Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [714,2,Mod(169,714)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(714, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("714.169");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 714.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: | \(5.70131870432\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.415622617344.23 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 21x^{6} + 104x^{4} + 21x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | yes |
| 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
\( x^{8} + 21x^{6} + 104x^{4} + 21x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -9\nu^{6} - 188\nu^{4} - 908\nu^{2} - 49 ) / 16 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -13\nu^{6} - 268\nu^{4} - 1276\nu^{2} - 101 ) / 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 9\nu^{7} + 188\nu^{5} + 916\nu^{3} + 97\nu ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -27\nu^{7} - 564\nu^{5} - 2740\nu^{3} - 179\nu ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -43\nu^{7} - 7\nu^{6} - 900\nu^{5} - 148\nu^{4} - 4404\nu^{3} - 740\nu^{2} - 531\nu - 111 ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -45\nu^{7} - 940\nu^{5} - 4572\nu^{3} - 405\nu ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -43\nu^{7} + 7\nu^{6} - 900\nu^{5} + 148\nu^{4} - 4404\nu^{3} + 740\nu^{2} - 531\nu + 111 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{6} + \beta_{4} - \beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - \beta_{5} - \beta_{2} + 3\beta _1 - 11 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 7\beta_{6} - 5\beta_{4} + 10\beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -10\beta_{7} + 10\beta_{5} + 19\beta_{2} - 43\beta _1 + 127 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -9\beta_{7} - 179\beta_{6} - 9\beta_{5} + 117\beta_{4} - 315\beta_{3} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 54\beta_{7} - 54\beta_{5} - 148\beta_{2} + 296\beta _1 - 777 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 188\beta_{7} + 2325\beta_{6} + 188\beta_{5} - 1437\beta_{4} + 4557\beta_{3} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/714\mathbb{Z}\right)^\times\).
| \(n\) | \(239\) | \(409\) | \(547\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 169.1 |
|
−1.00000 | − | 1.00000i | 1.00000 | − | 4.30207i | 1.00000i | 1.00000i | −1.00000 | −1.00000 | 4.30207i | ||||||||||||||||||||||||||||||||||||||||
| 169.2 | −1.00000 | − | 1.00000i | 1.00000 | − | 1.13914i | 1.00000i | 1.00000i | −1.00000 | −1.00000 | 1.13914i | |||||||||||||||||||||||||||||||||||||||||
| 169.3 | −1.00000 | − | 1.00000i | 1.00000 | 0.929786i | 1.00000i | 1.00000i | −1.00000 | −1.00000 | − | 0.929786i | |||||||||||||||||||||||||||||||||||||||||
| 169.4 | −1.00000 | − | 1.00000i | 1.00000 | 3.51142i | 1.00000i | 1.00000i | −1.00000 | −1.00000 | − | 3.51142i | |||||||||||||||||||||||||||||||||||||||||
| 169.5 | −1.00000 | 1.00000i | 1.00000 | − | 3.51142i | − | 1.00000i | − | 1.00000i | −1.00000 | −1.00000 | 3.51142i | ||||||||||||||||||||||||||||||||||||||||
| 169.6 | −1.00000 | 1.00000i | 1.00000 | − | 0.929786i | − | 1.00000i | − | 1.00000i | −1.00000 | −1.00000 | 0.929786i | ||||||||||||||||||||||||||||||||||||||||
| 169.7 | −1.00000 | 1.00000i | 1.00000 | 1.13914i | − | 1.00000i | − | 1.00000i | −1.00000 | −1.00000 | − | 1.13914i | ||||||||||||||||||||||||||||||||||||||||
| 169.8 | −1.00000 | 1.00000i | 1.00000 | 4.30207i | − | 1.00000i | − | 1.00000i | −1.00000 | −1.00000 | − | 4.30207i | ||||||||||||||||||||||||||||||||||||||||
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 | 714.2.b.f | ✓ | 8 |
| 3.b | odd | 2 | 1 | 2142.2.b.j | 8 | ||
| 17.b | even | 2 | 1 | inner | 714.2.b.f | ✓ | 8 |
| 51.c | odd | 2 | 1 | 2142.2.b.j | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 714.2.b.f | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 714.2.b.f | ✓ | 8 | 17.b | even | 2 | 1 | inner |
| 2142.2.b.j | 8 | 3.b | odd | 2 | 1 | ||
| 2142.2.b.j | 8 | 51.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\):
|
\( T_{5}^{8} + 33T_{5}^{6} + 296T_{5}^{4} + 528T_{5}^{2} + 256 \)
|
|
\( T_{11}^{8} + 93T_{11}^{6} + 2732T_{11}^{4} + 26448T_{11}^{2} + 73984 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( (T^{2} + 1)^{4} \)
|
| $5$ |
\( T^{8} + 33 T^{6} + \cdots + 256 \)
|
| $7$ |
\( (T^{2} + 1)^{4} \)
|
| $11$ |
\( T^{8} + 93 T^{6} + \cdots + 73984 \)
|
| $13$ |
\( (T^{4} - 3 T^{3} - 36 T^{2} + \cdots - 72)^{2} \)
|
| $17$ |
\( (T^{4} - 2 T^{3} + \cdots + 289)^{2} \)
|
| $19$ |
\( (T^{4} + 6 T^{3} + \cdots - 128)^{2} \)
|
| $23$ |
\( (T^{4} + 68 T^{2} + 1024)^{2} \)
|
| $29$ |
\( T^{8} + 144 T^{6} + \cdots + 1024 \)
|
| $31$ |
\( T^{8} + 132 T^{6} + \cdots + 65536 \)
|
| $37$ |
\( T^{8} + 109 T^{6} + \cdots + 215296 \)
|
| $41$ |
\( T^{8} + 264 T^{6} + \cdots + 1048576 \)
|
| $43$ |
\( (T^{4} - 5 T^{3} - 40 T^{2} + \cdots + 64)^{2} \)
|
| $47$ |
\( (T^{4} + 2 T^{3} + \cdots + 256)^{2} \)
|
| $53$ |
\( (T^{4} + 19 T^{3} + \cdots - 392)^{2} \)
|
| $59$ |
\( (T^{4} + 10 T^{3} + \cdots - 128)^{2} \)
|
| $61$ |
\( (T^{4} + 100 T^{2} + 2368)^{2} \)
|
| $67$ |
\( (T^{4} - 5 T^{3} - 40 T^{2} + \cdots + 64)^{2} \)
|
| $71$ |
\( T^{8} + 364 T^{6} + \cdots + 15745024 \)
|
| $73$ |
\( T^{8} + 297 T^{6} + \cdots + 1024 \)
|
| $79$ |
\( T^{8} + 249 T^{6} + \cdots + 5308416 \)
|
| $83$ |
\( (T^{4} - T^{3} - 178 T^{2} + \cdots + 592)^{2} \)
|
| $89$ |
\( (T^{4} - 11 T^{3} + \cdots + 544)^{2} \)
|
| $97$ |
\( T^{8} + 441 T^{6} + \cdots + 1327104 \)
|
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