Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [222,2,Mod(85,222)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(222, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("222.85");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 222 = 2 \cdot 3 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 222.j (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: | \(1.77267892487\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 8.0.303595776.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 32\nu^{4} + 16\nu^{2} + 45 ) / 144 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + 13\nu ) / 48 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} - 13 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5\nu^{7} + 16\nu^{5} + 80\nu^{3} + 225\nu ) / 144 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - 2\beta_{4} - \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} - 3\beta_{5} - 3\beta_{3} - 2\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 5\beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{7} + 15\beta_{3} \)
|
| \(\nu^{6}\) | \(=\) |
\( -16\beta_{6} - 13 \)
|
| \(\nu^{7}\) | \(=\) |
\( 48\beta_{5} - 13\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/222\mathbb{Z}\right)^\times\).
| \(n\) | \(149\) | \(187\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 85.1 |
|
−0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | −2.92048 | − | 1.68614i | 1.00000i | −0.103857 | + | 0.179885i | 1.00000i | −0.500000 | − | 0.866025i | 3.37228 | ||||||||||||||||||||||||||||||
| 85.2 | −0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | 2.05446 | + | 1.18614i | 1.00000i | −1.76217 | + | 3.05217i | 1.00000i | −0.500000 | − | 0.866025i | −2.37228 | |||||||||||||||||||||||||||||||
| 85.3 | 0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | −2.05446 | − | 1.18614i | − | 1.00000i | 0.762169 | − | 1.32012i | − | 1.00000i | −0.500000 | − | 0.866025i | −2.37228 | |||||||||||||||||||||||||||||
| 85.4 | 0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | 2.92048 | + | 1.68614i | − | 1.00000i | −0.896143 | + | 1.55217i | − | 1.00000i | −0.500000 | − | 0.866025i | 3.37228 | |||||||||||||||||||||||||||||
| 175.1 | −0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | −2.92048 | + | 1.68614i | − | 1.00000i | −0.103857 | − | 0.179885i | − | 1.00000i | −0.500000 | + | 0.866025i | 3.37228 | |||||||||||||||||||||||||||||
| 175.2 | −0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | 2.05446 | − | 1.18614i | − | 1.00000i | −1.76217 | − | 3.05217i | − | 1.00000i | −0.500000 | + | 0.866025i | −2.37228 | |||||||||||||||||||||||||||||
| 175.3 | 0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | −2.05446 | + | 1.18614i | 1.00000i | 0.762169 | + | 1.32012i | 1.00000i | −0.500000 | + | 0.866025i | −2.37228 | |||||||||||||||||||||||||||||||
| 175.4 | 0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | 2.92048 | − | 1.68614i | 1.00000i | −0.896143 | − | 1.55217i | 1.00000i | −0.500000 | + | 0.866025i | 3.37228 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.e | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 222.2.j.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 666.2.s.f | 8 | ||
| 4.b | odd | 2 | 1 | 1776.2.bz.i | 8 | ||
| 37.e | even | 6 | 1 | inner | 222.2.j.b | ✓ | 8 |
| 37.g | odd | 12 | 1 | 8214.2.a.z | 4 | ||
| 37.g | odd | 12 | 1 | 8214.2.a.ba | 4 | ||
| 111.h | odd | 6 | 1 | 666.2.s.f | 8 | ||
| 148.j | odd | 6 | 1 | 1776.2.bz.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 222.2.j.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 222.2.j.b | ✓ | 8 | 37.e | even | 6 | 1 | inner |
| 666.2.s.f | 8 | 3.b | odd | 2 | 1 | ||
| 666.2.s.f | 8 | 111.h | odd | 6 | 1 | ||
| 1776.2.bz.i | 8 | 4.b | odd | 2 | 1 | ||
| 1776.2.bz.i | 8 | 148.j | odd | 6 | 1 | ||
| 8214.2.a.z | 4 | 37.g | odd | 12 | 1 | ||
| 8214.2.a.ba | 4 | 37.g | odd | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 17T_{5}^{6} + 225T_{5}^{4} - 1088T_{5}^{2} + 4096 \)
acting on \(S_{2}^{\mathrm{new}}(222, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{2} + 1)^{2} \)
|
| $3$ |
\( (T^{2} - T + 1)^{4} \)
|
| $5$ |
\( T^{8} - 17 T^{6} + \cdots + 4096 \)
|
| $7$ |
\( T^{8} + 4 T^{7} + \cdots + 4 \)
|
| $11$ |
\( (T^{4} + 2 T^{3} - 22 T^{2} + \cdots - 8)^{2} \)
|
| $13$ |
\( T^{8} + 6 T^{7} + \cdots + 64 \)
|
| $17$ |
\( T^{8} - 6 T^{7} + \cdots + 64 \)
|
| $19$ |
\( T^{8} - 6 T^{7} + \cdots + 64 \)
|
| $23$ |
\( (T^{4} + 96 T^{2} + 1600)^{2} \)
|
| $29$ |
\( T^{8} + 130 T^{6} + \cdots + 110224 \)
|
| $31$ |
\( T^{8} + 70 T^{6} + \cdots + 5476 \)
|
| $37$ |
\( T^{8} - 12 T^{7} + \cdots + 1874161 \)
|
| $41$ |
\( T^{8} + 2 T^{7} + \cdots + 16 \)
|
| $43$ |
\( T^{8} + 302 T^{6} + \cdots + 4169764 \)
|
| $47$ |
\( (T^{4} + 14 T^{3} + \cdots - 3464)^{2} \)
|
| $53$ |
\( T^{8} - 4 T^{7} + \cdots + 160000 \)
|
| $59$ |
\( T^{8} - 12 T^{7} + \cdots + 16384 \)
|
| $61$ |
\( T^{8} + 36 T^{7} + \cdots + 28217344 \)
|
| $67$ |
\( T^{8} + 8 T^{7} + \cdots + 749956 \)
|
| $71$ |
\( T^{8} - 6 T^{7} + \cdots + 20286016 \)
|
| $73$ |
\( (T^{4} - 10 T^{3} + \cdots - 1832)^{2} \)
|
| $79$ |
\( T^{8} + 24 T^{7} + \cdots + 5635876 \)
|
| $83$ |
\( (T^{4} + 12 T^{3} + \cdots + 64)^{2} \)
|
| $89$ |
\( T^{8} + 42 T^{7} + \cdots + 515290000 \)
|
| $97$ |
\( T^{8} + 276 T^{6} + \cdots + 927369 \)
|
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