Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2166,4,Mod(1,2166)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2166, 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("2166.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2166 = 2 \cdot 3 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2166.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: | \(127.798137072\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.14457.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 32x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 114) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 32x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{2} - 6\nu - 85 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 88 ) / 4 \)
|
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 | −3.00000 | 4.00000 | −8.44242 | −6.00000 | −9.44242 | 8.00000 | 9.00000 | −16.8848 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | −3.00000 | 4.00000 | 4.18810 | −6.00000 | 3.18810 | 8.00000 | 9.00000 | 8.37621 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −3.00000 | 4.00000 | 14.2543 | −6.00000 | 13.2543 | 8.00000 | 9.00000 | 28.5086 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(19\) | \(1\) |
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 | 2166.4.a.u | 3 | |
| 19.b | odd | 2 | 1 | 2166.4.a.t | 3 | ||
| 19.c | even | 3 | 2 | 114.4.e.d | ✓ | 6 | |
| 57.h | odd | 6 | 2 | 342.4.g.h | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.4.e.d | ✓ | 6 | 19.c | even | 3 | 2 | |
| 342.4.g.h | 6 | 57.h | odd | 6 | 2 | ||
| 2166.4.a.t | 3 | 19.b | odd | 2 | 1 | ||
| 2166.4.a.u | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2166))\):
|
\( T_{5}^{3} - 10T_{5}^{2} - 96T_{5} + 504 \)
|
|
\( T_{13}^{3} + 9T_{13}^{2} - 4629T_{13} - 37685 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
|
| $3$ |
\( (T + 3)^{3} \)
|
| $5$ |
\( T^{3} - 10 T^{2} + \cdots + 504 \)
|
| $7$ |
\( T^{3} - 7 T^{2} + \cdots + 399 \)
|
| $11$ |
\( T^{3} + 44 T^{2} + \cdots - 217224 \)
|
| $13$ |
\( T^{3} + 9 T^{2} + \cdots - 37685 \)
|
| $17$ |
\( T^{3} + 84 T^{2} + \cdots - 820800 \)
|
| $19$ |
\( T^{3} \)
|
| $23$ |
\( T^{3} + 2 T^{2} + \cdots + 106776 \)
|
| $29$ |
\( T^{3} - 92 T^{2} + \cdots - 1302336 \)
|
| $31$ |
\( T^{3} + 109 T^{2} + \cdots - 9721957 \)
|
| $37$ |
\( T^{3} - 245 T^{2} + \cdots - 1317799 \)
|
| $41$ |
\( T^{3} + 688 T^{2} + \cdots - 10279872 \)
|
| $43$ |
\( T^{3} + 103 T^{2} + \cdots + 6249329 \)
|
| $47$ |
\( T^{3} - 322 T^{2} + \cdots - 11695896 \)
|
| $53$ |
\( T^{3} + 1322 T^{2} + \cdots + 66978360 \)
|
| $59$ |
\( T^{3} - 252 T^{2} + \cdots + 367416 \)
|
| $61$ |
\( T^{3} + 435 T^{2} + \cdots - 73871 \)
|
| $67$ |
\( T^{3} + 719 T^{2} + \cdots + 9614433 \)
|
| $71$ |
\( T^{3} + 62 T^{2} + \cdots - 111044664 \)
|
| $73$ |
\( T^{3} + 581 T^{2} + \cdots + 70657335 \)
|
| $79$ |
\( T^{3} + 489 T^{2} + \cdots - 51818777 \)
|
| $83$ |
\( T^{3} - 2496 T^{2} + \cdots - 372278592 \)
|
| $89$ |
\( T^{3} - 1584 T^{2} + \cdots + 395518680 \)
|
| $97$ |
\( T^{3} - 974 T^{2} + \cdots - 24200200 \)
|
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