Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [869,2,Mod(868,869)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(869, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("869.868");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 869 = 11 \cdot 79 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 869.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: | \(6.93899993565\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | 10.0.30049378896484375.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 7x^{5} + 32 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 7x^{5} + 32 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{8} + \nu^{3} ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{9} + 9\nu^{4} ) / 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{2} ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{8} + 7\nu^{3} + 8\nu ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{9} + 7\nu^{4} + 16\nu ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{6} - 5\nu ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{9} - 7\nu^{4} + 16\nu ) / 16 \)
|
| \(\beta_{9}\) | \(=\) |
\( \nu^{5} + \nu - 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{8} + \beta_{6} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{8} - \beta_{6} + 2\beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{8} - \beta_{6} + 2\beta_{5} + 2\beta_{2} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{8} + \beta_{6} + 2\beta_{3} ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{9} - \beta_{8} - \beta_{6} + 8 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 5\beta_{8} + 4\beta_{7} + 5\beta_{6} ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -3\beta_{8} - 3\beta_{6} + 8\beta_{4} + 6\beta_1 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( \beta_{8} + \beta_{6} - 2\beta_{5} + 14\beta_{2} ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 9\beta_{8} - 9\beta_{6} + 14\beta_{3} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/869\mathbb{Z}\right)^\times\).
| \(n\) | \(475\) | \(793\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 868.1 |
|
− | 2.80329i | 0 | −5.85843 | 1.65416 | 0 | 0 | 10.8163i | 3.00000 | − | 4.63709i | ||||||||||||||||||||||||||||||||||||||||||||||
| 868.2 | − | 2.48907i | 0 | −4.19545 | −3.44044 | 0 | 0 | 5.46463i | 3.00000 | 8.56349i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 868.3 | − | 2.04675i | 0 | −2.18919 | 4.46277 | 0 | 0 | 0.387225i | 3.00000 | − | 9.13418i | |||||||||||||||||||||||||||||||||||||||||||||||
| 868.4 | − | 1.22410i | 0 | 0.501568 | −3.78047 | 0 | 0 | − | 3.06218i | 3.00000 | 4.62770i | |||||||||||||||||||||||||||||||||||||||||||||||
| 868.5 | − | 0.508423i | 0 | 1.74151 | 1.10398 | 0 | 0 | − | 1.90227i | 3.00000 | − | 0.561291i | ||||||||||||||||||||||||||||||||||||||||||||||
| 868.6 | 0.508423i | 0 | 1.74151 | 1.10398 | 0 | 0 | 1.90227i | 3.00000 | 0.561291i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 868.7 | 1.22410i | 0 | 0.501568 | −3.78047 | 0 | 0 | 3.06218i | 3.00000 | − | 4.62770i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 868.8 | 2.04675i | 0 | −2.18919 | 4.46277 | 0 | 0 | − | 0.387225i | 3.00000 | 9.13418i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 868.9 | 2.48907i | 0 | −4.19545 | −3.44044 | 0 | 0 | − | 5.46463i | 3.00000 | − | 8.56349i | |||||||||||||||||||||||||||||||||||||||||||||||
| 868.10 | 2.80329i | 0 | −5.85843 | 1.65416 | 0 | 0 | − | 10.8163i | 3.00000 | 4.63709i | ||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 79.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-79}) \) |
| 11.b | odd | 2 | 1 | inner |
| 869.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 869.2.b.a | ✓ | 10 |
| 11.b | odd | 2 | 1 | inner | 869.2.b.a | ✓ | 10 |
| 79.b | odd | 2 | 1 | CM | 869.2.b.a | ✓ | 10 |
| 869.b | even | 2 | 1 | inner | 869.2.b.a | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 869.2.b.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 869.2.b.a | ✓ | 10 | 11.b | odd | 2 | 1 | inner |
| 869.2.b.a | ✓ | 10 | 79.b | odd | 2 | 1 | CM |
| 869.2.b.a | ✓ | 10 | 869.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 20T_{2}^{8} + 140T_{2}^{6} + 400T_{2}^{4} + 400T_{2}^{2} + 79 \)
acting on \(S_{2}^{\mathrm{new}}(869, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 20 T^{8} + \cdots + 79 \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( (T^{5} - 25 T^{3} + \cdots - 106)^{2} \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( T^{10} - 460 T^{5} + 161051 \)
|
| $13$ |
\( T^{10} + 130 T^{8} + \cdots + 11376 \)
|
| $17$ |
\( T^{10} \)
|
| $19$ |
\( T^{10} + 190 T^{8} + \cdots + 5759100 \)
|
| $23$ |
\( (T^{5} - 115 T^{3} + \cdots - 2416)^{2} \)
|
| $29$ |
\( T^{10} \)
|
| $31$ |
\( (T^{5} - 155 T^{3} + \cdots + 7752)^{2} \)
|
| $37$ |
\( T^{10} \)
|
| $41$ |
\( T^{10} \)
|
| $43$ |
\( T^{10} \)
|
| $47$ |
\( T^{10} \)
|
| $53$ |
\( T^{10} \)
|
| $59$ |
\( T^{10} \)
|
| $61$ |
\( T^{10} \)
|
| $67$ |
\( (T^{5} - 335 T^{3} + \cdots - 70572)^{2} \)
|
| $71$ |
\( T^{10} \)
|
| $73$ |
\( T^{10} + 730 T^{8} + \cdots + 2912256 \)
|
| $79$ |
\( (T^{2} + 79)^{5} \)
|
| $83$ |
\( (T^{2} + 316)^{5} \)
|
| $89$ |
\( (T^{5} - 445 T^{3} + \cdots - 132586)^{2} \)
|
| $97$ |
\( (T^{5} - 485 T^{3} + \cdots - 61662)^{2} \)
|
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