Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,6,Mod(28,81)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(81, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.28");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.c (of order \(3\), degree \(2\), not minimal) |
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: | no |
| Analytic conductor: | \(12.9910894049\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{17})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 5x^{2} + 4x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 27) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} + 5x^{2} + 4x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 5\nu^{2} + 65\nu - 16 ) / 20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{3} + 22 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 13\beta _1 - 14 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{3} - 22 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 28.1 |
|
−0.842329 | − | 1.45896i | 0 | 14.5810 | − | 25.2550i | 48.9233 | − | 84.7376i | 0 | −53.6619 | − | 92.9452i | −103.037 | 0 | −164.838 | ||||||||||||||||||||||
| 28.2 | 5.34233 | + | 9.25319i | 0 | −41.0810 | + | 71.1543i | −12.9233 | + | 22.3838i | 0 | 57.6619 | + | 99.8734i | −535.963 | 0 | −276.162 | |||||||||||||||||||||||
| 55.1 | −0.842329 | + | 1.45896i | 0 | 14.5810 | + | 25.2550i | 48.9233 | + | 84.7376i | 0 | −53.6619 | + | 92.9452i | −103.037 | 0 | −164.838 | |||||||||||||||||||||||
| 55.2 | 5.34233 | − | 9.25319i | 0 | −41.0810 | − | 71.1543i | −12.9233 | − | 22.3838i | 0 | 57.6619 | − | 99.8734i | −535.963 | 0 | −276.162 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 81.6.c.h | 4 | |
| 3.b | odd | 2 | 1 | 81.6.c.d | 4 | ||
| 9.c | even | 3 | 1 | 27.6.a.b | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 81.6.c.h | 4 | |
| 9.d | odd | 6 | 1 | 27.6.a.d | yes | 2 | |
| 9.d | odd | 6 | 1 | 81.6.c.d | 4 | ||
| 36.f | odd | 6 | 1 | 432.6.a.k | 2 | ||
| 36.h | even | 6 | 1 | 432.6.a.v | 2 | ||
| 45.h | odd | 6 | 1 | 675.6.a.f | 2 | ||
| 45.j | even | 6 | 1 | 675.6.a.n | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.6.a.b | ✓ | 2 | 9.c | even | 3 | 1 | |
| 27.6.a.d | yes | 2 | 9.d | odd | 6 | 1 | |
| 81.6.c.d | 4 | 3.b | odd | 2 | 1 | ||
| 81.6.c.d | 4 | 9.d | odd | 6 | 1 | ||
| 81.6.c.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 81.6.c.h | 4 | 9.c | even | 3 | 1 | inner | |
| 432.6.a.k | 2 | 36.f | odd | 6 | 1 | ||
| 432.6.a.v | 2 | 36.h | even | 6 | 1 | ||
| 675.6.a.f | 2 | 45.h | odd | 6 | 1 | ||
| 675.6.a.n | 2 | 45.j | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 9T_{2}^{3} + 99T_{2}^{2} + 162T_{2} + 324 \)
acting on \(S_{6}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 9 T^{3} + \cdots + 324 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 72 T^{3} + \cdots + 6395841 \)
|
| $7$ |
\( T^{4} - 8 T^{3} + \cdots + 153190129 \)
|
| $11$ |
\( T^{4} - 522 T^{3} + \cdots + 838276209 \)
|
| $13$ |
\( T^{4} + \cdots + 5532979456 \)
|
| $17$ |
\( (T^{2} + 216 T - 1067904)^{2} \)
|
| $19$ |
\( (T^{2} + 2840 T + 1570252)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 3705887376 \)
|
| $29$ |
\( T^{4} + \cdots + 13\!\cdots\!44 \)
|
| $31$ |
\( T^{4} + \cdots + 102809237181121 \)
|
| $37$ |
\( (T^{2} - 9004 T - 24346796)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 24\!\cdots\!24 \)
|
| $43$ |
\( T^{4} + \cdots + 83\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots + 15\!\cdots\!84 \)
|
| $53$ |
\( (T^{2} + 4536 T - 176302089)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 11\!\cdots\!44 \)
|
| $61$ |
\( T^{4} + \cdots + 17\!\cdots\!64 \)
|
| $67$ |
\( T^{4} + \cdots + 43\!\cdots\!84 \)
|
| $71$ |
\( (T^{2} - 43848 T - 374378112)^{2} \)
|
| $73$ |
\( (T^{2} - 47218 T + 300403633)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 37\!\cdots\!36 \)
|
| $83$ |
\( T^{4} + \cdots + 94\!\cdots\!81 \)
|
| $89$ |
\( (T^{2} - 35856 T - 3686257188)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 40\!\cdots\!21 \)
|
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