Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,11,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, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.26");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| 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: | \(51.4639374666\) |
| 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: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 26.1 |
|
45.0000 | + | 25.9808i | 0 | 838.000 | + | 1451.46i | 720.000 | − | 415.692i | 0 | 8046.50 | − | 13936.9i | 33878.9i | 0 | 43200.0 | ||||||||||||||||
| 53.1 | 45.0000 | − | 25.9808i | 0 | 838.000 | − | 1451.46i | 720.000 | + | 415.692i | 0 | 8046.50 | + | 13936.9i | − | 33878.9i | 0 | 43200.0 | ||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 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 | 81.11.d.c | 2 | |
| 3.b | odd | 2 | 1 | 81.11.d.a | 2 | ||
| 9.c | even | 3 | 1 | 27.11.b.b | ✓ | 2 | |
| 9.c | even | 3 | 1 | 81.11.d.a | 2 | ||
| 9.d | odd | 6 | 1 | 27.11.b.b | ✓ | 2 | |
| 9.d | odd | 6 | 1 | inner | 81.11.d.c | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.11.b.b | ✓ | 2 | 9.c | even | 3 | 1 | |
| 27.11.b.b | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 81.11.d.a | 2 | 3.b | odd | 2 | 1 | ||
| 81.11.d.a | 2 | 9.c | even | 3 | 1 | ||
| 81.11.d.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 81.11.d.c | 2 | 9.d | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 90T_{2} + 2700 \)
acting on \(S_{11}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 90T + 2700 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 1440 T + 691200 \)
|
| $7$ |
\( T^{2} - 16093 T + 258984649 \)
|
| $11$ |
\( T^{2} + \cdots + 86129740800 \)
|
| $13$ |
\( T^{2} + \cdots + 370441432321 \)
|
| $17$ |
\( T^{2} + 304819200 \)
|
| $19$ |
\( (T + 2104549)^{2} \)
|
| $23$ |
\( T^{2} + \cdots + 21007758412800 \)
|
| $29$ |
\( T^{2} + \cdots + 990228672000000 \)
|
| $31$ |
\( T^{2} + \cdots + 676734751543876 \)
|
| $37$ |
\( (T - 51946607)^{2} \)
|
| $41$ |
\( T^{2} + \cdots + 32\!\cdots\!00 \)
|
| $43$ |
\( T^{2} + \cdots + 51\!\cdots\!24 \)
|
| $47$ |
\( T^{2} + \cdots + 22\!\cdots\!00 \)
|
| $53$ |
\( T^{2} + 11\!\cdots\!00 \)
|
| $59$ |
\( T^{2} + \cdots + 36\!\cdots\!00 \)
|
| $61$ |
\( T^{2} + \cdots + 12\!\cdots\!01 \)
|
| $67$ |
\( T^{2} + \cdots + 34\!\cdots\!21 \)
|
| $71$ |
\( T^{2} + 95\!\cdots\!00 \)
|
| $73$ |
\( (T - 413274143)^{2} \)
|
| $79$ |
\( T^{2} + \cdots + 11\!\cdots\!01 \)
|
| $83$ |
\( T^{2} + \cdots + 32\!\cdots\!00 \)
|
| $89$ |
\( T^{2} + 81\!\cdots\!00 \)
|
| $97$ |
\( T^{2} + \cdots + 18\!\cdots\!49 \)
|
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