Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [806,2,Mod(1,806)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(806, 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("806.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 806 = 2 \cdot 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 806.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: | \(6.43594240292\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.58446133.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 6x^{2} - 3x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 6x^{2} - 3x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{5} + 2\nu^{4} + 7\nu^{3} - 11\nu^{2} - 6\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 12\nu^{2} + 5\nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 12\nu^{2} - \nu - 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} + 3\nu^{4} + 5\nu^{3} - 18\nu^{2} + 6\nu + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{5} - 5\nu^{4} - 12\nu^{3} + 30\nu^{2} + \nu - 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta_{2} + 2\beta _1 + 7 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{5} + 3\beta_{4} + \beta_{3} - 4\beta_{2} + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 7\beta_{5} + 9\beta_{4} + 4\beta_{3} + 3\beta_{2} + 12\beta _1 + 41 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 39\beta_{5} + 43\beta_{4} + 22\beta_{3} - 55\beta_{2} + 39 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
1.00000 | −2.01503 | 1.00000 | 3.28061 | −2.01503 | 3.80073 | 1.00000 | 1.06034 | 3.28061 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.50975 | 1.00000 | −0.355042 | −1.50975 | −4.18990 | 1.00000 | −0.720660 | −0.355042 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 1.31037 | 1.00000 | 2.77389 | 1.31037 | 2.04275 | 1.00000 | −1.28294 | 2.77389 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.54560 | 1.00000 | −4.31786 | 1.54560 | 5.05813 | 1.00000 | −0.611133 | −4.31786 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.34112 | 1.00000 | 2.30312 | 2.34112 | −1.88736 | 1.00000 | 2.48084 | 2.30312 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 3.32770 | 1.00000 | −0.684721 | 3.32770 | −0.824344 | 1.00000 | 8.07356 | −0.684721 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(13\) | \(1\) |
| \(31\) | \(1\) |
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 | 806.2.a.l | ✓ | 6 |
| 3.b | odd | 2 | 1 | 7254.2.a.bk | 6 | ||
| 4.b | odd | 2 | 1 | 6448.2.a.v | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 806.2.a.l | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 6448.2.a.v | 6 | 4.b | odd | 2 | 1 | ||
| 7254.2.a.bk | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(806))\):
|
\( T_{3}^{6} - 5T_{3}^{5} - T_{3}^{4} + 32T_{3}^{3} - 24T_{3}^{2} - 47T_{3} + 48 \)
|
|
\( T_{5}^{6} - 3T_{5}^{5} - 17T_{5}^{4} + 64T_{5}^{3} - 12T_{5}^{2} - 75T_{5} - 22 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} - 5 T^{5} + \cdots + 48 \)
|
| $5$ |
\( T^{6} - 3 T^{5} + \cdots - 22 \)
|
| $7$ |
\( T^{6} - 4 T^{5} + \cdots - 256 \)
|
| $11$ |
\( T^{6} - 6 T^{5} + \cdots + 512 \)
|
| $13$ |
\( (T + 1)^{6} \)
|
| $17$ |
\( T^{6} - 10 T^{5} + \cdots - 2928 \)
|
| $19$ |
\( T^{6} - 15 T^{5} + \cdots + 14592 \)
|
| $23$ |
\( T^{6} - 2 T^{5} + \cdots + 2048 \)
|
| $29$ |
\( T^{6} + 3 T^{5} + \cdots - 1256 \)
|
| $31$ |
\( (T + 1)^{6} \)
|
| $37$ |
\( T^{6} + 26 T^{5} + \cdots - 17904 \)
|
| $41$ |
\( T^{6} + 2 T^{5} + \cdots + 576 \)
|
| $43$ |
\( T^{6} - 3 T^{5} + \cdots + 2776 \)
|
| $47$ |
\( T^{6} - 12 T^{5} + \cdots + 21568 \)
|
| $53$ |
\( T^{6} + 25 T^{5} + \cdots - 8328 \)
|
| $59$ |
\( T^{6} - 3 T^{5} + \cdots - 760784 \)
|
| $61$ |
\( T^{6} - 17 T^{5} + \cdots + 39688 \)
|
| $67$ |
\( T^{6} - 3 T^{5} + \cdots - 53584 \)
|
| $71$ |
\( T^{6} - 8 T^{5} + \cdots - 512 \)
|
| $73$ |
\( T^{6} + 19 T^{5} + \cdots + 114344 \)
|
| $79$ |
\( T^{6} + 8 T^{5} + \cdots + 38912 \)
|
| $83$ |
\( T^{6} + 4 T^{5} + \cdots + 421632 \)
|
| $89$ |
\( T^{6} + 5 T^{5} + \cdots - 67128 \)
|
| $97$ |
\( T^{6} - 196 T^{4} + \cdots + 256 \)
|
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