Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [867,4,Mod(1,867)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(867, 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("867.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 867 = 3 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 867.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: | \(51.1546559750\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 23x^{2} + 31x + 28 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} - x^{3} - 23x^{2} + 31x + 28 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 19\nu + 10 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 19\beta _1 - 10 \)
|
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 |
|
−4.83514 | −3.00000 | 15.3786 | −13.6282 | 14.5054 | −30.2985 | −35.6765 | 9.00000 | 65.8942 | ||||||||||||||||||||||||||||||
| 1.2 | −0.625648 | −3.00000 | −7.60856 | −2.83829 | 1.87695 | −11.0896 | 9.76547 | 9.00000 | 1.77577 | |||||||||||||||||||||||||||||||
| 1.3 | 2.14433 | −3.00000 | −3.40187 | 16.9873 | −6.43298 | 14.2760 | −24.4493 | 9.00000 | 36.4264 | |||||||||||||||||||||||||||||||
| 1.4 | 4.31646 | −3.00000 | 10.6319 | −9.52085 | −12.9494 | −7.88792 | 11.3603 | 9.00000 | −41.0964 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(17\) | \( +1 \) |
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 | 867.4.a.l | ✓ | 4 |
| 17.b | even | 2 | 1 | 867.4.a.m | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 867.4.a.l | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 867.4.a.m | yes | 4 | 17.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(867))\):
|
\( T_{2}^{4} - T_{2}^{3} - 23T_{2}^{2} + 31T_{2} + 28 \)
|
|
\( T_{5}^{4} + 9T_{5}^{3} - 246T_{5}^{2} - 2952T_{5} - 6256 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} + \cdots + 28 \)
|
| $3$ |
\( (T + 3)^{4} \)
|
| $5$ |
\( T^{4} + 9 T^{3} + \cdots - 6256 \)
|
| $7$ |
\( T^{4} + 35 T^{3} + \cdots - 37836 \)
|
| $11$ |
\( T^{4} - 47 T^{3} + \cdots + 2366928 \)
|
| $13$ |
\( T^{4} - 85 T^{3} + \cdots - 8589442 \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} + 29 T^{3} + \cdots - 16333452 \)
|
| $23$ |
\( T^{4} + 60 T^{3} + \cdots + 247294304 \)
|
| $29$ |
\( T^{4} - 39 T^{3} + \cdots + 800114112 \)
|
| $31$ |
\( T^{4} + 342 T^{3} + \cdots - 623524321 \)
|
| $37$ |
\( T^{4} + 549 T^{3} + \cdots + 55504448 \)
|
| $41$ |
\( T^{4} - 102 T^{3} + \cdots - 95161824 \)
|
| $43$ |
\( T^{4} + \cdots - 5524621524 \)
|
| $47$ |
\( T^{4} + \cdots - 5909013504 \)
|
| $53$ |
\( T^{4} + \cdots + 3943759104 \)
|
| $59$ |
\( T^{4} - 549 T^{3} + \cdots - 52551504 \)
|
| $61$ |
\( T^{4} + \cdots + 1908804914 \)
|
| $67$ |
\( T^{4} + \cdots - 5530703342 \)
|
| $71$ |
\( T^{4} + \cdots + 27022765888 \)
|
| $73$ |
\( T^{4} + \cdots - 7734146328 \)
|
| $79$ |
\( T^{4} + \cdots + 36705426048 \)
|
| $83$ |
\( T^{4} + \cdots - 915482243264 \)
|
| $89$ |
\( T^{4} + \cdots - 235655540704 \)
|
| $97$ |
\( T^{4} + \cdots - 370622096929 \)
|
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