Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [714,2,Mod(83,714)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(714, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([4, 4, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("714.83");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 714.w (of order \(8\), degree \(4\), 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: | \(5.70131870432\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{8})\) |
| Coefficient field: | \(\Q(\zeta_{16})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/714\mathbb{Z}\right)^\times\).
| \(n\) | \(239\) | \(409\) | \(547\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-\zeta_{16}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 83.1 |
|
0.707107 | − | 0.707107i | −1.46508 | − | 0.923880i | − | 1.00000i | −3.15432 | + | 1.30656i | −1.68925 | + | 0.382683i | 2.25570 | + | 1.38268i | −0.707107 | − | 0.707107i | 1.29289 | + | 2.70711i | −1.30656 | + | 3.15432i | |||||||||||||||||||||||||
| 83.2 | 0.707107 | − | 0.707107i | 1.46508 | + | 0.923880i | − | 1.00000i | 3.15432 | − | 1.30656i | 1.68925 | − | 0.382683i | 2.57273 | + | 0.617317i | −0.707107 | − | 0.707107i | 1.29289 | + | 2.70711i | 1.30656 | − | 3.15432i | ||||||||||||||||||||||||||
| 461.1 | −0.707107 | + | 0.707107i | −1.68925 | − | 0.382683i | − | 1.00000i | −0.224171 | − | 0.541196i | 1.46508 | − | 0.923880i | −2.64466 | + | 0.0761205i | 0.707107 | + | 0.707107i | 2.70711 | + | 1.29289i | 0.541196 | + | 0.224171i | ||||||||||||||||||||||||||
| 461.2 | −0.707107 | + | 0.707107i | 1.68925 | + | 0.382683i | − | 1.00000i | 0.224171 | + | 0.541196i | −1.46508 | + | 0.923880i | 1.81623 | + | 1.92388i | 0.707107 | + | 0.707107i | 2.70711 | + | 1.29289i | −0.541196 | − | 0.224171i | ||||||||||||||||||||||||||
| 587.1 | −0.707107 | − | 0.707107i | −1.68925 | + | 0.382683i | 1.00000i | −0.224171 | + | 0.541196i | 1.46508 | + | 0.923880i | −2.64466 | − | 0.0761205i | 0.707107 | − | 0.707107i | 2.70711 | − | 1.29289i | 0.541196 | − | 0.224171i | |||||||||||||||||||||||||||
| 587.2 | −0.707107 | − | 0.707107i | 1.68925 | − | 0.382683i | 1.00000i | 0.224171 | − | 0.541196i | −1.46508 | − | 0.923880i | 1.81623 | − | 1.92388i | 0.707107 | − | 0.707107i | 2.70711 | − | 1.29289i | −0.541196 | + | 0.224171i | |||||||||||||||||||||||||||
| 671.1 | 0.707107 | + | 0.707107i | −1.46508 | + | 0.923880i | 1.00000i | −3.15432 | − | 1.30656i | −1.68925 | − | 0.382683i | 2.25570 | − | 1.38268i | −0.707107 | + | 0.707107i | 1.29289 | − | 2.70711i | −1.30656 | − | 3.15432i | |||||||||||||||||||||||||||
| 671.2 | 0.707107 | + | 0.707107i | 1.46508 | − | 0.923880i | 1.00000i | 3.15432 | + | 1.30656i | 1.68925 | + | 0.382683i | 2.57273 | − | 0.617317i | −0.707107 | + | 0.707107i | 1.29289 | − | 2.70711i | 1.30656 | + | 3.15432i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 51.g | odd | 8 | 1 | inner |
| 357.w | even | 8 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 714.2.w.d | yes | 8 |
| 3.b | odd | 2 | 1 | 714.2.w.a | ✓ | 8 | |
| 7.b | odd | 2 | 1 | inner | 714.2.w.d | yes | 8 |
| 17.d | even | 8 | 1 | 714.2.w.a | ✓ | 8 | |
| 21.c | even | 2 | 1 | 714.2.w.a | ✓ | 8 | |
| 51.g | odd | 8 | 1 | inner | 714.2.w.d | yes | 8 |
| 119.l | odd | 8 | 1 | 714.2.w.a | ✓ | 8 | |
| 357.w | even | 8 | 1 | inner | 714.2.w.d | yes | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 714.2.w.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 714.2.w.a | ✓ | 8 | 17.d | even | 8 | 1 | |
| 714.2.w.a | ✓ | 8 | 21.c | even | 2 | 1 | |
| 714.2.w.a | ✓ | 8 | 119.l | odd | 8 | 1 | |
| 714.2.w.d | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 714.2.w.d | yes | 8 | 7.b | odd | 2 | 1 | inner |
| 714.2.w.d | yes | 8 | 51.g | odd | 8 | 1 | inner |
| 714.2.w.d | yes | 8 | 357.w | even | 8 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\):
|
\( T_{5}^{8} - 16T_{5}^{6} + 128T_{5}^{4} + 64T_{5}^{2} + 16 \)
|
|
\( T_{11}^{4} + 4T_{11}^{3} + 22T_{11}^{2} + 12T_{11} + 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 1)^{2} \)
|
| $3$ |
\( T^{8} - 8 T^{6} + \cdots + 81 \)
|
| $5$ |
\( T^{8} - 16 T^{6} + \cdots + 16 \)
|
| $7$ |
\( T^{8} - 8 T^{7} + \cdots + 2401 \)
|
| $11$ |
\( (T^{4} + 4 T^{3} + 22 T^{2} + \cdots + 2)^{2} \)
|
| $13$ |
\( (T^{4} - 8 T^{2} + 8)^{2} \)
|
| $17$ |
\( T^{8} - 48 T^{6} + \cdots + 83521 \)
|
| $19$ |
\( T^{8} + 2508 T^{4} + 9604 \)
|
| $23$ |
\( (T^{4} + 4 T^{3} + \cdots + 1058)^{2} \)
|
| $29$ |
\( (T^{4} + 20 T^{3} + \cdots + 392)^{2} \)
|
| $31$ |
\( T^{8} + 48 T^{6} + \cdots + 38416 \)
|
| $37$ |
\( (T^{4} - 16 T^{3} + \cdots + 1058)^{2} \)
|
| $41$ |
\( T^{8} - 96 T^{6} + \cdots + 614656 \)
|
| $43$ |
\( (T^{4} + 16)^{2} \)
|
| $47$ |
\( (T^{4} + 160 T^{2} + 6272)^{2} \)
|
| $53$ |
\( (T^{2} - 16 T + 128)^{4} \)
|
| $59$ |
\( T^{8} + 6192 T^{4} + 5345344 \)
|
| $61$ |
\( T^{8} - 192 T^{6} + \cdots + 342102016 \)
|
| $67$ |
\( (T^{2} - 24 T + 136)^{4} \)
|
| $71$ |
\( (T^{4} - 28 T^{3} + \cdots + 1058)^{2} \)
|
| $73$ |
\( T^{8} + 26873856 \)
|
| $79$ |
\( (T^{4} + 24 T^{3} + \cdots + 6272)^{2} \)
|
| $83$ |
\( T^{8} + 3888 T^{4} + 419904 \)
|
| $89$ |
\( (T^{4} + 68 T^{2} + 1058)^{2} \)
|
| $97$ |
\( T^{8} - 128 T^{6} + \cdots + 236421376 \)
|
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