Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [111,2,Mod(10,111)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(111, 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("111.10");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 111 = 3 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 111.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: | \(0.886339462436\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.1415907.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 4x^{4} - 2x^{3} + 16x^{2} - 4x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 4x^{4} - 2x^{3} + 16x^{2} - 4x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 1 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 4\nu^{2} - \nu + 12 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 4\nu^{3} - \nu^{2} + 16\nu - 4 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{5} - 12\nu^{3} + 7\nu^{2} - 48\nu + 12 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{2} + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -4\beta_{5} - 12\beta_{4} + 4\beta_{3} + \beta _1 - 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 7\beta_{4} - 16\beta_{2} - 16\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/111\mathbb{Z}\right)^\times\).
| \(n\) | \(38\) | \(76\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 10.1 |
|
−1.05745 | + | 1.83156i | 0.500000 | + | 0.866025i | −1.23642 | − | 2.14154i | 1.23642 | + | 2.14154i | −2.11491 | −0.178963 | − | 0.309973i | 1.00000 | −0.500000 | + | 0.866025i | −5.22982 | ||||||||||||||||||||||||
| 10.2 | 0.127051 | − | 0.220059i | 0.500000 | + | 0.866025i | 0.967716 | + | 1.67613i | −0.967716 | − | 1.67613i | 0.254102 | 0.840665 | + | 1.45608i | 1.00000 | −0.500000 | + | 0.866025i | −0.491797 | |||||||||||||||||||||||||
| 10.3 | 0.930403 | − | 1.61151i | 0.500000 | + | 0.866025i | −0.731299 | − | 1.26665i | 0.731299 | + | 1.26665i | 1.86081 | −1.66170 | − | 2.87815i | 1.00000 | −0.500000 | + | 0.866025i | 2.72161 | |||||||||||||||||||||||||
| 100.1 | −1.05745 | − | 1.83156i | 0.500000 | − | 0.866025i | −1.23642 | + | 2.14154i | 1.23642 | − | 2.14154i | −2.11491 | −0.178963 | + | 0.309973i | 1.00000 | −0.500000 | − | 0.866025i | −5.22982 | |||||||||||||||||||||||||
| 100.2 | 0.127051 | + | 0.220059i | 0.500000 | − | 0.866025i | 0.967716 | − | 1.67613i | −0.967716 | + | 1.67613i | 0.254102 | 0.840665 | − | 1.45608i | 1.00000 | −0.500000 | − | 0.866025i | −0.491797 | |||||||||||||||||||||||||
| 100.3 | 0.930403 | + | 1.61151i | 0.500000 | − | 0.866025i | −0.731299 | + | 1.26665i | 0.731299 | − | 1.26665i | 1.86081 | −1.66170 | + | 2.87815i | 1.00000 | −0.500000 | − | 0.866025i | 2.72161 | |||||||||||||||||||||||||
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 | 111.2.e.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 333.2.f.b | 6 | ||
| 4.b | odd | 2 | 1 | 1776.2.q.l | 6 | ||
| 37.c | even | 3 | 1 | inner | 111.2.e.a | ✓ | 6 |
| 37.c | even | 3 | 1 | 4107.2.a.e | 3 | ||
| 37.e | even | 6 | 1 | 4107.2.a.f | 3 | ||
| 111.i | odd | 6 | 1 | 333.2.f.b | 6 | ||
| 148.i | odd | 6 | 1 | 1776.2.q.l | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 111.2.e.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 111.2.e.a | ✓ | 6 | 37.c | even | 3 | 1 | inner |
| 333.2.f.b | 6 | 3.b | odd | 2 | 1 | ||
| 333.2.f.b | 6 | 111.i | odd | 6 | 1 | ||
| 1776.2.q.l | 6 | 4.b | odd | 2 | 1 | ||
| 1776.2.q.l | 6 | 148.i | odd | 6 | 1 | ||
| 4107.2.a.e | 3 | 37.c | even | 3 | 1 | ||
| 4107.2.a.f | 3 | 37.e | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 4T_{2}^{4} - 2T_{2}^{3} + 16T_{2}^{2} - 4T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(111, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 4 T^{4} + \cdots + 1 \)
|
| $3$ |
\( (T^{2} - T + 1)^{3} \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots + 49 \)
|
| $7$ |
\( T^{6} + 2 T^{5} + \cdots + 4 \)
|
| $11$ |
\( (T^{3} + 5 T^{2} - 7 T + 2)^{2} \)
|
| $13$ |
\( T^{6} + 7 T^{5} + \cdots + 2116 \)
|
| $17$ |
\( T^{6} + T^{5} + \cdots + 49 \)
|
| $19$ |
\( T^{6} - 5 T^{5} + \cdots + 196 \)
|
| $23$ |
\( (T^{3} - T^{2} - 19 T + 32)^{2} \)
|
| $29$ |
\( (T^{3} + 11 T^{2} + 19 T + 1)^{2} \)
|
| $31$ |
\( (T^{3} - 16 T^{2} + 64 T - 8)^{2} \)
|
| $37$ |
\( T^{6} + 3 T^{5} + \cdots + 50653 \)
|
| $41$ |
\( T^{6} + 36 T^{4} + \cdots + 729 \)
|
| $43$ |
\( (T^{3} + 3 T^{2} - 79 T + 26)^{2} \)
|
| $47$ |
\( (T^{3} + 4 T^{2} + \cdots - 514)^{2} \)
|
| $53$ |
\( T^{6} - 6 T^{5} + \cdots + 21316 \)
|
| $59$ |
\( T^{6} + 3 T^{5} + \cdots + 571536 \)
|
| $61$ |
\( T^{6} - 9 T^{5} + \cdots + 38416 \)
|
| $67$ |
\( T^{6} + 17 T^{5} + \cdots + 232324 \)
|
| $71$ |
\( T^{6} - 18 T^{5} + \cdots + 53824 \)
|
| $73$ |
\( (T^{3} - 6 T^{2} + \cdots + 184)^{2} \)
|
| $79$ |
\( T^{6} - 3 T^{5} + \cdots + 3844 \)
|
| $83$ |
\( T^{6} - 12 T^{5} + \cdots + 16384 \)
|
| $89$ |
\( T^{6} + 5 T^{5} + \cdots + 99856 \)
|
| $97$ |
\( (T^{3} - 10 T^{2} + \cdots + 1141)^{2} \)
|
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