Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [92,6,Mod(91,92)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(92, 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("92.91");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 92 = 2^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 92.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: | \(14.7553114228\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.8869743.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{3} + 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{3} + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 2\nu^{4} - 3\nu^{2} - 2\nu ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} + 3\nu^{2} - 6\nu ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} + 3\nu^{2} + 2\nu ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{3} \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{5} - \beta_{4} + 2\beta_{3} + 3\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/92\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(47\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 91.1 |
|
−5.19240 | − | 2.24476i | 4.19304i | 21.9221 | + | 23.3114i | 0 | 9.41237 | − | 21.7720i | 0 | −61.5000 | − | 170.252i | 225.418 | 0 | ||||||||||||||||||||||||||||
| 91.2 | −5.19240 | + | 2.24476i | − | 4.19304i | 21.9221 | − | 23.3114i | 0 | 9.41237 | + | 21.7720i | 0 | −61.5000 | + | 170.252i | 225.418 | 0 | ||||||||||||||||||||||||||||
| 91.3 | 0.652183 | − | 5.61913i | − | 24.6582i | −31.1493 | − | 7.32941i | 0 | −138.558 | − | 16.0817i | 0 | −61.5000 | + | 170.252i | −365.026 | 0 | ||||||||||||||||||||||||||||
| 91.4 | 0.652183 | + | 5.61913i | 24.6582i | −31.1493 | + | 7.32941i | 0 | −138.558 | + | 16.0817i | 0 | −61.5000 | − | 170.252i | −365.026 | 0 | |||||||||||||||||||||||||||||
| 91.5 | 4.54022 | − | 3.37437i | − | 28.8512i | 9.22720 | − | 30.6408i | 0 | −97.3548 | − | 130.991i | 0 | −61.5000 | − | 170.252i | −589.393 | 0 | ||||||||||||||||||||||||||||
| 91.6 | 4.54022 | + | 3.37437i | 28.8512i | 9.22720 | + | 30.6408i | 0 | −97.3548 | + | 130.991i | 0 | −61.5000 | + | 170.252i | −589.393 | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-23}) \) |
| 4.b | odd | 2 | 1 | inner |
| 92.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 92.6.b.a | ✓ | 6 |
| 4.b | odd | 2 | 1 | inner | 92.6.b.a | ✓ | 6 |
| 23.b | odd | 2 | 1 | CM | 92.6.b.a | ✓ | 6 |
| 92.b | even | 2 | 1 | inner | 92.6.b.a | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 92.6.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 92.6.b.a | ✓ | 6 | 4.b | odd | 2 | 1 | inner |
| 92.6.b.a | ✓ | 6 | 23.b | odd | 2 | 1 | CM |
| 92.6.b.a | ✓ | 6 | 92.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 1458T_{3}^{4} + 531441T_{3}^{2} + 8898332 \)
acting on \(S_{6}^{\mathrm{new}}(92, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 123 T^{3} + 32768 \)
|
| $3$ |
\( T^{6} + 1458 T^{4} + \cdots + 8898332 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( (T^{3} - 1113879 T - 446459686)^{2} \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( (T^{2} + 6436343)^{3} \)
|
| $29$ |
\( (T^{3} - 61533447 T - 112622621718)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 15\!\cdots\!28 \)
|
| $37$ |
\( T^{6} \)
|
| $41$ |
\( (T^{3} + \cdots + 2493581286774)^{2} \)
|
| $43$ |
\( T^{6} \)
|
| $47$ |
\( T^{6} + \cdots + 41\!\cdots\!68 \)
|
| $53$ |
\( T^{6} \)
|
| $59$ |
\( (T^{2} + 148620572)^{3} \)
|
| $61$ |
\( T^{6} \)
|
| $67$ |
\( T^{6} \)
|
| $71$ |
\( T^{6} + \cdots + 15\!\cdots\!28 \)
|
| $73$ |
\( (T^{3} + \cdots - 113157362418986)^{2} \)
|
| $79$ |
\( T^{6} \)
|
| $83$ |
\( T^{6} \)
|
| $89$ |
\( T^{6} \)
|
| $97$ |
\( T^{6} \)
|
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