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: | \(0\) |
| 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 216) |
| 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 | 2.00000 | + | 3.46410i | 0 | −1.50000 | + | 2.59808i | 0 | 0 | 0 | ||||||||||||||||||||
| 433.1 | 0 | 0 | 0 | 2.00000 | − | 3.46410i | 0 | −1.50000 | − | 2.59808i | 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.i | 2 | |
| 3.b | odd | 2 | 1 | 648.4.i.d | 2 | ||
| 9.c | even | 3 | 1 | 216.4.a.a | ✓ | 1 | |
| 9.c | even | 3 | 1 | inner | 648.4.i.i | 2 | |
| 9.d | odd | 6 | 1 | 216.4.a.d | yes | 1 | |
| 9.d | odd | 6 | 1 | 648.4.i.d | 2 | ||
| 36.f | odd | 6 | 1 | 432.4.a.d | 1 | ||
| 36.h | even | 6 | 1 | 432.4.a.k | 1 | ||
| 72.j | odd | 6 | 1 | 1728.4.a.j | 1 | ||
| 72.l | even | 6 | 1 | 1728.4.a.i | 1 | ||
| 72.n | even | 6 | 1 | 1728.4.a.x | 1 | ||
| 72.p | odd | 6 | 1 | 1728.4.a.w | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 216.4.a.a | ✓ | 1 | 9.c | even | 3 | 1 | |
| 216.4.a.d | yes | 1 | 9.d | odd | 6 | 1 | |
| 432.4.a.d | 1 | 36.f | odd | 6 | 1 | ||
| 432.4.a.k | 1 | 36.h | even | 6 | 1 | ||
| 648.4.i.d | 2 | 3.b | odd | 2 | 1 | ||
| 648.4.i.d | 2 | 9.d | odd | 6 | 1 | ||
| 648.4.i.i | 2 | 1.a | even | 1 | 1 | trivial | |
| 648.4.i.i | 2 | 9.c | even | 3 | 1 | inner | |
| 1728.4.a.i | 1 | 72.l | even | 6 | 1 | ||
| 1728.4.a.j | 1 | 72.j | odd | 6 | 1 | ||
| 1728.4.a.w | 1 | 72.p | odd | 6 | 1 | ||
| 1728.4.a.x | 1 | 72.n | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 4T_{5} + 16 \)
acting on \(S_{4}^{\mathrm{new}}(648, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 4T + 16 \)
|
| $7$ |
\( T^{2} + 3T + 9 \)
|
| $11$ |
\( T^{2} - 28T + 784 \)
|
| $13$ |
\( T^{2} - 11T + 121 \)
|
| $17$ |
\( (T + 44)^{2} \)
|
| $19$ |
\( (T - 29)^{2} \)
|
| $23$ |
\( T^{2} - 172T + 29584 \)
|
| $29$ |
\( T^{2} - 192T + 36864 \)
|
| $31$ |
\( T^{2} + 116T + 13456 \)
|
| $37$ |
\( (T + 69)^{2} \)
|
| $41$ |
\( T^{2} - 384T + 147456 \)
|
| $43$ |
\( T^{2} + 328T + 107584 \)
|
| $47$ |
\( T^{2} - 156T + 24336 \)
|
| $53$ |
\( (T - 392)^{2} \)
|
| $59$ |
\( T^{2} - 412T + 169744 \)
|
| $61$ |
\( T^{2} - 425T + 180625 \)
|
| $67$ |
\( T^{2} + 257T + 66049 \)
|
| $71$ |
\( (T - 1000)^{2} \)
|
| $73$ |
\( (T + 359)^{2} \)
|
| $79$ |
\( T^{2} + 877T + 769129 \)
|
| $83$ |
\( T^{2} + 328T + 107584 \)
|
| $89$ |
\( (T - 1572)^{2} \)
|
| $97$ |
\( T^{2} - 1483 T + 2199289 \)
|
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