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: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.11196169353.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 87x^{4} + 179x^{3} + 2574x^{2} - 2664x - 25992 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 87x^{4} + 179x^{3} + 2574x^{2} - 2664x - 25992 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 59\nu^{2} + 60\nu + 876 ) / 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 65\nu^{2} + 66\nu + 1056 ) / 12 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 65\nu^{3} + 66\nu^{2} + 1056\nu ) / 12 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 62\nu^{3} + 63\nu^{2} + 966\nu ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{3} + 2\beta_{2} + \beta _1 + 30 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} - 4\beta_{4} - 2\beta_{3} + 2\beta_{2} + 31\beta _1 + 30 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{5} - 8\beta_{4} - 122\beta_{3} + 134\beta_{2} + 61\beta _1 + 954 \)
|
| \(\nu^{5}\) | \(=\) |
\( 138\beta_{5} - 264\beta_{4} - 242\beta_{3} + 266\beta_{2} + 1015\beta _1 + 1878 \)
|
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 | −11.4115 | −6.00000 | −8.90590 | −8.00000 | 9.00000 | 22.8230 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | 3.00000 | 4.00000 | −0.304840 | −6.00000 | 19.7612 | −8.00000 | 9.00000 | 0.609681 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 3.00000 | 4.00000 | 3.90567 | −6.00000 | 10.7383 | −8.00000 | 9.00000 | −7.81134 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 3.00000 | 4.00000 | 6.58796 | −6.00000 | −12.5977 | −8.00000 | 9.00000 | −13.1759 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 3.00000 | 4.00000 | 10.3852 | −6.00000 | −35.0342 | −8.00000 | 9.00000 | −20.7704 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | 3.00000 | 4.00000 | 17.8375 | −6.00000 | −15.9616 | −8.00000 | 9.00000 | −35.6750 | |||||||||||||||||||||||||||||||||||||
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.bd | 6 | |
| 19.b | odd | 2 | 1 | 2166.4.a.bf | 6 | ||
| 19.f | odd | 18 | 2 | 114.4.i.a | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.4.i.a | ✓ | 12 | 19.f | odd | 18 | 2 | |
| 2166.4.a.bd | 6 | 1.a | even | 1 | 1 | trivial | |
| 2166.4.a.bf | 6 | 19.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2166))\):
|
\( T_{5}^{6} - 27T_{5}^{5} + 57T_{5}^{4} + 3137T_{5}^{3} - 24753T_{5}^{2} + 46557T_{5} + 16581 \)
|
|
\( T_{13}^{6} + 21T_{13}^{5} - 9426T_{13}^{4} - 67637T_{13}^{3} + 19362624T_{13}^{2} - 400534872T_{13} + 2306444401 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{6} \)
|
| $3$ |
\( (T - 3)^{6} \)
|
| $5$ |
\( T^{6} - 27 T^{5} + \cdots + 16581 \)
|
| $7$ |
\( T^{6} + 42 T^{5} + \cdots + 13313319 \)
|
| $11$ |
\( T^{6} + 57 T^{5} + \cdots - 2432541 \)
|
| $13$ |
\( T^{6} + \cdots + 2306444401 \)
|
| $17$ |
\( T^{6} + \cdots - 9820042461 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} + \cdots + 622767323499 \)
|
| $29$ |
\( T^{6} + \cdots - 289887396429 \)
|
| $31$ |
\( T^{6} + \cdots + 2345132975453 \)
|
| $37$ |
\( T^{6} + \cdots + 57885526637 \)
|
| $41$ |
\( T^{6} + \cdots + 765019289763 \)
|
| $43$ |
\( T^{6} + \cdots - 545155113675673 \)
|
| $47$ |
\( T^{6} + \cdots + 10007606341767 \)
|
| $53$ |
\( T^{6} + \cdots + 13593723912537 \)
|
| $59$ |
\( T^{6} + \cdots - 32\!\cdots\!93 \)
|
| $61$ |
\( T^{6} + \cdots - 468771871433759 \)
|
| $67$ |
\( T^{6} + \cdots + 10\!\cdots\!92 \)
|
| $71$ |
\( T^{6} + \cdots + 12\!\cdots\!71 \)
|
| $73$ |
\( T^{6} + \cdots - 259014155862003 \)
|
| $79$ |
\( T^{6} + \cdots - 28\!\cdots\!67 \)
|
| $83$ |
\( T^{6} + \cdots + 77\!\cdots\!67 \)
|
| $89$ |
\( T^{6} + \cdots + 56728115322777 \)
|
| $97$ |
\( T^{6} + \cdots + 449305474162913 \)
|
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