Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [648,4,Mod(217,648)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(648, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("648.217");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 648 = 2^{3} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 648.i (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: | \(38.2332376837\) |
| Analytic rank: | \(1\) |
| 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: | 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 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 \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).
| \(n\) | \(325\) | \(487\) | \(569\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 217.1 |
|
0 | 0 | 0 | −2.50000 | − | 4.33013i | 0 | −18.0000 | + | 31.1769i | 0 | 0 | 0 | ||||||||||||||||||||
| 433.1 | 0 | 0 | 0 | −2.50000 | + | 4.33013i | 0 | −18.0000 | − | 31.1769i | 0 | 0 | 0 | |||||||||||||||||||||
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 | 648.4.i.c | 2 | |
| 3.b | odd | 2 | 1 | 648.4.i.j | 2 | ||
| 9.c | even | 3 | 1 | 648.4.a.b | yes | 1 | |
| 9.c | even | 3 | 1 | inner | 648.4.i.c | 2 | |
| 9.d | odd | 6 | 1 | 648.4.a.a | ✓ | 1 | |
| 9.d | odd | 6 | 1 | 648.4.i.j | 2 | ||
| 36.f | odd | 6 | 1 | 1296.4.a.f | 1 | ||
| 36.h | even | 6 | 1 | 1296.4.a.c | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 648.4.a.a | ✓ | 1 | 9.d | odd | 6 | 1 | |
| 648.4.a.b | yes | 1 | 9.c | even | 3 | 1 | |
| 648.4.i.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 648.4.i.c | 2 | 9.c | even | 3 | 1 | inner | |
| 648.4.i.j | 2 | 3.b | odd | 2 | 1 | ||
| 648.4.i.j | 2 | 9.d | odd | 6 | 1 | ||
| 1296.4.a.c | 1 | 36.h | even | 6 | 1 | ||
| 1296.4.a.f | 1 | 36.f | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 5T_{5} + 25 \)
acting on \(S_{4}^{\mathrm{new}}(648, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 5T + 25 \)
|
| $7$ |
\( T^{2} + 36T + 1296 \)
|
| $11$ |
\( T^{2} - 64T + 4096 \)
|
| $13$ |
\( T^{2} - 65T + 4225 \)
|
| $17$ |
\( (T + 59)^{2} \)
|
| $19$ |
\( (T + 28)^{2} \)
|
| $23$ |
\( T^{2} - 160T + 25600 \)
|
| $29$ |
\( T^{2} + 57T + 3249 \)
|
| $31$ |
\( T^{2} + 164T + 26896 \)
|
| $37$ |
\( (T + 321)^{2} \)
|
| $41$ |
\( T^{2} + 246T + 60516 \)
|
| $43$ |
\( T^{2} - 8T + 64 \)
|
| $47$ |
\( T^{2} - 84T + 7056 \)
|
| $53$ |
\( (T + 478)^{2} \)
|
| $59$ |
\( T^{2} + 32T + 1024 \)
|
| $61$ |
\( T^{2} + 415T + 172225 \)
|
| $67$ |
\( T^{2} - 220T + 48400 \)
|
| $71$ |
\( (T + 884)^{2} \)
|
| $73$ |
\( (T + 77)^{2} \)
|
| $79$ |
\( T^{2} - 80T + 6400 \)
|
| $83$ |
\( T^{2} - 1268 T + 1607824 \)
|
| $89$ |
\( (T + 123)^{2} \)
|
| $97$ |
\( T^{2} + 1346 T + 1811716 \)
|
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