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: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-11})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{3} \) |
| 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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 2\nu^{2} - 2\nu - 9 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( -2\nu^{3} + 2\nu^{2} + 10\nu + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{3} + 2\nu^{2} + 10\nu - 33 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 9\beta_1 ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + 45\beta _1 + 45 ) / 18 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{3} - \beta_{2} + 36 ) / 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).
| \(n\) | \(325\) | \(487\) | \(569\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1 - \beta_{1}\) |
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 | −10.6168 | − | 18.3889i | 0 | −14.6168 | + | 25.3171i | 0 | 0 | 0 | ||||||||||||||||||||||||||
| 217.2 | 0 | 0 | 0 | 6.61684 | + | 11.4607i | 0 | 2.61684 | − | 4.53251i | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 433.1 | 0 | 0 | 0 | −10.6168 | + | 18.3889i | 0 | −14.6168 | − | 25.3171i | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 433.2 | 0 | 0 | 0 | 6.61684 | − | 11.4607i | 0 | 2.61684 | + | 4.53251i | 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.m | 4 | |
| 3.b | odd | 2 | 1 | 648.4.i.s | 4 | ||
| 9.c | even | 3 | 1 | 216.4.a.h | yes | 2 | |
| 9.c | even | 3 | 1 | inner | 648.4.i.m | 4 | |
| 9.d | odd | 6 | 1 | 216.4.a.e | ✓ | 2 | |
| 9.d | odd | 6 | 1 | 648.4.i.s | 4 | ||
| 36.f | odd | 6 | 1 | 432.4.a.s | 2 | ||
| 36.h | even | 6 | 1 | 432.4.a.o | 2 | ||
| 72.j | odd | 6 | 1 | 1728.4.a.bt | 2 | ||
| 72.l | even | 6 | 1 | 1728.4.a.bs | 2 | ||
| 72.n | even | 6 | 1 | 1728.4.a.bh | 2 | ||
| 72.p | odd | 6 | 1 | 1728.4.a.bg | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 216.4.a.e | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 216.4.a.h | yes | 2 | 9.c | even | 3 | 1 | |
| 432.4.a.o | 2 | 36.h | even | 6 | 1 | ||
| 432.4.a.s | 2 | 36.f | odd | 6 | 1 | ||
| 648.4.i.m | 4 | 1.a | even | 1 | 1 | trivial | |
| 648.4.i.m | 4 | 9.c | even | 3 | 1 | inner | |
| 648.4.i.s | 4 | 3.b | odd | 2 | 1 | ||
| 648.4.i.s | 4 | 9.d | odd | 6 | 1 | ||
| 1728.4.a.bg | 2 | 72.p | odd | 6 | 1 | ||
| 1728.4.a.bh | 2 | 72.n | even | 6 | 1 | ||
| 1728.4.a.bs | 2 | 72.l | even | 6 | 1 | ||
| 1728.4.a.bt | 2 | 72.j | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 8T_{5}^{3} + 345T_{5}^{2} - 2248T_{5} + 78961 \)
acting on \(S_{4}^{\mathrm{new}}(648, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 8 T^{3} + \cdots + 78961 \)
|
| $7$ |
\( T^{4} + 24 T^{3} + \cdots + 23409 \)
|
| $11$ |
\( (T^{2} + T + 1)^{2} \)
|
| $13$ |
\( T^{4} + 32 T^{3} + \cdots + 20214016 \)
|
| $17$ |
\( (T^{2} + 56 T - 3968)^{2} \)
|
| $19$ |
\( (T^{2} - 184 T + 7276)^{2} \)
|
| $23$ |
\( T^{4} + 92 T^{3} + \cdots + 6948496 \)
|
| $29$ |
\( T^{4} - 336 T^{3} + \cdots + 730945296 \)
|
| $31$ |
\( T^{4} + 376 T^{3} + \cdots + 779470561 \)
|
| $37$ |
\( (T^{2} - 348 T + 25524)^{2} \)
|
| $41$ |
\( T^{4} - 312 T^{3} + \cdots + 28772496 \)
|
| $43$ |
\( T^{4} + \cdots + 3204918544 \)
|
| $47$ |
\( T^{4} - 228 T^{3} + \cdots + 36144144 \)
|
| $53$ |
\( (T^{2} - 152 T - 44417)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 3077142784 \)
|
| $61$ |
\( T^{4} + \cdots + 90596184064 \)
|
| $67$ |
\( T^{4} + \cdots + 97566270736 \)
|
| $71$ |
\( (T^{2} - 1816 T + 781696)^{2} \)
|
| $73$ |
\( (T + 287)^{4} \)
|
| $79$ |
\( T^{4} + \cdots + 25050025984 \)
|
| $83$ |
\( T^{4} + \cdots + 22875655009 \)
|
| $89$ |
\( (T^{2} - 240 T - 81828)^{2} \)
|
| $97$ |
\( T^{4} - 338 T^{3} + \cdots + 91259809 \)
|
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