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(500, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("500.249");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 500 = 2^{2} \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 500.c (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(3.99252010106\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.12400.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 12x^{2} + 31 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 12x^{2} + 31 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + 7 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 7\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} - 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} - 7\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/500\mathbb{Z}\right)^\times\).
| \(n\) | \(251\) | \(377\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 249.1 |
|
0 | − | 3.13912i | 0 | 0 | 0 | 1.19904i | 0 | −6.85410 | 0 | |||||||||||||||||||||||||||||
| 249.2 | 0 | − | 1.77367i | 0 | 0 | 0 | 4.64352i | 0 | −0.145898 | 0 | ||||||||||||||||||||||||||||||
| 249.3 | 0 | 1.77367i | 0 | 0 | 0 | − | 4.64352i | 0 | −0.145898 | 0 | ||||||||||||||||||||||||||||||
| 249.4 | 0 | 3.13912i | 0 | 0 | 0 | − | 1.19904i | 0 | −6.85410 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 500.2.c.a | 4 | |
| 3.b | odd | 2 | 1 | 4500.2.d.e | 4 | ||
| 4.b | odd | 2 | 1 | 2000.2.c.a | 4 | ||
| 5.b | even | 2 | 1 | inner | 500.2.c.a | 4 | |
| 5.c | odd | 4 | 2 | 500.2.a.c | ✓ | 4 | |
| 15.d | odd | 2 | 1 | 4500.2.d.e | 4 | ||
| 15.e | even | 4 | 2 | 4500.2.a.q | 4 | ||
| 20.d | odd | 2 | 1 | 2000.2.c.a | 4 | ||
| 20.e | even | 4 | 2 | 2000.2.a.p | 4 | ||
| 40.i | odd | 4 | 2 | 8000.2.a.bl | 4 | ||
| 40.k | even | 4 | 2 | 8000.2.a.bm | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 500.2.a.c | ✓ | 4 | 5.c | odd | 4 | 2 | |
| 500.2.c.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 500.2.c.a | 4 | 5.b | even | 2 | 1 | inner | |
| 2000.2.a.p | 4 | 20.e | even | 4 | 2 | ||
| 2000.2.c.a | 4 | 4.b | odd | 2 | 1 | ||
| 2000.2.c.a | 4 | 20.d | odd | 2 | 1 | ||
| 4500.2.a.q | 4 | 15.e | even | 4 | 2 | ||
| 4500.2.d.e | 4 | 3.b | odd | 2 | 1 | ||
| 4500.2.d.e | 4 | 15.d | odd | 2 | 1 | ||
| 8000.2.a.bl | 4 | 40.i | odd | 4 | 2 | ||
| 8000.2.a.bm | 4 | 40.k | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 13T_{3}^{2} + 31 \)
acting on \(S_{2}^{\mathrm{new}}(500, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 13T^{2} + 31 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 23T^{2} + 31 \)
|
| $11$ |
\( (T^{2} - 20)^{2} \)
|
| $13$ |
\( T^{4} + 48T^{2} + 496 \)
|
| $17$ |
\( T^{4} + 52T^{2} + 496 \)
|
| $19$ |
\( (T^{2} + 2 T - 4)^{2} \)
|
| $23$ |
\( T^{4} + 27T^{2} + 31 \)
|
| $29$ |
\( (T^{2} + 7 T + 11)^{2} \)
|
| $31$ |
\( (T - 6)^{4} \)
|
| $37$ |
\( T^{4} + 108T^{2} + 496 \)
|
| $41$ |
\( (T^{2} + 3 T + 1)^{2} \)
|
| $43$ |
\( T^{4} + 65T^{2} + 775 \)
|
| $47$ |
\( T^{4} + 57T^{2} + 31 \)
|
| $53$ |
\( T^{4} + 192T^{2} + 7936 \)
|
| $59$ |
\( (T^{2} - 16 T + 44)^{2} \)
|
| $61$ |
\( (T^{2} - 9 T + 9)^{2} \)
|
| $67$ |
\( T^{4} + 108T^{2} + 496 \)
|
| $71$ |
\( (T^{2} - 6 T + 4)^{2} \)
|
| $73$ |
\( T^{4} + 48T^{2} + 496 \)
|
| $79$ |
\( (T^{2} + 6 T - 36)^{2} \)
|
| $83$ |
\( T^{4} + 127T^{2} + 3751 \)
|
| $89$ |
\( (T^{2} - 7 T - 49)^{2} \)
|
| $97$ |
\( T^{4} + 48T^{2} + 496 \)
|
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