Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [531,2,Mod(1,531)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(531, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("531.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 531 = 3^{2} \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 531.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: | \(4.24005634733\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.138136.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 4x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 59) |
| 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
\( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 4x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} - 3\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} - 2\nu + 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\nu^{4} - \nu^{3} - 12\nu^{2} - \nu + 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + 2\beta_{2} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6\beta_{3} + 7\beta_{2} + 9\beta _1 + 16 \)
|
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 |
|
−2.46927 | 0 | 4.09728 | 0.0566491 | 0 | −4.07193 | −5.17874 | 0 | −0.139882 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.31167 | 0 | −0.279526 | 3.47936 | 0 | 2.38080 | 2.98998 | 0 | −4.56377 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.554026 | 0 | −1.69306 | −3.72868 | 0 | −1.22679 | 2.04605 | 0 | 2.06579 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.67791 | 0 | 0.815372 | 0.571895 | 0 | 2.73575 | −1.98770 | 0 | 0.959587 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.65705 | 0 | 5.05993 | −2.37922 | 0 | 2.18217 | 8.13040 | 0 | −6.32172 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(59\) | \(1\) |
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 | 531.2.a.f | 5 | |
| 3.b | odd | 2 | 1 | 59.2.a.a | ✓ | 5 | |
| 4.b | odd | 2 | 1 | 8496.2.a.bv | 5 | ||
| 12.b | even | 2 | 1 | 944.2.a.n | 5 | ||
| 15.d | odd | 2 | 1 | 1475.2.a.i | 5 | ||
| 15.e | even | 4 | 2 | 1475.2.b.e | 10 | ||
| 21.c | even | 2 | 1 | 2891.2.a.f | 5 | ||
| 24.f | even | 2 | 1 | 3776.2.a.bl | 5 | ||
| 24.h | odd | 2 | 1 | 3776.2.a.bn | 5 | ||
| 33.d | even | 2 | 1 | 7139.2.a.k | 5 | ||
| 39.d | odd | 2 | 1 | 9971.2.a.d | 5 | ||
| 177.d | even | 2 | 1 | 3481.2.a.d | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 59.2.a.a | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 531.2.a.f | 5 | 1.a | even | 1 | 1 | trivial | |
| 944.2.a.n | 5 | 12.b | even | 2 | 1 | ||
| 1475.2.a.i | 5 | 15.d | odd | 2 | 1 | ||
| 1475.2.b.e | 10 | 15.e | even | 4 | 2 | ||
| 2891.2.a.f | 5 | 21.c | even | 2 | 1 | ||
| 3481.2.a.d | 5 | 177.d | even | 2 | 1 | ||
| 3776.2.a.bl | 5 | 24.f | even | 2 | 1 | ||
| 3776.2.a.bn | 5 | 24.h | odd | 2 | 1 | ||
| 7139.2.a.k | 5 | 33.d | even | 2 | 1 | ||
| 8496.2.a.bv | 5 | 4.b | odd | 2 | 1 | ||
| 9971.2.a.d | 5 | 39.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 9T_{2}^{3} - 2T_{2}^{2} + 16T_{2} + 8 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(531))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 9 T^{3} + \cdots + 8 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + 2 T^{4} + \cdots - 1 \)
|
| $7$ |
\( T^{5} - 2 T^{4} + \cdots - 71 \)
|
| $11$ |
\( T^{5} - 2 T^{4} + \cdots + 64 \)
|
| $13$ |
\( T^{5} - 8 T^{4} + \cdots - 224 \)
|
| $17$ |
\( T^{5} - T^{4} + \cdots - 412 \)
|
| $19$ |
\( T^{5} - 6 T^{4} + \cdots - 469 \)
|
| $23$ |
\( T^{5} - 8 T^{4} + \cdots + 32 \)
|
| $29$ |
\( T^{5} + 14 T^{4} + \cdots + 1757 \)
|
| $31$ |
\( T^{5} - 116 T^{3} + \cdots + 256 \)
|
| $37$ |
\( T^{5} - 18 T^{4} + \cdots + 32 \)
|
| $41$ |
\( T^{5} - 10 T^{4} + \cdots - 217 \)
|
| $43$ |
\( T^{5} + 4 T^{4} + \cdots - 128 \)
|
| $47$ |
\( T^{5} - 20 T^{4} + \cdots + 256 \)
|
| $53$ |
\( T^{5} - 10 T^{4} + \cdots - 73 \)
|
| $59$ |
\( (T + 1)^{5} \)
|
| $61$ |
\( T^{5} - 22 T^{4} + \cdots + 11072 \)
|
| $67$ |
\( T^{5} - 188 T^{3} + \cdots - 8896 \)
|
| $71$ |
\( T^{5} + 3 T^{4} + \cdots - 3424 \)
|
| $73$ |
\( T^{5} + 8 T^{4} + \cdots + 9952 \)
|
| $79$ |
\( T^{5} - 10 T^{4} + \cdots + 3923 \)
|
| $83$ |
\( T^{5} + 6 T^{4} + \cdots - 29152 \)
|
| $89$ |
\( T^{5} + 10 T^{4} + \cdots + 1984 \)
|
| $97$ |
\( T^{5} + 22 T^{4} + \cdots + 2656 \)
|
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