Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1134,4,Mod(1,1134)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1134, 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("1134.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1134 = 2 \cdot 3^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1134.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: | \(66.9081659465\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.1065321.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 20x^{2} - 9x + 48 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 126) |
| 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
\( x^{4} - x^{3} - 20x^{2} - 9x + 48 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 5\nu^{2} - 9\nu + 31 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{3} - 7\nu^{2} - 21\nu + 33 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 2\beta_{2} + \beta _1 + 30 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{3} - 7\beta_{2} + 14\beta _1 + 66 ) / 3 \)
|
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 |
|
2.00000 | 0 | 4.00000 | −19.5010 | 0 | −7.00000 | 8.00000 | 0 | −39.0021 | ||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | 0 | 4.00000 | 1.44283 | 0 | −7.00000 | 8.00000 | 0 | 2.88565 | |||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 0 | 4.00000 | 5.96620 | 0 | −7.00000 | 8.00000 | 0 | 11.9324 | |||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 0 | 4.00000 | 11.0920 | 0 | −7.00000 | 8.00000 | 0 | 22.1840 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(7\) | \(1\) |
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 | 1134.4.a.o | 4 | |
| 3.b | odd | 2 | 1 | 1134.4.a.l | 4 | ||
| 9.c | even | 3 | 2 | 126.4.f.b | ✓ | 8 | |
| 9.d | odd | 6 | 2 | 378.4.f.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 126.4.f.b | ✓ | 8 | 9.c | even | 3 | 2 | |
| 378.4.f.b | 8 | 9.d | odd | 6 | 2 | ||
| 1134.4.a.l | 4 | 3.b | odd | 2 | 1 | ||
| 1134.4.a.o | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + T_{5}^{3} - 270T_{5}^{2} + 1675T_{5} - 1862 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1134))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + T^{3} + \cdots - 1862 \)
|
| $7$ |
\( (T + 7)^{4} \)
|
| $11$ |
\( T^{4} + 5 T^{3} + \cdots - 312605 \)
|
| $13$ |
\( T^{4} + 21 T^{3} + \cdots + 79974 \)
|
| $17$ |
\( T^{4} + 23 T^{3} + \cdots + 20106721 \)
|
| $19$ |
\( T^{4} + 94 T^{3} + \cdots + 36369301 \)
|
| $23$ |
\( T^{4} + 374 T^{3} + \cdots + 40203820 \)
|
| $29$ |
\( T^{4} + 271 T^{3} + \cdots - 62499788 \)
|
| $31$ |
\( T^{4} + \cdots - 1190859570 \)
|
| $37$ |
\( T^{4} + \cdots - 2370848354 \)
|
| $41$ |
\( T^{4} + \cdots + 1684483803 \)
|
| $43$ |
\( T^{4} + \cdots + 5935839493 \)
|
| $47$ |
\( T^{4} + \cdots + 2242593864 \)
|
| $53$ |
\( T^{4} + \cdots + 7638055812 \)
|
| $59$ |
\( T^{4} + \cdots + 3831600169 \)
|
| $61$ |
\( T^{4} + \cdots - 32276693916 \)
|
| $67$ |
\( T^{4} + \cdots - 2415036551 \)
|
| $71$ |
\( T^{4} + \cdots - 17389723904 \)
|
| $73$ |
\( T^{4} + \cdots - 41833011071 \)
|
| $79$ |
\( T^{4} + \cdots - 60211065906 \)
|
| $83$ |
\( T^{4} + \cdots - 168514277576 \)
|
| $89$ |
\( T^{4} + \cdots - 127304987850 \)
|
| $97$ |
\( T^{4} + \cdots + 2071130706613 \)
|
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