Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [92,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("92.91");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 92 = 2^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(47.3832969271\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-23})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 5x^{2} - 6x + 36 \)
|
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 5x^{2} - 6x + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 5\nu^{2} + 10\nu - 21 ) / 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} - 5\nu^{2} + 20\nu + 21 ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 6 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 2\beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -5\beta_{3} + 6 \)
|
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 |
|
−13.3267 | − | 18.2866i | − | 172.412i | −156.800 | + | 487.399i | 0 | −3152.84 | + | 2297.68i | 0 | 11002.5 | − | 3628.05i | −10043.1 | 0 | |||||||||||||||||||||
| 91.2 | −13.3267 | + | 18.2866i | 172.412i | −156.800 | − | 487.399i | 0 | −3152.84 | − | 2297.68i | 0 | 11002.5 | + | 3628.05i | −10043.1 | 0 | |||||||||||||||||||||||
| 91.3 | −9.17334 | − | 20.6845i | − | 277.921i | −343.700 | + | 379.493i | 0 | −5748.66 | + | 2549.46i | 0 | 11002.5 | + | 3628.05i | −57556.9 | 0 | ||||||||||||||||||||||
| 91.4 | −9.17334 | + | 20.6845i | 277.921i | −343.700 | − | 379.493i | 0 | −5748.66 | − | 2549.46i | 0 | 11002.5 | − | 3628.05i | −57556.9 | 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 | 92.10.b.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | inner | 92.10.b.b | ✓ | 4 |
| 23.b | odd | 2 | 1 | CM | 92.10.b.b | ✓ | 4 |
| 92.b | even | 2 | 1 | inner | 92.10.b.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 92.10.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 92.10.b.b | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 92.10.b.b | ✓ | 4 | 23.b | odd | 2 | 1 | CM |
| 92.10.b.b | ✓ | 4 | 92.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 106966T_{3}^{2} + 2296038889 \)
acting on \(S_{10}^{\mathrm{new}}(92, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 45 T^{3} + \cdots + 262144 \)
|
| $3$ |
\( T^{4} + \cdots + 2296038889 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( (T^{2} + 82510 T - 25005598019)^{2} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( (T^{2} + 1801152661463)^{2} \)
|
| $29$ |
\( (T^{2} + \cdots - 40307757857331)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 53\!\cdots\!69 \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( (T^{2} + \cdots - 866104174585239)^{2} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} + \cdots + 80\!\cdots\!89 \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( (T^{2} + 15\!\cdots\!00)^{2} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} + \cdots + 74\!\cdots\!49 \)
|
| $73$ |
\( (T^{2} + \cdots - 68\!\cdots\!39)^{2} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
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