Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.28");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(25.3031870642\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{25}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 3) |
| 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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-\zeta_{6}\) |
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 |
|
−3.00000 | − | 5.19615i | 0 | 46.0000 | − | 79.6743i | −195.000 | + | 337.750i | 0 | 32.0000 | + | 55.4256i | −1320.00 | 0 | 2340.00 | ||||||||||||||||
| 55.1 | −3.00000 | + | 5.19615i | 0 | 46.0000 | + | 79.6743i | −195.000 | − | 337.750i | 0 | 32.0000 | − | 55.4256i | −1320.00 | 0 | 2340.00 | |||||||||||||||||
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 | 81.8.c.a | 2 | |
| 3.b | odd | 2 | 1 | 81.8.c.c | 2 | ||
| 9.c | even | 3 | 1 | 3.8.a.a | ✓ | 1 | |
| 9.c | even | 3 | 1 | inner | 81.8.c.a | 2 | |
| 9.d | odd | 6 | 1 | 9.8.a.a | 1 | ||
| 9.d | odd | 6 | 1 | 81.8.c.c | 2 | ||
| 36.f | odd | 6 | 1 | 48.8.a.g | 1 | ||
| 36.h | even | 6 | 1 | 144.8.a.b | 1 | ||
| 45.h | odd | 6 | 1 | 225.8.a.i | 1 | ||
| 45.j | even | 6 | 1 | 75.8.a.a | 1 | ||
| 45.k | odd | 12 | 2 | 75.8.b.c | 2 | ||
| 45.l | even | 12 | 2 | 225.8.b.f | 2 | ||
| 63.g | even | 3 | 1 | 147.8.e.b | 2 | ||
| 63.h | even | 3 | 1 | 147.8.e.b | 2 | ||
| 63.k | odd | 6 | 1 | 147.8.e.a | 2 | ||
| 63.l | odd | 6 | 1 | 147.8.a.b | 1 | ||
| 63.o | even | 6 | 1 | 441.8.a.a | 1 | ||
| 63.t | odd | 6 | 1 | 147.8.e.a | 2 | ||
| 72.j | odd | 6 | 1 | 576.8.a.w | 1 | ||
| 72.l | even | 6 | 1 | 576.8.a.x | 1 | ||
| 72.n | even | 6 | 1 | 192.8.a.i | 1 | ||
| 72.p | odd | 6 | 1 | 192.8.a.a | 1 | ||
| 99.h | odd | 6 | 1 | 363.8.a.b | 1 | ||
| 117.t | even | 6 | 1 | 507.8.a.a | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.8.a.a | ✓ | 1 | 9.c | even | 3 | 1 | |
| 9.8.a.a | 1 | 9.d | odd | 6 | 1 | ||
| 48.8.a.g | 1 | 36.f | odd | 6 | 1 | ||
| 75.8.a.a | 1 | 45.j | even | 6 | 1 | ||
| 75.8.b.c | 2 | 45.k | odd | 12 | 2 | ||
| 81.8.c.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 81.8.c.a | 2 | 9.c | even | 3 | 1 | inner | |
| 81.8.c.c | 2 | 3.b | odd | 2 | 1 | ||
| 81.8.c.c | 2 | 9.d | odd | 6 | 1 | ||
| 144.8.a.b | 1 | 36.h | even | 6 | 1 | ||
| 147.8.a.b | 1 | 63.l | odd | 6 | 1 | ||
| 147.8.e.a | 2 | 63.k | odd | 6 | 1 | ||
| 147.8.e.a | 2 | 63.t | odd | 6 | 1 | ||
| 147.8.e.b | 2 | 63.g | even | 3 | 1 | ||
| 147.8.e.b | 2 | 63.h | even | 3 | 1 | ||
| 192.8.a.a | 1 | 72.p | odd | 6 | 1 | ||
| 192.8.a.i | 1 | 72.n | even | 6 | 1 | ||
| 225.8.a.i | 1 | 45.h | odd | 6 | 1 | ||
| 225.8.b.f | 2 | 45.l | even | 12 | 2 | ||
| 363.8.a.b | 1 | 99.h | odd | 6 | 1 | ||
| 441.8.a.a | 1 | 63.o | even | 6 | 1 | ||
| 507.8.a.a | 1 | 117.t | even | 6 | 1 | ||
| 576.8.a.w | 1 | 72.j | odd | 6 | 1 | ||
| 576.8.a.x | 1 | 72.l | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 6T_{2} + 36 \)
acting on \(S_{8}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 6T + 36 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 390T + 152100 \)
|
| $7$ |
\( T^{2} - 64T + 4096 \)
|
| $11$ |
\( T^{2} - 948T + 898704 \)
|
| $13$ |
\( T^{2} - 5098 T + 25989604 \)
|
| $17$ |
\( (T - 28386)^{2} \)
|
| $19$ |
\( (T + 8620)^{2} \)
|
| $23$ |
\( T^{2} - 15288 T + 233722944 \)
|
| $29$ |
\( T^{2} + \cdots + 1332980100 \)
|
| $31$ |
\( T^{2} + \cdots + 76622668864 \)
|
| $37$ |
\( (T - 268526)^{2} \)
|
| $41$ |
\( T^{2} + \cdots + 396544759524 \)
|
| $43$ |
\( T^{2} + \cdots + 470283235984 \)
|
| $47$ |
\( T^{2} + \cdots + 340234223616 \)
|
| $53$ |
\( (T + 428058)^{2} \)
|
| $59$ |
\( T^{2} + \cdots + 1706628704400 \)
|
| $61$ |
\( T^{2} + \cdots + 90397638244 \)
|
| $67$ |
\( T^{2} + \cdots + 257296475536 \)
|
| $71$ |
\( (T - 5560632)^{2} \)
|
| $73$ |
\( (T - 1369082)^{2} \)
|
| $79$ |
\( T^{2} + \cdots + 47799524238400 \)
|
| $83$ |
\( T^{2} + \cdots + 19155923055504 \)
|
| $89$ |
\( (T + 8528310)^{2} \)
|
| $97$ |
\( T^{2} + \cdots + 77912645390596 \)
|
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