Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [538,2,Mod(1,538)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(538, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("538.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 538 = 2 \cdot 269 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 538.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: | \(4.29595162874\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.4913.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 6x^{2} + x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} - x^{3} - 6x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 2\nu^{2} + 4\nu - 3 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 6\nu - 3 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 6\beta _1 + 3 \)
|
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 |
|
−1.00000 | −1.90570 | 1.00000 | 0.655849 | 1.90570 | 2.04948 | −1.00000 | 0.631706 | −0.655849 | ||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 0.512072 | 1.00000 | −1.04948 | −0.512072 | −2.90570 | −1.00000 | −2.73778 | 1.04948 | |||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 1.34415 | 1.00000 | 3.90570 | −1.34415 | −0.487928 | −1.00000 | −1.19326 | −3.90570 | |||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 3.04948 | 1.00000 | 1.48793 | −3.04948 | 0.344151 | −1.00000 | 6.29934 | −1.48793 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(269\) | \( -1 \) |
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 | 538.2.a.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 4842.2.a.j | 4 | ||
| 4.b | odd | 2 | 1 | 4304.2.a.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 538.2.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 4304.2.a.e | 4 | 4.b | odd | 2 | 1 | ||
| 4842.2.a.j | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 3T_{3}^{3} - 3T_{3}^{2} + 10T_{3} - 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(538))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{4} \)
|
| $3$ |
\( T^{4} - 3 T^{3} + \cdots - 4 \)
|
| $5$ |
\( T^{4} - 5 T^{3} + \cdots - 4 \)
|
| $7$ |
\( T^{4} + T^{3} - 6 T^{2} + \cdots + 1 \)
|
| $11$ |
\( T^{4} - 7 T^{3} + \cdots - 13 \)
|
| $13$ |
\( T^{4} - 4 T^{3} + \cdots + 52 \)
|
| $17$ |
\( T^{4} - 4 T^{3} + \cdots - 16 \)
|
| $19$ |
\( T^{4} - 2 T^{3} + \cdots + 16 \)
|
| $23$ |
\( T^{4} - 16 T^{3} + \cdots + 52 \)
|
| $29$ |
\( T^{4} - 51 T^{2} + \cdots + 17 \)
|
| $31$ |
\( T^{4} + T^{3} + \cdots - 67 \)
|
| $37$ |
\( T^{4} + 2 T^{3} + \cdots - 188 \)
|
| $41$ |
\( T^{4} + T^{3} + \cdots + 52 \)
|
| $43$ |
\( T^{4} - 9 T^{3} + \cdots + 1937 \)
|
| $47$ |
\( T^{4} - 13 T^{3} + \cdots - 1172 \)
|
| $53$ |
\( (T^{2} - 14 T + 32)^{2} \)
|
| $59$ |
\( T^{4} - 19 T^{3} + \cdots - 188 \)
|
| $61$ |
\( T^{4} + 20 T^{3} + \cdots + 268 \)
|
| $67$ |
\( T^{4} - 15 T^{3} + \cdots + 101 \)
|
| $71$ |
\( T^{4} - 15 T^{3} + \cdots - 2228 \)
|
| $73$ |
\( T^{4} + 7 T^{3} + \cdots + 6481 \)
|
| $79$ |
\( T^{4} + 17 T^{3} + \cdots + 68 \)
|
| $83$ |
\( T^{4} - 6 T^{3} + \cdots + 1024 \)
|
| $89$ |
\( T^{4} - 20 T^{3} + \cdots - 191 \)
|
| $97$ |
\( T^{4} - 3 T^{3} + \cdots + 27196 \)
|
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