Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [799,2,Mod(1,799)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(799, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("799.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 799 = 17 \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 799.a (trivial) |
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: | yes |
| Analytic conductor: | \(6.38004712150\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 8x^{6} + 3x^{5} + 18x^{4} - 10x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{8} - x^{7} - 8x^{6} + 3x^{5} + 18x^{4} - 10x^{2} - x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{7} - 2\nu^{6} - 6\nu^{5} + 9\nu^{4} + 9\nu^{3} - 9\nu^{2} - \nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{7} - \nu^{6} - 8\nu^{5} + 4\nu^{4} + 16\nu^{3} - 4\nu^{2} - 5\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{7} - \nu^{6} - 8\nu^{5} + 3\nu^{4} + 18\nu^{3} - 10\nu - 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{7} - 2\nu^{6} - 6\nu^{5} + 9\nu^{4} + 10\nu^{3} - 11\nu^{2} - 4\nu + 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{7} + 2\nu^{6} + 7\nu^{5} - 11\nu^{4} - 14\nu^{3} + 16\nu^{2} + 6\nu - 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( -2\nu^{7} + 3\nu^{6} + 14\nu^{5} - 12\nu^{4} - 27\nu^{3} + 10\nu^{2} + 10\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{4} + \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} + \beta_{5} + 2\beta_{4} + \beta_{2} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{7} + 2\beta_{5} + 7\beta_{4} + \beta_{3} + 6\beta_{2} + 9\beta _1 + 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( 19\beta_{7} + \beta_{6} + 9\beta_{5} + 17\beta_{4} + 2\beta_{3} + 11\beta_{2} + 31\beta _1 + 19 \)
|
| \(\nu^{6}\) | \(=\) |
\( 59\beta_{7} + 2\beta_{6} + 21\beta_{5} + 50\beta_{4} + 10\beta_{3} + 39\beta_{2} + 71\beta _1 + 64 \)
|
| \(\nu^{7}\) | \(=\) |
\( 151\beta_{7} + 10\beta_{6} + 69\beta_{5} + 130\beta_{4} + 23\beta_{3} + 91\beta_{2} + 212\beta _1 + 151 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.21326 | −1.77884 | 2.89853 | −4.00774 | 3.93704 | 2.27264 | −1.98868 | 0.164272 | 8.87017 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.97697 | −0.201661 | 1.90839 | 1.83089 | 0.398678 | −2.43371 | 0.181104 | −2.95933 | −3.61961 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.39501 | −2.49338 | −0.0539349 | −3.75291 | 3.47830 | −3.19565 | 2.86527 | 3.21694 | 5.23536 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.0304470 | 1.70906 | −1.99907 | −0.713351 | 0.0520358 | −1.24737 | −0.121760 | −0.0791178 | −0.0217194 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.887509 | −2.74404 | −1.21233 | −0.0255318 | −2.43536 | 2.42435 | −2.85097 | 4.52975 | −0.0226597 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.32352 | −0.710480 | −0.248291 | 2.07080 | −0.940335 | −1.47833 | −2.97566 | −2.49522 | 2.74075 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.80272 | 0.686238 | 1.24980 | −3.44678 | 1.23710 | −0.240642 | −1.35239 | −2.52908 | −6.21358 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.54104 | −1.46690 | 4.45690 | −1.95538 | −3.72745 | −5.10130 | 6.24309 | −0.848209 | −4.96872 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(17\) | \(-1\) |
| \(47\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 799.2.a.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | 7191.2.a.u | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 799.2.a.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 7191.2.a.u | 8 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(799))\):
|
\( T_{2}^{8} - T_{2}^{7} - 11T_{2}^{6} + 10T_{2}^{5} + 37T_{2}^{4} - 33T_{2}^{3} - 37T_{2}^{2} + 34T_{2} - 1 \)
|
|
\( T_{5}^{8} + 10T_{5}^{7} + 23T_{5}^{6} - 53T_{5}^{5} - 226T_{5}^{4} - 22T_{5}^{3} + 463T_{5}^{2} + 286T_{5} + 7 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{7} - 11 T^{6} + \cdots - 1 \)
|
| $3$ |
\( T^{8} + 7 T^{7} + \cdots + 3 \)
|
| $5$ |
\( T^{8} + 10 T^{7} + \cdots + 7 \)
|
| $7$ |
\( T^{8} + 9 T^{7} + \cdots + 97 \)
|
| $11$ |
\( T^{8} + 2 T^{7} + \cdots - 5061 \)
|
| $13$ |
\( T^{8} + 3 T^{7} + \cdots - 41 \)
|
| $17$ |
\( (T - 1)^{8} \)
|
| $19$ |
\( T^{8} + 8 T^{7} + \cdots - 138169 \)
|
| $23$ |
\( T^{8} + 9 T^{7} + \cdots - 115631 \)
|
| $29$ |
\( T^{8} + 2 T^{7} + \cdots - 12927 \)
|
| $31$ |
\( T^{8} + 7 T^{7} + \cdots + 37247 \)
|
| $37$ |
\( T^{8} + 13 T^{7} + \cdots - 4537 \)
|
| $41$ |
\( T^{8} + 39 T^{7} + \cdots - 16173 \)
|
| $43$ |
\( T^{8} - 35 T^{7} + \cdots + 25637 \)
|
| $47$ |
\( (T - 1)^{8} \)
|
| $53$ |
\( T^{8} + 6 T^{7} + \cdots + 127317 \)
|
| $59$ |
\( T^{8} + 19 T^{7} + \cdots - 952167 \)
|
| $61$ |
\( T^{8} + 15 T^{7} + \cdots + 64043 \)
|
| $67$ |
\( T^{8} - 15 T^{7} + \cdots + 49697 \)
|
| $71$ |
\( T^{8} + 7 T^{7} + \cdots - 6426029 \)
|
| $73$ |
\( T^{8} + 22 T^{7} + \cdots + 695361 \)
|
| $79$ |
\( T^{8} + 43 T^{7} + \cdots + 4430157 \)
|
| $83$ |
\( T^{8} + 3 T^{7} + \cdots - 8487209 \)
|
| $89$ |
\( T^{8} + 53 T^{7} + \cdots - 65651297 \)
|
| $97$ |
\( T^{8} + 40 T^{7} + \cdots + 21108341 \)
|
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