Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.28");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(4.77915471046\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| 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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{12}^{3} + \zeta_{12} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 3 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} ) / 3 \)
|
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 |
|
−0.866025 | − | 1.50000i | 0 | 2.50000 | − | 4.33013i | 6.06218 | − | 10.5000i | 0 | 11.0000 | + | 19.0526i | −22.5167 | 0 | −21.0000 | ||||||||||||||||||||||
| 28.2 | 0.866025 | + | 1.50000i | 0 | 2.50000 | − | 4.33013i | −6.06218 | + | 10.5000i | 0 | 11.0000 | + | 19.0526i | 22.5167 | 0 | −21.0000 | |||||||||||||||||||||||
| 55.1 | −0.866025 | + | 1.50000i | 0 | 2.50000 | + | 4.33013i | 6.06218 | + | 10.5000i | 0 | 11.0000 | − | 19.0526i | −22.5167 | 0 | −21.0000 | |||||||||||||||||||||||
| 55.2 | 0.866025 | − | 1.50000i | 0 | 2.50000 | + | 4.33013i | −6.06218 | − | 10.5000i | 0 | 11.0000 | − | 19.0526i | 22.5167 | 0 | −21.0000 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 9.c | even | 3 | 1 | inner |
| 9.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 81.4.c.f | 4 | |
| 3.b | odd | 2 | 1 | inner | 81.4.c.f | 4 | |
| 9.c | even | 3 | 1 | 81.4.a.c | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 81.4.c.f | 4 | |
| 9.d | odd | 6 | 1 | 81.4.a.c | ✓ | 2 | |
| 9.d | odd | 6 | 1 | inner | 81.4.c.f | 4 | |
| 36.f | odd | 6 | 1 | 1296.4.a.o | 2 | ||
| 36.h | even | 6 | 1 | 1296.4.a.o | 2 | ||
| 45.h | odd | 6 | 1 | 2025.4.a.k | 2 | ||
| 45.j | even | 6 | 1 | 2025.4.a.k | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 81.4.a.c | ✓ | 2 | 9.c | even | 3 | 1 | |
| 81.4.a.c | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 81.4.c.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 81.4.c.f | 4 | 3.b | odd | 2 | 1 | inner | |
| 81.4.c.f | 4 | 9.c | even | 3 | 1 | inner | |
| 81.4.c.f | 4 | 9.d | odd | 6 | 1 | inner | |
| 1296.4.a.o | 2 | 36.f | odd | 6 | 1 | ||
| 1296.4.a.o | 2 | 36.h | even | 6 | 1 | ||
| 2025.4.a.k | 2 | 45.h | odd | 6 | 1 | ||
| 2025.4.a.k | 2 | 45.j | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 3T_{2}^{2} + 9 \)
acting on \(S_{4}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 3T^{2} + 9 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 147 T^{2} + 21609 \)
|
| $7$ |
\( (T^{2} - 22 T + 484)^{2} \)
|
| $11$ |
\( T^{4} + 3468 T^{2} + 12027024 \)
|
| $13$ |
\( (T^{2} - 49 T + 2401)^{2} \)
|
| $17$ |
\( (T^{2} - 1323)^{2} \)
|
| $19$ |
\( (T + 70)^{4} \)
|
| $23$ |
\( T^{4} + 588 T^{2} + 345744 \)
|
| $29$ |
\( T^{4} + \cdots + 6047084169 \)
|
| $31$ |
\( (T^{2} - 112 T + 12544)^{2} \)
|
| $37$ |
\( (T - 281)^{4} \)
|
| $41$ |
\( T^{4} + 2352 T^{2} + 5531904 \)
|
| $43$ |
\( (T^{2} + 50 T + 2500)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 3457440000 \)
|
| $53$ |
\( (T^{2} - 139968)^{2} \)
|
| $59$ |
\( T^{4} + 9408 T^{2} + 88510464 \)
|
| $61$ |
\( (T^{2} - 679 T + 461041)^{2} \)
|
| $67$ |
\( (T^{2} - 274 T + 75076)^{2} \)
|
| $71$ |
\( (T^{2} - 199692)^{2} \)
|
| $73$ |
\( (T + 511)^{4} \)
|
| $79$ |
\( (T^{2} - 526 T + 276676)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 22658678784 \)
|
| $89$ |
\( (T^{2} - 1323)^{2} \)
|
| $97$ |
\( (T^{2} + 1778 T + 3161284)^{2} \)
|
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