Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1143,4,Mod(1,1143)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1143, 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("1143.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1143 = 3^{2} \cdot 127 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1143.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: | \(67.4391831366\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{13}, \sqrt{73})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 43x^{2} + 225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3\cdot 7 \) |
| 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
\( x^{4} - 43x^{2} + 225 \)
:
| \(\beta_{1}\) | \(=\) |
\( 7\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 33\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 22 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 7 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 22 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 35\beta_{2} + 33\beta_1 ) / 7 \)
|
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 | 0 | −8.00000 | −13.2859 | 0 | 4.90292 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | −8.00000 | −4.74188 | 0 | −25.9029 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | −8.00000 | 4.74188 | 0 | −25.9029 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | −8.00000 | 13.2859 | 0 | 4.90292 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(127\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1143.4.a.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 1143.4.a.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1143.4.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1143.4.a.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1143))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 199T^{2} + 3969 \)
|
| $7$ |
\( (T^{2} + 21 T - 127)^{2} \)
|
| $11$ |
\( T^{4} - 2107 T^{2} + 540225 \)
|
| $13$ |
\( (T^{2} - 47 T + 315)^{2} \)
|
| $17$ |
\( T^{4} - 12060 T^{2} + 33593616 \)
|
| $19$ |
\( (T^{2} + 26 T - 8372)^{2} \)
|
| $23$ |
\( T^{4} - 36000 T^{2} + 9144576 \)
|
| $29$ |
\( T^{4} - 65575 T^{2} + 804913641 \)
|
| $31$ |
\( (T^{2} - 234 T + 12740)^{2} \)
|
| $37$ |
\( (T^{2} - 391 T + 19003)^{2} \)
|
| $41$ |
\( T^{4} - 69580 T^{2} + 763748496 \)
|
| $43$ |
\( (T^{2} - 439 T - 5201)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 15045720921 \)
|
| $53$ |
\( T^{4} + \cdots + 6519755025 \)
|
| $59$ |
\( T^{4} + \cdots + 109534521600 \)
|
| $61$ |
\( (T^{2} + 243 T - 423913)^{2} \)
|
| $67$ |
\( (T^{2} + 507 T - 226369)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 62523502209 \)
|
| $73$ |
\( (T^{2} + 917 T + 105595)^{2} \)
|
| $79$ |
\( (T^{2} + 1646 T + 668788)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 105840910224 \)
|
| $89$ |
\( T^{4} - 33631 T^{2} + 113358609 \)
|
| $97$ |
\( (T^{2} - 144 T - 55552)^{2} \)
|
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