Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [216,3,Mod(53,216)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(216, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("216.53");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 216 = 2^{3} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 216.h (of order \(2\), degree \(1\), 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: | \(5.88557371018\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.121670000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 4x^{4} - 6x^{3} + 16x^{2} - 16x + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 4x^{4} - 6x^{3} + 16x^{2} - 16x + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} + 4\nu^{3} - 2\nu^{2} - 8 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + \nu^{3} - 4\nu^{2} + 2\nu - 8 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} + 4\nu^{3} - 6\nu^{2} + 8\nu - 8 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{3} + \beta_{2} - \beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - 4\beta_{4} + \beta_{3} - 3\beta_{2} + \beta _1 - 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( 5\beta_{5} - 4\beta_{4} - 3\beta_{3} - \beta_{2} - 3\beta _1 - 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).
| \(n\) | \(55\) | \(109\) | \(137\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 53.1 |
|
−1.63177 | − | 1.15643i | 0 | 1.32536 | + | 3.77405i | −2.38717 | 0 | 2.13992 | 2.20173 | − | 7.69106i | 0 | 3.89531 | + | 2.76059i | ||||||||||||||||||||||||||||
| 53.2 | −1.63177 | + | 1.15643i | 0 | 1.32536 | − | 3.77405i | −2.38717 | 0 | 2.13992 | 2.20173 | + | 7.69106i | 0 | 3.89531 | − | 2.76059i | |||||||||||||||||||||||||||||
| 53.3 | −0.122180 | − | 1.99626i | 0 | −3.97014 | + | 0.487807i | 5.18465 | 0 | 3.67337 | 1.45886 | + | 7.86586i | 0 | −0.633460 | − | 10.3499i | |||||||||||||||||||||||||||||
| 53.4 | −0.122180 | + | 1.99626i | 0 | −3.97014 | − | 0.487807i | 5.18465 | 0 | 3.67337 | 1.45886 | − | 7.86586i | 0 | −0.633460 | + | 10.3499i | |||||||||||||||||||||||||||||
| 53.5 | 1.25395 | − | 1.55808i | 0 | −0.855212 | − | 3.90751i | −3.79748 | 0 | −10.8133 | −7.16059 | − | 3.56734i | 0 | −4.76185 | + | 5.91677i | |||||||||||||||||||||||||||||
| 53.6 | 1.25395 | + | 1.55808i | 0 | −0.855212 | + | 3.90751i | −3.79748 | 0 | −10.8133 | −7.16059 | + | 3.56734i | 0 | −4.76185 | − | 5.91677i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 24.h | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 216.3.h.c | ✓ | 6 |
| 3.b | odd | 2 | 1 | 216.3.h.d | yes | 6 | |
| 4.b | odd | 2 | 1 | 864.3.h.c | 6 | ||
| 8.b | even | 2 | 1 | 216.3.h.d | yes | 6 | |
| 8.d | odd | 2 | 1 | 864.3.h.d | 6 | ||
| 12.b | even | 2 | 1 | 864.3.h.d | 6 | ||
| 24.f | even | 2 | 1 | 864.3.h.c | 6 | ||
| 24.h | odd | 2 | 1 | inner | 216.3.h.c | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 216.3.h.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 216.3.h.c | ✓ | 6 | 24.h | odd | 2 | 1 | inner |
| 216.3.h.d | yes | 6 | 3.b | odd | 2 | 1 | |
| 216.3.h.d | yes | 6 | 8.b | even | 2 | 1 | |
| 864.3.h.c | 6 | 4.b | odd | 2 | 1 | ||
| 864.3.h.c | 6 | 24.f | even | 2 | 1 | ||
| 864.3.h.d | 6 | 8.d | odd | 2 | 1 | ||
| 864.3.h.d | 6 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + T_{5}^{2} - 23T_{5} - 47 \)
acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} + \cdots + 64 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{3} + T^{2} - 23 T - 47)^{2} \)
|
| $7$ |
\( (T^{3} + 5 T^{2} - 55 T + 85)^{2} \)
|
| $11$ |
\( (T^{3} - 5 T^{2} - 105 T + 45)^{2} \)
|
| $13$ |
\( T^{6} + 680 T^{4} + \cdots + 5299200 \)
|
| $17$ |
\( T^{6} + 1336 T^{4} + \cdots + 17169408 \)
|
| $19$ |
\( T^{6} + 1304 T^{4} + \cdots + 30523392 \)
|
| $23$ |
\( T^{6} + 2056 T^{4} + \cdots + 245035008 \)
|
| $29$ |
\( (T^{3} - 50 T^{2} + \cdots - 2880)^{2} \)
|
| $31$ |
\( (T^{3} - 3 T^{2} + \cdots - 15971)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 1716940800 \)
|
| $41$ |
\( T^{6} + 7000 T^{4} + \cdots + 529920000 \)
|
| $43$ |
\( T^{6} + 6080 T^{4} + \cdots + 339148800 \)
|
| $47$ |
\( T^{6} + \cdots + 6867763200 \)
|
| $53$ |
\( (T^{3} + T^{2} + \cdots + 32553)^{2} \)
|
| $59$ |
\( (T^{3} + 10 T^{2} + \cdots + 190360)^{2} \)
|
| $61$ |
\( T^{6} + \cdots + 89006211072 \)
|
| $67$ |
\( T^{6} + \cdots + 27471052800 \)
|
| $71$ |
\( T^{6} + \cdots + 3863116800 \)
|
| $73$ |
\( (T^{3} - 65 T^{2} + \cdots + 983025)^{2} \)
|
| $79$ |
\( (T^{3} + 38 T^{2} + \cdots - 323744)^{2} \)
|
| $83$ |
\( (T^{3} + 19 T^{2} + \cdots - 378683)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 61809868800 \)
|
| $97$ |
\( (T^{3} + 35 T^{2} + \cdots + 193825)^{2} \)
|
| show more | |
| show less |