Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [839,1,Mod(838,839)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(839, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("839.838");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 839 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 839.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: | yes |
| Analytic conductor: | \(0.418715545599\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\Q(\zeta_{33})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{9} - 10x^{8} + 10x^{7} + 34x^{6} - 34x^{5} - 43x^{4} + 43x^{3} + 12x^{2} - 12x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{33}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{33} - \cdots)\) |
$q$-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 \(\nu = \zeta_{33} + \zeta_{33}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} + 5\nu \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 6\nu^{4} + 9\nu^{2} - 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 7\nu^{5} + \nu^{4} + 14\nu^{3} - 4\nu^{2} - 7\nu + 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( \nu^{8} - 8\nu^{6} + 20\nu^{4} - 16\nu^{2} + 2 \)
|
| \(\beta_{9}\) | \(=\) |
\( \nu^{9} - 9\nu^{7} + 27\nu^{5} - 30\nu^{3} + 9\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 4\beta_{2} + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 5\beta_{3} + 10\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 6\beta_{4} + 15\beta_{2} + 20 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{7} + 7\beta_{5} - \beta_{4} + 21\beta_{3} + 35\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{8} + 8\beta_{6} + 28\beta_{4} + 56\beta_{2} + 70 \)
|
| \(\nu^{9}\) | \(=\) |
\( \beta_{9} + 9\beta_{7} + 36\beta_{5} - 9\beta_{4} + 84\beta_{3} + 126\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/839\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 838.1 |
|
−1.99094 | 1.16011 | 2.96386 | 1.68251 | −2.30972 | −0.284630 | −3.90993 | 0.345864 | −3.34978 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 838.2 | −1.77767 | 0.0951638 | 2.16011 | −1.91899 | −0.169170 | −1.30972 | −2.06230 | −0.990944 | 3.41133 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 838.3 | −1.57211 | 1.85674 | 1.47152 | −1.30972 | −2.91899 | 1.68251 | −0.741276 | 2.44747 | 2.05902 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 838.4 | −0.654136 | 1.96386 | −0.572106 | 0.830830 | −1.28463 | −1.91899 | 1.02837 | 2.85674 | −0.543476 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 838.5 | 0.0951638 | −1.77767 | −0.990944 | −1.91899 | −0.169170 | −1.30972 | −0.189466 | 2.16011 | −0.182618 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 838.6 | 0.471518 | 1.44747 | −0.777671 | −0.284630 | 0.682507 | 0.830830 | −0.838204 | 1.09516 | −0.134208 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 838.7 | 1.16011 | −1.99094 | 0.345864 | 1.68251 | −2.30972 | −0.284630 | −0.758872 | 2.96386 | 1.95190 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 838.8 | 1.44747 | 0.471518 | 1.09516 | −0.284630 | 0.682507 | 0.830830 | 0.137747 | −0.777671 | −0.411992 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 838.9 | 1.85674 | −1.57211 | 2.44747 | −1.30972 | −2.91899 | 1.68251 | 2.68757 | 1.47152 | −2.43181 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 838.10 | 1.96386 | −0.654136 | 2.85674 | 0.830830 | −1.28463 | −1.91899 | 3.64636 | −0.572106 | 1.63163 | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 839.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-839}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 839.1.b.c | ✓ | 10 |
| 839.b | odd | 2 | 1 | CM | 839.1.b.c | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 839.1.b.c | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 839.1.b.c | ✓ | 10 | 839.b | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - T_{2}^{9} - 10T_{2}^{8} + 10T_{2}^{7} + 34T_{2}^{6} - 34T_{2}^{5} - 43T_{2}^{4} + 43T_{2}^{3} + 12T_{2}^{2} - 12T_{2} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(839, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - T^{9} + \cdots + 1 \)
|
| $3$ |
\( T^{10} - T^{9} + \cdots + 1 \)
|
| $5$ |
\( (T^{5} + T^{4} - 4 T^{3} + \cdots + 1)^{2} \)
|
| $7$ |
\( (T^{5} + T^{4} - 4 T^{3} + \cdots + 1)^{2} \)
|
| $11$ |
\( T^{10} \)
|
| $13$ |
\( T^{10} \)
|
| $17$ |
\( T^{10} \)
|
| $19$ |
\( T^{10} - T^{9} + \cdots + 1 \)
|
| $23$ |
\( T^{10} - T^{9} + \cdots + 1 \)
|
| $29$ |
\( T^{10} \)
|
| $31$ |
\( T^{10} \)
|
| $37$ |
\( (T + 1)^{10} \)
|
| $41$ |
\( T^{10} \)
|
| $43$ |
\( T^{10} - T^{9} + \cdots + 1 \)
|
| $47$ |
\( (T + 1)^{10} \)
|
| $53$ |
\( T^{10} - T^{9} + \cdots + 1 \)
|
| $59$ |
\( T^{10} - T^{9} + \cdots + 1 \)
|
| $61$ |
\( (T^{5} + T^{4} - 4 T^{3} + \cdots + 1)^{2} \)
|
| $67$ |
\( T^{10} \)
|
| $71$ |
\( T^{10} \)
|
| $73$ |
\( T^{10} - T^{9} + \cdots + 1 \)
|
| $79$ |
\( T^{10} \)
|
| $83$ |
\( T^{10} \)
|
| $89$ |
\( T^{10} \)
|
| $97$ |
\( T^{10} \)
|
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