Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1666,2,Mod(883,1666)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1666, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1666.883");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1666 = 2 \cdot 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1666.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: | \(13.3030769767\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.84345856.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 13x^{4} + 41x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 238) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} + 13x^{4} + 41x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 8\nu^{2} + 5 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 12\nu^{3} + 35\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 14\nu^{3} + 47\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - \beta_{4} - 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{3} - 8\beta_{2} + 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( -12\beta_{5} + 14\beta_{4} + 37\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1666\mathbb{Z}\right)^\times\).
| \(n\) | \(785\) | \(885\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 883.1 |
|
1.00000 | − | 2.89028i | 1.00000 | 3.32463i | − | 2.89028i | 0 | 1.00000 | −5.35371 | 3.32463i | ||||||||||||||||||||||||||||||||||
| 883.2 | 1.00000 | − | 2.56387i | 1.00000 | − | 3.81430i | − | 2.56387i | 0 | 1.00000 | −3.57344 | − | 3.81430i | |||||||||||||||||||||||||||||||||
| 883.3 | 1.00000 | − | 0.269894i | 1.00000 | 0.630859i | − | 0.269894i | 0 | 1.00000 | 2.92716 | 0.630859i | |||||||||||||||||||||||||||||||||||
| 883.4 | 1.00000 | 0.269894i | 1.00000 | − | 0.630859i | 0.269894i | 0 | 1.00000 | 2.92716 | − | 0.630859i | |||||||||||||||||||||||||||||||||||
| 883.5 | 1.00000 | 2.56387i | 1.00000 | 3.81430i | 2.56387i | 0 | 1.00000 | −3.57344 | 3.81430i | |||||||||||||||||||||||||||||||||||||
| 883.6 | 1.00000 | 2.89028i | 1.00000 | − | 3.32463i | 2.89028i | 0 | 1.00000 | −5.35371 | − | 3.32463i | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1666.2.b.k | 6 | |
| 7.b | odd | 2 | 1 | 1666.2.b.l | 6 | ||
| 7.c | even | 3 | 2 | 238.2.j.c | ✓ | 12 | |
| 17.b | even | 2 | 1 | inner | 1666.2.b.k | 6 | |
| 119.d | odd | 2 | 1 | 1666.2.b.l | 6 | ||
| 119.j | even | 6 | 2 | 238.2.j.c | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 238.2.j.c | ✓ | 12 | 7.c | even | 3 | 2 | |
| 238.2.j.c | ✓ | 12 | 119.j | even | 6 | 2 | |
| 1666.2.b.k | 6 | 1.a | even | 1 | 1 | trivial | |
| 1666.2.b.k | 6 | 17.b | even | 2 | 1 | inner | |
| 1666.2.b.l | 6 | 7.b | odd | 2 | 1 | ||
| 1666.2.b.l | 6 | 119.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1666, [\chi])\):
|
\( T_{3}^{6} + 15T_{3}^{4} + 56T_{3}^{2} + 4 \)
|
|
\( T_{5}^{6} + 26T_{5}^{4} + 171T_{5}^{2} + 64 \)
|
|
\( T_{11}^{6} + 63T_{11}^{4} + 1304T_{11}^{2} + 8836 \)
|
|
\( T_{13}^{3} + T_{13}^{2} - 14T_{13} + 14 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} + 15 T^{4} + \cdots + 4 \)
|
| $5$ |
\( T^{6} + 26 T^{4} + \cdots + 64 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + 63 T^{4} + \cdots + 8836 \)
|
| $13$ |
\( (T^{3} + T^{2} - 14 T + 14)^{2} \)
|
| $17$ |
\( T^{6} + 2 T^{5} + \cdots + 4913 \)
|
| $19$ |
\( (T^{3} - 2 T^{2} - 13 T + 28)^{2} \)
|
| $23$ |
\( T^{6} + 42 T^{4} + \cdots + 1444 \)
|
| $29$ |
\( T^{6} + 60 T^{4} + \cdots + 3136 \)
|
| $31$ |
\( T^{6} + 48 T^{4} + \cdots + 784 \)
|
| $37$ |
\( T^{6} + 170 T^{4} + \cdots + 141376 \)
|
| $41$ |
\( T^{6} + 92 T^{4} + \cdots + 23104 \)
|
| $43$ |
\( (T^{3} - 10 T^{2} + \cdots + 32)^{2} \)
|
| $47$ |
\( (T^{3} + 4 T^{2} - 46 T - 8)^{2} \)
|
| $53$ |
\( (T^{3} - 11 T^{2} + 164)^{2} \)
|
| $59$ |
\( (T^{3} + 6 T^{2} - 7 T - 4)^{2} \)
|
| $61$ |
\( T^{6} + 196 T^{4} + \cdots + 1024 \)
|
| $67$ |
\( (T^{3} - 14 T^{2} + \cdots - 16)^{2} \)
|
| $71$ |
\( T^{6} + 429 T^{4} + \cdots + 2798929 \)
|
| $73$ |
\( T^{6} + 204 T^{4} + \cdots + 16384 \)
|
| $79$ |
\( T^{6} + 119 T^{4} + \cdots + 1444 \)
|
| $83$ |
\( (T^{3} - 12 T^{2} + \cdots + 448)^{2} \)
|
| $89$ |
\( (T^{3} - 9 T^{2} + \cdots + 763)^{2} \)
|
| $97$ |
\( T^{6} + 364 T^{4} + \cdots + 135424 \)
|
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