Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [99,6,Mod(32,99)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(99, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5, 3]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("99.32");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 99.g (of order \(6\), degree \(2\), 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: | \(15.8779981615\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| 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: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{6}]$ |
$q$-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
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 2\nu + 3 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{3} + 2\beta _1 + 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).
| \(n\) | \(46\) | \(56\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 32.1 |
|
0 | −9.18614 | + | 12.5942i | 16.0000 | − | 27.7128i | 84.3981 | + | 48.7273i | 0 | 0 | 0 | −74.2296 | − | 231.385i | 0 | ||||||||||||||||||||||
| 32.2 | 0 | −6.31386 | + | 14.2525i | 16.0000 | − | 27.7128i | 1.10192 | + | 0.636194i | 0 | 0 | 0 | −163.270 | − | 179.977i | 0 | |||||||||||||||||||||||
| 65.1 | 0 | −9.18614 | − | 12.5942i | 16.0000 | + | 27.7128i | 84.3981 | − | 48.7273i | 0 | 0 | 0 | −74.2296 | + | 231.385i | 0 | |||||||||||||||||||||||
| 65.2 | 0 | −6.31386 | − | 14.2525i | 16.0000 | + | 27.7128i | 1.10192 | − | 0.636194i | 0 | 0 | 0 | −163.270 | + | 179.977i | 0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
| 9.d | odd | 6 | 1 | inner |
| 99.g | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 99.6.g.a | ✓ | 4 |
| 9.d | odd | 6 | 1 | inner | 99.6.g.a | ✓ | 4 |
| 11.b | odd | 2 | 1 | CM | 99.6.g.a | ✓ | 4 |
| 99.g | even | 6 | 1 | inner | 99.6.g.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 99.6.g.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 99.6.g.a | ✓ | 4 | 9.d | odd | 6 | 1 | inner |
| 99.6.g.a | ✓ | 4 | 11.b | odd | 2 | 1 | CM |
| 99.6.g.a | ✓ | 4 | 99.g | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{6}^{\mathrm{new}}(99, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 31 T^{3} + \cdots + 59049 \)
|
| $5$ |
\( T^{4} - 171 T^{3} + \cdots + 15376 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 25937424601 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 614197634226121 \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} + \cdots + 647032218701584 \)
|
| $37$ |
\( (T^{2} - 1267 T - 206426582)^{2} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} + \cdots + 14\!\cdots\!49 \)
|
| $53$ |
\( T^{4} + \cdots + 62\!\cdots\!49 \)
|
| $59$ |
\( T^{4} + \cdots + 97\!\cdots\!76 \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} + \cdots + 16\!\cdots\!24 \)
|
| $71$ |
\( T^{4} + \cdots + 66\!\cdots\!84 \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( (T^{2} + 14039031875)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 75\!\cdots\!24 \)
|
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