Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1296,3,Mod(593,1296)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1296, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1296.593");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1296 = 2^{4} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1296.q (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: | \(35.3134422611\) |
| 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 12) |
| Sato-Tate group: | $\mathrm{U}(1)[D_{6}]$ |
$q$-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 \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).
| \(n\) | \(325\) | \(1135\) | \(1217\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 593.1 |
|
0 | 0 | 0 | 0 | 0 | 1.00000 | − | 1.73205i | 0 | 0 | 0 | ||||||||||||||||||||||
| 1025.1 | 0 | 0 | 0 | 0 | 0 | 1.00000 | + | 1.73205i | 0 | 0 | 0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
| 9.c | even | 3 | 1 | inner |
| 9.d | odd | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1296, [\chi])\):
|
\( T_{5} \)
|
|
\( T_{7}^{2} - 2T_{7} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} - 2T + 4 \)
|
| $11$ |
\( T^{2} \)
|
| $13$ |
\( T^{2} - 22T + 484 \)
|
| $17$ |
\( T^{2} \)
|
| $19$ |
\( (T + 26)^{2} \)
|
| $23$ |
\( T^{2} \)
|
| $29$ |
\( T^{2} \)
|
| $31$ |
\( T^{2} + 46T + 2116 \)
|
| $37$ |
\( (T - 26)^{2} \)
|
| $41$ |
\( T^{2} \)
|
| $43$ |
\( T^{2} + 22T + 484 \)
|
| $47$ |
\( T^{2} \)
|
| $53$ |
\( T^{2} \)
|
| $59$ |
\( T^{2} \)
|
| $61$ |
\( T^{2} + 74T + 5476 \)
|
| $67$ |
\( T^{2} - 122T + 14884 \)
|
| $71$ |
\( T^{2} \)
|
| $73$ |
\( (T + 46)^{2} \)
|
| $79$ |
\( T^{2} + 142T + 20164 \)
|
| $83$ |
\( T^{2} \)
|
| $89$ |
\( T^{2} \)
|
| $97$ |
\( T^{2} + 2T + 4 \)
|
| show more | |
| show less |