Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [108,6,Mod(107,108)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(108, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("108.107");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 108 = 2^{2} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 108.b (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: | \(17.3214525398\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{3}, \sqrt{-29})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 13x^{2} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 13x^{2} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 5\nu ) / 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 21\nu ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} + 13 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 13 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{2} + 21\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(55\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 107.1 |
|
−1.73205 | − | 5.38516i | 0 | −26.0000 | + | 18.6548i | − | 91.5478i | 0 | 158.565i | 145.492 | + | 107.703i | 0 | −493.000 | + | 158.565i | |||||||||||||||||||||
| 107.2 | −1.73205 | + | 5.38516i | 0 | −26.0000 | − | 18.6548i | 91.5478i | 0 | − | 158.565i | 145.492 | − | 107.703i | 0 | −493.000 | − | 158.565i | ||||||||||||||||||||||
| 107.3 | 1.73205 | − | 5.38516i | 0 | −26.0000 | − | 18.6548i | − | 91.5478i | 0 | − | 158.565i | −145.492 | + | 107.703i | 0 | −493.000 | − | 158.565i | |||||||||||||||||||||
| 107.4 | 1.73205 | + | 5.38516i | 0 | −26.0000 | + | 18.6548i | 91.5478i | 0 | 158.565i | −145.492 | − | 107.703i | 0 | −493.000 | + | 158.565i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 108.6.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 108.6.b.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | inner | 108.6.b.a | ✓ | 4 |
| 12.b | even | 2 | 1 | inner | 108.6.b.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 108.6.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 108.6.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 108.6.b.a | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 108.6.b.a | ✓ | 4 | 12.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 8381 \)
acting on \(S_{6}^{\mathrm{new}}(108, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 52T^{2} + 1024 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 8381)^{2} \)
|
| $7$ |
\( (T^{2} + 25143)^{2} \)
|
| $11$ |
\( (T^{2} - 336675)^{2} \)
|
| $13$ |
\( (T + 166)^{4} \)
|
| $17$ |
\( (T^{2} + 687764)^{2} \)
|
| $19$ |
\( (T^{2} + 451008)^{2} \)
|
| $23$ |
\( (T^{2} - 14891952)^{2} \)
|
| $29$ |
\( (T^{2} + 11656724)^{2} \)
|
| $31$ |
\( (T^{2} + 40110567)^{2} \)
|
| $37$ |
\( (T - 15332)^{4} \)
|
| $41$ |
\( (T^{2} + 102061904)^{2} \)
|
| $43$ |
\( (T^{2} + 9020508)^{2} \)
|
| $47$ |
\( (T^{2} - 49988172)^{2} \)
|
| $53$ |
\( (T^{2} + 298632749)^{2} \)
|
| $59$ |
\( (T^{2} - 808258188)^{2} \)
|
| $61$ |
\( (T + 53188)^{4} \)
|
| $67$ |
\( (T^{2} + 1685851548)^{2} \)
|
| $71$ |
\( (T^{2} - 719448588)^{2} \)
|
| $73$ |
\( (T + 30739)^{4} \)
|
| $79$ |
\( (T^{2} + 4927738812)^{2} \)
|
| $83$ |
\( (T^{2} - 130007667)^{2} \)
|
| $89$ |
\( (T^{2} + 4717270964)^{2} \)
|
| $97$ |
\( (T + 13717)^{4} \)
|
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