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: | \(5\) |
| Coefficient field: | 5.5.170701.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 6x^{3} + 4x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 6x^{3} + 4x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 6\nu + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -2\nu^{4} + 3\nu^{3} + 10\nu^{2} - 4\nu - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\nu^{4} - 2\nu^{3} - 12\nu^{2} + 2\nu + 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + 2\beta_{3} + 2\beta_{2} + 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + 2\beta_{3} + 2\beta_{2} + 6\beta _1 + 8 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + 3\beta_{3} + 2\beta_{2} + 4\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 15\beta_{4} + 22\beta_{3} + 18\beta_{2} + 50\beta _1 + 44 ) / 4 \)
|
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 | −2.80813 | 1.00000 | 2.07745 | 2.80813 | −5.14956 | −1.00000 | 4.88558 | −2.07745 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.61453 | 1.00000 | −2.00782 | 1.61453 | 4.21034 | −1.00000 | −0.393292 | 2.00782 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −0.602352 | 1.00000 | −3.23952 | 0.602352 | −3.54318 | −1.00000 | −2.63717 | 3.23952 | ||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.86328 | 1.00000 | 2.33510 | −1.86328 | 3.89825 | −1.00000 | 0.471822 | −2.33510 | ||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 2.16173 | 1.00000 | 3.83479 | −2.16173 | −1.41585 | −1.00000 | 1.67306 | −3.83479 | ||||||||||||||||||||||||||||||||||
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.k | ✓ | 5 |
| 3.b | odd | 2 | 1 | 7254.2.a.bg | 5 | ||
| 4.b | odd | 2 | 1 | 6448.2.a.t | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 806.2.a.k | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 6448.2.a.t | 5 | 4.b | odd | 2 | 1 | ||
| 7254.2.a.bg | 5 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(806))\):
|
\( T_{3}^{5} + T_{3}^{4} - 9T_{3}^{3} - 6T_{3}^{2} + 18T_{3} + 11 \)
|
|
\( T_{5}^{5} - 3T_{5}^{4} - 15T_{5}^{3} + 42T_{5}^{2} + 44T_{5} - 121 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{5} \)
|
| $3$ |
\( T^{5} + T^{4} + \cdots + 11 \)
|
| $5$ |
\( T^{5} - 3 T^{4} + \cdots - 121 \)
|
| $7$ |
\( T^{5} + 2 T^{4} + \cdots + 424 \)
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| $11$ |
\( T^{5} - 28 T^{3} + \cdots - 128 \)
|
| $13$ |
\( (T + 1)^{5} \)
|
| $17$ |
\( T^{5} - 47 T^{3} + \cdots + 8 \)
|
| $19$ |
\( T^{5} - 9 T^{4} + \cdots - 1232 \)
|
| $23$ |
\( T^{5} \)
|
| $29$ |
\( T^{5} - 7 T^{4} + \cdots - 4 \)
|
| $31$ |
\( (T - 1)^{5} \)
|
| $37$ |
\( T^{5} + 8 T^{4} + \cdots + 8488 \)
|
| $41$ |
\( T^{5} + 2 T^{4} + \cdots + 2944 \)
|
| $43$ |
\( T^{5} - 9 T^{4} + \cdots - 1043 \)
|
| $47$ |
\( T^{5} + 2 T^{4} + \cdots + 56 \)
|
| $53$ |
\( T^{5} - 7 T^{4} + \cdots + 496 \)
|
| $59$ |
\( T^{5} + 5 T^{4} + \cdots + 8368 \)
|
| $61$ |
\( T^{5} - 33 T^{4} + \cdots + 1252 \)
|
| $67$ |
\( T^{5} + 17 T^{4} + \cdots - 908 \)
|
| $71$ |
\( T^{5} + 18 T^{4} + \cdots + 12608 \)
|
| $73$ |
\( T^{5} - 33 T^{4} + \cdots - 2012 \)
|
| $79$ |
\( T^{5} - 28 T^{4} + \cdots + 110848 \)
|
| $83$ |
\( T^{5} - 4 T^{4} + \cdots - 992 \)
|
| $89$ |
\( T^{5} + 19 T^{4} + \cdots - 22076 \)
|
| $97$ |
\( T^{5} - 14 T^{4} + \cdots - 25216 \)
|
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