Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [216,4,Mod(109,216)]
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, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("216.109");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 216 = 2^{3} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 216.d (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: | \(12.7444125612\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( 9\zeta_{8}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{8}^{3} + \zeta_{8} \)
|
| \(\beta_{3}\) | \(=\) |
\( -9\zeta_{8}^{3} + 9\zeta_{8} \)
|
| \(\zeta_{8}\) | \(=\) |
\( ( \beta_{3} + 9\beta_{2} ) / 18 \)
|
| \(\zeta_{8}^{2}\) | \(=\) |
\( ( \beta_1 ) / 9 \)
|
| \(\zeta_{8}^{3}\) | \(=\) |
\( ( -\beta_{3} + 9\beta_{2} ) / 18 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 109.1 |
|
− | 2.82843i | 0 | −8.00000 | − | 18.8995i | 0 | 29.7279 | 22.6274i | 0 | −53.4558 | ||||||||||||||||||||||||||||
| 109.2 | − | 2.82843i | 0 | −8.00000 | − | 0.899495i | 0 | 4.27208 | 22.6274i | 0 | −2.54416 | |||||||||||||||||||||||||||||
| 109.3 | 2.82843i | 0 | −8.00000 | 0.899495i | 0 | 4.27208 | − | 22.6274i | 0 | −2.54416 | ||||||||||||||||||||||||||||||
| 109.4 | 2.82843i | 0 | −8.00000 | 18.8995i | 0 | 29.7279 | − | 22.6274i | 0 | −53.4558 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 24.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-6}) \) |
| 3.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 216.4.d.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 216.4.d.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 864.4.d.b | 4 | ||
| 8.b | even | 2 | 1 | inner | 216.4.d.a | ✓ | 4 |
| 8.d | odd | 2 | 1 | 864.4.d.b | 4 | ||
| 12.b | even | 2 | 1 | 864.4.d.b | 4 | ||
| 24.f | even | 2 | 1 | 864.4.d.b | 4 | ||
| 24.h | odd | 2 | 1 | CM | 216.4.d.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 216.4.d.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 216.4.d.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 216.4.d.a | ✓ | 4 | 8.b | even | 2 | 1 | inner |
| 216.4.d.a | ✓ | 4 | 24.h | odd | 2 | 1 | CM |
| 864.4.d.b | 4 | 4.b | odd | 2 | 1 | ||
| 864.4.d.b | 4 | 8.d | odd | 2 | 1 | ||
| 864.4.d.b | 4 | 12.b | even | 2 | 1 | ||
| 864.4.d.b | 4 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 358T_{5}^{2} + 289 \)
acting on \(S_{4}^{\mathrm{new}}(216, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 358T^{2} + 289 \)
|
| $7$ |
\( (T^{2} - 34 T + 127)^{2} \)
|
| $11$ |
\( T^{4} + 7954 T^{2} + 15689521 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( (T^{2} + 49928)^{2} \)
|
| $31$ |
\( (T^{2} - 70 T - 84473)^{2} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} + \cdots + 12195005761 \)
|
| $59$ |
\( (T^{2} + 307328)^{2} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( (T^{2} - 322 T - 1063367)^{2} \)
|
| $79$ |
\( (T - 1370)^{4} \)
|
| $83$ |
\( T^{4} + \cdots + 43477671169 \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( (T^{2} - 574 T - 2408543)^{2} \)
|
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