Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1680,2,Mod(961,1680)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1680, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1680.961");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1680.bg (of order \(3\), degree \(2\), not 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: | \(13.4148675396\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.29428272.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 6x^{4} - 4x^{3} - 42x^{2} + 343 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 840) |
| 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
\( x^{6} - 6x^{4} - 4x^{3} - 42x^{2} + 343 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3\nu^{5} + 35\nu^{4} + 31\nu^{3} + 121\nu^{2} - 217\nu - 1519 ) / 490 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{5} + 7\nu^{4} + 19\nu^{3} - 13\nu^{2} + 203\nu - 147 ) / 490 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{5} - 7\nu^{4} - 19\nu^{3} + 13\nu^{2} + 287\nu + 147 ) / 490 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} - 7\nu^{4} - \nu^{3} + 11\nu^{2} + 91\nu + 245 ) / 70 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -13\nu^{5} - 35\nu^{4} + 29\nu^{3} - 81\nu^{2} + 637\nu + 1519 ) / 490 \)
|
| \(\nu\) | \(=\) |
\( \beta_{3} + \beta_{2} \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + 2\beta_{4} - \beta_{2} + 2\beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{5} + \beta_{4} - 11\beta_{3} + \beta_{2} + 4\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - 9\beta_{4} + 18\beta_{3} + 5\beta_{2} + 5\beta _1 + 40 \)
|
| \(\nu^{5}\) | \(=\) |
\( -23\beta_{5} + 14\beta_{4} - 24\beta_{3} + 44\beta_{2} - 17\beta _1 - 10 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).
| \(n\) | \(241\) | \(337\) | \(421\) | \(1121\) | \(1471\) |
| \(\chi(n)\) | \(-1 - \beta_{3}\) | \(1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 961.1 |
|
0 | 0.500000 | + | 0.866025i | 0 | 0.500000 | − | 0.866025i | 0 | −2.63435 | + | 0.245357i | 0 | −0.500000 | + | 0.866025i | 0 | ||||||||||||||||||||||||||||
| 961.2 | 0 | 0.500000 | + | 0.866025i | 0 | 0.500000 | − | 0.866025i | 0 | 0.0741344 | − | 2.64471i | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||
| 961.3 | 0 | 0.500000 | + | 0.866025i | 0 | 0.500000 | − | 0.866025i | 0 | 2.56022 | + | 0.667305i | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||
| 1201.1 | 0 | 0.500000 | − | 0.866025i | 0 | 0.500000 | + | 0.866025i | 0 | −2.63435 | − | 0.245357i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||
| 1201.2 | 0 | 0.500000 | − | 0.866025i | 0 | 0.500000 | + | 0.866025i | 0 | 0.0741344 | + | 2.64471i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||
| 1201.3 | 0 | 0.500000 | − | 0.866025i | 0 | 0.500000 | + | 0.866025i | 0 | 2.56022 | − | 0.667305i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1680.2.bg.v | 6 | |
| 4.b | odd | 2 | 1 | 840.2.bg.j | ✓ | 6 | |
| 7.c | even | 3 | 1 | inner | 1680.2.bg.v | 6 | |
| 12.b | even | 2 | 1 | 2520.2.bi.n | 6 | ||
| 28.f | even | 6 | 1 | 5880.2.a.bu | 3 | ||
| 28.g | odd | 6 | 1 | 840.2.bg.j | ✓ | 6 | |
| 28.g | odd | 6 | 1 | 5880.2.a.bv | 3 | ||
| 84.n | even | 6 | 1 | 2520.2.bi.n | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 840.2.bg.j | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 840.2.bg.j | ✓ | 6 | 28.g | odd | 6 | 1 | |
| 1680.2.bg.v | 6 | 1.a | even | 1 | 1 | trivial | |
| 1680.2.bg.v | 6 | 7.c | even | 3 | 1 | inner | |
| 2520.2.bi.n | 6 | 12.b | even | 2 | 1 | ||
| 2520.2.bi.n | 6 | 84.n | even | 6 | 1 | ||
| 5880.2.a.bu | 3 | 28.f | even | 6 | 1 | ||
| 5880.2.a.bv | 3 | 28.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1680, [\chi])\):
|
\( T_{11}^{6} + 3T_{11}^{5} + 45T_{11}^{4} + 144T_{11}^{3} + 1674T_{11}^{2} + 4536T_{11} + 15876 \)
|
|
\( T_{13}^{3} + 3T_{13}^{2} - 15T_{13} + 3 \)
|
|
\( T_{17}^{6} + 6T_{17}^{5} + 90T_{17}^{4} + 316T_{17}^{3} + 4836T_{17}^{2} + 17280T_{17} + 102400 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} - T + 1)^{3} \)
|
| $5$ |
\( (T^{2} - T + 1)^{3} \)
|
| $7$ |
\( T^{6} - 6 T^{4} + \cdots + 343 \)
|
| $11$ |
\( T^{6} + 3 T^{5} + \cdots + 15876 \)
|
| $13$ |
\( (T^{3} + 3 T^{2} - 15 T + 3)^{2} \)
|
| $17$ |
\( T^{6} + 6 T^{5} + \cdots + 102400 \)
|
| $19$ |
\( (T^{2} - T + 1)^{3} \)
|
| $23$ |
\( T^{6} + 3 T^{5} + \cdots + 484 \)
|
| $29$ |
\( (T^{3} - 6 T^{2} - 6 T + 48)^{2} \)
|
| $31$ |
\( T^{6} - 12 T^{5} + \cdots + 202500 \)
|
| $37$ |
\( T^{6} + 3 T^{5} + \cdots + 145161 \)
|
| $41$ |
\( (T^{3} + 9 T^{2} - 58)^{2} \)
|
| $43$ |
\( (T^{3} - 39 T - 88)^{2} \)
|
| $47$ |
\( T^{6} - 3 T^{5} + \cdots + 19600 \)
|
| $53$ |
\( T^{6} - 15 T^{5} + \cdots + 3136 \)
|
| $59$ |
\( T^{6} - 30 T^{5} + \cdots + 705600 \)
|
| $61$ |
\( T^{6} \)
|
| $67$ |
\( T^{6} + 39 T^{4} + \cdots + 7744 \)
|
| $71$ |
\( (T^{3} - 12 T^{2} + \cdots + 2100)^{2} \)
|
| $73$ |
\( T^{6} + 18 T^{5} + \cdots + 3364 \)
|
| $79$ |
\( T^{6} - 6 T^{5} + \cdots + 1024 \)
|
| $83$ |
\( (T^{3} + 6 T^{2} - 6 T - 48)^{2} \)
|
| $89$ |
\( T^{6} + 90 T^{4} + \cdots + 85264 \)
|
| $97$ |
\( (T - 8)^{6} \)
|
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