Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [216,6,Mod(1,216)]
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, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("216.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 216 = 2^{3} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 216.a (trivial) |
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: | yes |
| Analytic conductor: | \(34.6429050796\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{85}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 21 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = 6\sqrt{85}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 0 | 0 | −41.3173 | 0 | −83.6345 | 0 | 0 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 69.3173 | 0 | 137.635 | 0 | 0 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 216.6.a.g | yes | 2 |
| 3.b | odd | 2 | 1 | 216.6.a.d | ✓ | 2 | |
| 4.b | odd | 2 | 1 | 432.6.a.t | 2 | ||
| 12.b | even | 2 | 1 | 432.6.a.m | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 216.6.a.d | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 216.6.a.g | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 432.6.a.m | 2 | 12.b | even | 2 | 1 | ||
| 432.6.a.t | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 28T_{5} - 2864 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(216))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 28T - 2864 \)
|
| $7$ |
\( T^{2} - 54T - 11511 \)
|
| $11$ |
\( T^{2} - 148T - 22064 \)
|
| $13$ |
\( T^{2} - 242T - 291359 \)
|
| $17$ |
\( T^{2} - 212T - 505904 \)
|
| $19$ |
\( T^{2} + 62T - 194879 \)
|
| $23$ |
\( T^{2} - 2308 T - 287024 \)
|
| $29$ |
\( T^{2} - 2400 T - 22256640 \)
|
| $31$ |
\( T^{2} + 968 T - 23462384 \)
|
| $37$ |
\( T^{2} + 8178 T - 98446239 \)
|
| $41$ |
\( T^{2} - 19200 T + 13824000 \)
|
| $43$ |
\( T^{2} - 11264 T - 138710336 \)
|
| $47$ |
\( T^{2} - 28116 T + 183891024 \)
|
| $53$ |
\( T^{2} - 46984 T + 531298624 \)
|
| $59$ |
\( T^{2} + \cdots - 1165001264 \)
|
| $61$ |
\( T^{2} + 11770 T - 181880135 \)
|
| $67$ |
\( T^{2} - 34522 T + 297158761 \)
|
| $71$ |
\( T^{2} + \cdots + 2594799424 \)
|
| $73$ |
\( T^{2} + 56158 T + 787989601 \)
|
| $79$ |
\( T^{2} + \cdots - 5283833015 \)
|
| $83$ |
\( T^{2} - 86216 T - 765576896 \)
|
| $89$ |
\( T^{2} + \cdots + 6252742224 \)
|
| $97$ |
\( T^{2} + 139430 T + 394833385 \)
|
| show more | |
| show less |