Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2016,2,Mod(1009,2016)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2016, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2016.1009");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2016.c (of order \(2\), degree \(1\), 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: | \(16.0978410475\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.72339481600.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{6} - 2x^{4} - 4x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{10} \) |
| Twist minimal: | no (minimal twist has level 504) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{6} - 2x^{4} - 4x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} + \nu^{5} - 4\nu ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} - \nu^{3} + 4\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} - \nu^{5} + 8\nu^{3} + 12\nu ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - \nu^{4} + 2\nu^{2} - 4 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{5} - 4\nu^{3} + 12\nu ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} + \nu^{4} + 6\nu^{2} + 2 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{6} + 3\nu^{4} - 2\nu^{2} - 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{3} + 2\beta_{2} + 2\beta_1 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + \beta_{4} + 1 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{5} + 3\beta_{3} - 2\beta_{2} + 2\beta_1 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{7} + \beta_{6} - \beta_{4} + 5 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 5\beta_{5} + \beta_{3} - 6\beta_{2} + 6\beta_1 ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 2\beta_{7} - \beta_{6} + 5\beta_{4} + 19 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -\beta_{5} + 3\beta_{3} + 14\beta_{2} + 34\beta_1 ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(577\) | \(1765\) | \(1793\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1009.1 |
|
0 | 0 | 0 | − | 3.66103i | 0 | 1.00000 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1009.2 | 0 | 0 | 0 | − | 3.66103i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1009.3 | 0 | 0 | 0 | − | 0.772577i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1009.4 | 0 | 0 | 0 | − | 0.772577i | 0 | 1.00000 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1009.5 | 0 | 0 | 0 | 0.772577i | 0 | 1.00000 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1009.6 | 0 | 0 | 0 | 0.772577i | 0 | 1.00000 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1009.7 | 0 | 0 | 0 | 3.66103i | 0 | 1.00000 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1009.8 | 0 | 0 | 0 | 3.66103i | 0 | 1.00000 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 24.h | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2016.2.c.f | 8 | |
| 3.b | odd | 2 | 1 | inner | 2016.2.c.f | 8 | |
| 4.b | odd | 2 | 1 | 504.2.c.e | ✓ | 8 | |
| 8.b | even | 2 | 1 | inner | 2016.2.c.f | 8 | |
| 8.d | odd | 2 | 1 | 504.2.c.e | ✓ | 8 | |
| 12.b | even | 2 | 1 | 504.2.c.e | ✓ | 8 | |
| 24.f | even | 2 | 1 | 504.2.c.e | ✓ | 8 | |
| 24.h | odd | 2 | 1 | inner | 2016.2.c.f | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 504.2.c.e | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 504.2.c.e | ✓ | 8 | 8.d | odd | 2 | 1 | |
| 504.2.c.e | ✓ | 8 | 12.b | even | 2 | 1 | |
| 504.2.c.e | ✓ | 8 | 24.f | even | 2 | 1 | |
| 2016.2.c.f | 8 | 1.a | even | 1 | 1 | trivial | |
| 2016.2.c.f | 8 | 3.b | odd | 2 | 1 | inner | |
| 2016.2.c.f | 8 | 8.b | even | 2 | 1 | inner | |
| 2016.2.c.f | 8 | 24.h | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2016, [\chi])\):
|
\( T_{5}^{4} + 14T_{5}^{2} + 8 \)
|
|
\( T_{11}^{4} + 26T_{11}^{2} + 128 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} + 14 T^{2} + 8)^{2} \)
|
| $7$ |
\( (T - 1)^{8} \)
|
| $11$ |
\( (T^{4} + 26 T^{2} + 128)^{2} \)
|
| $13$ |
\( (T^{4} + 44 T^{2} + 320)^{2} \)
|
| $17$ |
\( (T^{4} - 74 T^{2} + 1000)^{2} \)
|
| $19$ |
\( (T^{4} + 76 T^{2} + 1280)^{2} \)
|
| $23$ |
\( (T^{4} - 90 T^{2} + 1000)^{2} \)
|
| $29$ |
\( (T^{4} + 52 T^{2} + 512)^{2} \)
|
| $31$ |
\( (T^{2} + 2 T - 40)^{4} \)
|
| $37$ |
\( (T^{4} + 76 T^{2} + 1280)^{2} \)
|
| $41$ |
\( (T^{4} - 90 T^{2} + 1000)^{2} \)
|
| $43$ |
\( (T^{4} + 104 T^{2} + 80)^{2} \)
|
| $47$ |
\( T^{8} \)
|
| $53$ |
\( (T^{4} + 116 T^{2} + 3200)^{2} \)
|
| $59$ |
\( (T^{4} + 112 T^{2} + 512)^{2} \)
|
| $61$ |
\( (T^{4} + 44 T^{2} + 320)^{2} \)
|
| $67$ |
\( (T^{4} + 104 T^{2} + 80)^{2} \)
|
| $71$ |
\( (T^{4} - 74 T^{2} + 1000)^{2} \)
|
| $73$ |
\( (T + 6)^{8} \)
|
| $79$ |
\( T^{8} \)
|
| $83$ |
\( (T^{4} + 56 T^{2} + 128)^{2} \)
|
| $89$ |
\( (T^{4} - 90 T^{2} + 1000)^{2} \)
|
| $97$ |
\( (T^{2} - 8 T - 148)^{4} \)
|
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