Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [500,4,Mod(1,500)]
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, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("500.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 500 = 2^{2} \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 500.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: | \(29.5009550029\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.13525.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 12x^{2} + 8x + 29 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} - x^{3} - 12x^{2} + 8x + 29 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 7\nu + 1 ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 17\nu - 1 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 10\nu^{2} - 7\nu - 64 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - \beta _1 + 13 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{2} + 17\beta _1 - 2 ) / 2 \)
|
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 |
|
0 | −5.18209 | 0 | 0 | 0 | 13.5669 | 0 | −0.145898 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −4.48842 | 0 | 0 | 0 | 1.71442 | 0 | −6.85410 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 4.48842 | 0 | 0 | 0 | −1.71442 | 0 | −6.85410 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 5.18209 | 0 | 0 | 0 | −13.5669 | 0 | −0.145898 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
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 | 500.4.a.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 2000.4.a.h | 4 | ||
| 5.b | even | 2 | 1 | inner | 500.4.a.a | ✓ | 4 |
| 5.c | odd | 4 | 2 | 500.4.c.a | 4 | ||
| 20.d | odd | 2 | 1 | 2000.4.a.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 500.4.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 500.4.a.a | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 500.4.c.a | 4 | 5.c | odd | 4 | 2 | ||
| 2000.4.a.h | 4 | 4.b | odd | 2 | 1 | ||
| 2000.4.a.h | 4 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 47T_{3}^{2} + 541 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(500))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 47T^{2} + 541 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 187T^{2} + 541 \)
|
| $11$ |
\( (T^{2} + 10 T - 20)^{2} \)
|
| $13$ |
\( T^{4} - 4732 T^{2} + 1047376 \)
|
| $17$ |
\( T^{4} - 6748 T^{2} + 7279696 \)
|
| $19$ |
\( (T^{2} + 42 T - 164)^{2} \)
|
| $23$ |
\( T^{4} - 15843 T^{2} + 4285261 \)
|
| $29$ |
\( (T^{2} + 157 T + 3161)^{2} \)
|
| $31$ |
\( (T^{2} + 178 T + 1796)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 2349593296 \)
|
| $41$ |
\( (T^{2} + 283 T + 7271)^{2} \)
|
| $43$ |
\( T^{4} - 62935 T^{2} + 308383525 \)
|
| $47$ |
\( T^{4} + \cdots + 2298017061 \)
|
| $53$ |
\( T^{4} + \cdots + 5320462336 \)
|
| $59$ |
\( (T^{2} + 454 T - 123316)^{2} \)
|
| $61$ |
\( (T^{2} + 731 T + 47129)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 120042931456 \)
|
| $71$ |
\( (T^{2} + 854 T - 57476)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 454855496656 \)
|
| $79$ |
\( (T^{2} + 836 T + 87604)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 1562459438821 \)
|
| $89$ |
\( (T^{2} + 803 T - 226609)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 1480697446096 \)
|
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