Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,5,Mod(26,81)]
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([1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.26");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.d (of order \(6\), 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: | \(8.37296700979\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{3} \) |
| Twist minimal: | no (minimal twist has level 27) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{3} + 12\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{3} + 6\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 9 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -4\beta_{3} + 2\beta_{2} ) / 9 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 26.1 |
|
−6.36396 | − | 3.67423i | 0 | 19.0000 | + | 32.9090i | 12.7279 | − | 7.34847i | 0 | −8.50000 | + | 14.7224i | − | 161.666i | 0 | −108.000 | |||||||||||||||||||||
| 26.2 | 6.36396 | + | 3.67423i | 0 | 19.0000 | + | 32.9090i | −12.7279 | + | 7.34847i | 0 | −8.50000 | + | 14.7224i | 161.666i | 0 | −108.000 | |||||||||||||||||||||||
| 53.1 | −6.36396 | + | 3.67423i | 0 | 19.0000 | − | 32.9090i | 12.7279 | + | 7.34847i | 0 | −8.50000 | − | 14.7224i | 161.666i | 0 | −108.000 | |||||||||||||||||||||||
| 53.2 | 6.36396 | − | 3.67423i | 0 | 19.0000 | − | 32.9090i | −12.7279 | − | 7.34847i | 0 | −8.50000 | − | 14.7224i | − | 161.666i | 0 | −108.000 | ||||||||||||||||||||||
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 | 81.5.d.d | 4 | |
| 3.b | odd | 2 | 1 | inner | 81.5.d.d | 4 | |
| 9.c | even | 3 | 1 | 27.5.b.b | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 81.5.d.d | 4 | |
| 9.d | odd | 6 | 1 | 27.5.b.b | ✓ | 2 | |
| 9.d | odd | 6 | 1 | inner | 81.5.d.d | 4 | |
| 36.f | odd | 6 | 1 | 432.5.e.c | 2 | ||
| 36.h | even | 6 | 1 | 432.5.e.c | 2 | ||
| 45.h | odd | 6 | 1 | 675.5.c.d | 2 | ||
| 45.j | even | 6 | 1 | 675.5.c.d | 2 | ||
| 45.k | odd | 12 | 2 | 675.5.d.g | 4 | ||
| 45.l | even | 12 | 2 | 675.5.d.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.5.b.b | ✓ | 2 | 9.c | even | 3 | 1 | |
| 27.5.b.b | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 81.5.d.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 81.5.d.d | 4 | 3.b | odd | 2 | 1 | inner | |
| 81.5.d.d | 4 | 9.c | even | 3 | 1 | inner | |
| 81.5.d.d | 4 | 9.d | odd | 6 | 1 | inner | |
| 432.5.e.c | 2 | 36.f | odd | 6 | 1 | ||
| 432.5.e.c | 2 | 36.h | even | 6 | 1 | ||
| 675.5.c.d | 2 | 45.h | odd | 6 | 1 | ||
| 675.5.c.d | 2 | 45.j | even | 6 | 1 | ||
| 675.5.d.g | 4 | 45.k | odd | 12 | 2 | ||
| 675.5.d.g | 4 | 45.l | even | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 54T_{2}^{2} + 2916 \)
acting on \(S_{5}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 54T^{2} + 2916 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 216 T^{2} + 46656 \)
|
| $7$ |
\( (T^{2} + 17 T + 289)^{2} \)
|
| $11$ |
\( T^{4} - 26136 T^{2} + 683090496 \)
|
| $13$ |
\( (T^{2} + 95 T + 9025)^{2} \)
|
| $17$ |
\( (T^{2} + 95256)^{2} \)
|
| $19$ |
\( (T - 209)^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 565347594816 \)
|
| $29$ |
\( T^{4} + \cdots + 10929447936 \)
|
| $31$ |
\( (T^{2} + 950 T + 902500)^{2} \)
|
| $37$ |
\( (T + 1177)^{4} \)
|
| $41$ |
\( T^{4} + \cdots + 21199173313536 \)
|
| $43$ |
\( (T^{2} + 1430 T + 2044900)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 6115649864256 \)
|
| $53$ |
\( (T^{2} + 8468064)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 20624313960000 \)
|
| $61$ |
\( (T^{2} - 1441 T + 2076481)^{2} \)
|
| $67$ |
\( (T^{2} + 3497 T + 12229009)^{2} \)
|
| $71$ |
\( (T^{2} + 3763584)^{2} \)
|
| $73$ |
\( (T + 9025)^{4} \)
|
| $79$ |
\( (T^{2} + 5273 T + 27804529)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 14\!\cdots\!56 \)
|
| $89$ |
\( (T^{2} + 124433496)^{2} \)
|
| $97$ |
\( (T^{2} - 2809 T + 7890481)^{2} \)
|
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