Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [222,2,Mod(121,222)]
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, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("222.121");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 222 = 2 \cdot 3 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 222.e (of order \(3\), 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: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| 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: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 5\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/222\mathbb{Z}\right)^\times\).
| \(n\) | \(149\) | \(187\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 121.1 |
|
0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −1.68614 | − | 2.92048i | 1.00000 | −2.18614 | − | 3.78651i | −1.00000 | −0.500000 | + | 0.866025i | −3.37228 | ||||||||||||||||||
| 121.2 | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | 1.18614 | + | 2.05446i | 1.00000 | 0.686141 | + | 1.18843i | −1.00000 | −0.500000 | + | 0.866025i | 2.37228 | |||||||||||||||||||
| 211.1 | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −1.68614 | + | 2.92048i | 1.00000 | −2.18614 | + | 3.78651i | −1.00000 | −0.500000 | − | 0.866025i | −3.37228 | |||||||||||||||||||
| 211.2 | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | 1.18614 | − | 2.05446i | 1.00000 | 0.686141 | − | 1.18843i | −1.00000 | −0.500000 | − | 0.866025i | 2.37228 | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 222.2.e.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 666.2.f.g | 4 | ||
| 4.b | odd | 2 | 1 | 1776.2.q.h | 4 | ||
| 37.c | even | 3 | 1 | inner | 222.2.e.c | ✓ | 4 |
| 37.c | even | 3 | 1 | 8214.2.a.m | 2 | ||
| 37.e | even | 6 | 1 | 8214.2.a.o | 2 | ||
| 111.i | odd | 6 | 1 | 666.2.f.g | 4 | ||
| 148.i | odd | 6 | 1 | 1776.2.q.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 222.2.e.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 222.2.e.c | ✓ | 4 | 37.c | even | 3 | 1 | inner |
| 666.2.f.g | 4 | 3.b | odd | 2 | 1 | ||
| 666.2.f.g | 4 | 111.i | odd | 6 | 1 | ||
| 1776.2.q.h | 4 | 4.b | odd | 2 | 1 | ||
| 1776.2.q.h | 4 | 148.i | odd | 6 | 1 | ||
| 8214.2.a.m | 2 | 37.c | even | 3 | 1 | ||
| 8214.2.a.o | 2 | 37.e | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + T_{5}^{3} + 9T_{5}^{2} - 8T_{5} + 64 \)
acting on \(S_{2}^{\mathrm{new}}(222, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{2} \)
|
| $3$ |
\( (T^{2} - T + 1)^{2} \)
|
| $5$ |
\( T^{4} + T^{3} + \cdots + 64 \)
|
| $7$ |
\( T^{4} + 3 T^{3} + \cdots + 36 \)
|
| $11$ |
\( (T - 2)^{4} \)
|
| $13$ |
\( T^{4} - 3 T^{3} + \cdots + 36 \)
|
| $17$ |
\( T^{4} - 9 T^{3} + \cdots + 144 \)
|
| $19$ |
\( T^{4} - 2 T^{3} + \cdots + 1024 \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( (T^{2} - 3 T - 6)^{2} \)
|
| $31$ |
\( (T^{2} + 13 T + 34)^{2} \)
|
| $37$ |
\( T^{4} - 7 T^{3} + \cdots + 1369 \)
|
| $41$ |
\( T^{4} - 3 T^{3} + \cdots + 36 \)
|
| $43$ |
\( (T^{2} + 13 T + 34)^{2} \)
|
| $47$ |
\( (T - 2)^{4} \)
|
| $53$ |
\( T^{4} + 132 T^{2} + 17424 \)
|
| $59$ |
\( (T^{2} - 4 T + 16)^{2} \)
|
| $61$ |
\( T^{4} - T^{3} + \cdots + 5476 \)
|
| $67$ |
\( T^{4} - 9 T^{3} + \cdots + 2916 \)
|
| $71$ |
\( T^{4} + 132 T^{2} + 17424 \)
|
| $73$ |
\( (T^{2} - 11 T + 22)^{2} \)
|
| $79$ |
\( T^{4} - 3 T^{3} + \cdots + 36 \)
|
| $83$ |
\( (T^{2} + 8 T + 64)^{2} \)
|
| $89$ |
\( T^{4} + 13 T^{3} + \cdots + 1156 \)
|
| $97$ |
\( (T^{2} - 33)^{2} \)
|
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