Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [731,2,Mod(1,731)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(731, 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("731.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 731 = 17 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 731.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: | \(5.83706438776\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.2460365.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 6x^{4} + 6x^{3} + 7x^{2} - 7x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 6x^{4} + 6x^{3} + 7x^{2} - 7x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{5} + 6\nu^{3} - 7\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 6\nu^{3} + \nu^{2} + 7\nu - 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + \nu^{4} + 6\nu^{3} - 5\nu^{2} - 7\nu + 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 5\beta_{4} + 4\beta_{3} + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{3} + 6\beta_{2} + 11\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.35826 | 1.68213 | 3.56138 | 0.252087 | −3.96689 | −3.41152 | −3.68213 | −0.170443 | −0.594485 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.24167 | 0.297851 | 3.02506 | 1.49772 | −0.667683 | 2.00334 | −2.29785 | −2.91128 | −3.35738 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.212277 | −2.83954 | −1.95494 | −1.65901 | 0.602769 | 2.53919 | 0.839542 | 5.06300 | 0.352169 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.291653 | −0.858197 | −1.91494 | 3.99528 | −0.250296 | −2.74049 | −1.14180 | −2.26350 | 1.16523 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.55253 | 0.467968 | 0.410361 | −1.37639 | 0.726536 | −3.35157 | −2.46797 | −2.78101 | −2.13690 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.96801 | −1.75021 | 1.87307 | 0.290323 | −3.44443 | −2.03895 | −0.249790 | 0.0632334 | 0.571360 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(17\) | \( +1 \) |
| \(43\) | \( +1 \) |
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 | 731.2.a.c | ✓ | 6 |
| 3.b | odd | 2 | 1 | 6579.2.a.j | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 731.2.a.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 6579.2.a.j | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + T_{2}^{5} - 8T_{2}^{4} - 4T_{2}^{3} + 17T_{2}^{2} - T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(731))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 8 T^{4} + \cdots - 1 \)
|
| $3$ |
\( T^{6} + 3 T^{5} + \cdots - 1 \)
|
| $5$ |
\( T^{6} - 3 T^{5} + \cdots + 1 \)
|
| $7$ |
\( T^{6} + 7 T^{5} + \cdots + 325 \)
|
| $11$ |
\( T^{6} - 4 T^{5} + \cdots + 1 \)
|
| $13$ |
\( T^{6} + 10 T^{5} + \cdots + 995 \)
|
| $17$ |
\( (T + 1)^{6} \)
|
| $19$ |
\( T^{6} + 20 T^{5} + \cdots - 9239 \)
|
| $23$ |
\( T^{6} + 3 T^{5} + \cdots - 281 \)
|
| $29$ |
\( T^{6} + 15 T^{5} + \cdots - 4679 \)
|
| $31$ |
\( T^{6} - 12 T^{5} + \cdots + 2767 \)
|
| $37$ |
\( T^{6} + 14 T^{5} + \cdots - 131765 \)
|
| $41$ |
\( T^{6} + 2 T^{5} + \cdots - 442115 \)
|
| $43$ |
\( (T + 1)^{6} \)
|
| $47$ |
\( T^{6} + 11 T^{5} + \cdots - 299 \)
|
| $53$ |
\( T^{6} - 3 T^{5} + \cdots + 115 \)
|
| $59$ |
\( T^{6} - 2 T^{5} + \cdots - 64283 \)
|
| $61$ |
\( T^{6} + 20 T^{5} + \cdots - 23 \)
|
| $67$ |
\( T^{6} + 2 T^{5} + \cdots - 558599 \)
|
| $71$ |
\( T^{6} - T^{5} + \cdots + 14543 \)
|
| $73$ |
\( T^{6} - 13 T^{5} + \cdots - 2137 \)
|
| $79$ |
\( T^{6} + 26 T^{5} + \cdots - 114701 \)
|
| $83$ |
\( T^{6} - 10 T^{5} + \cdots + 7360 \)
|
| $89$ |
\( T^{6} + 15 T^{5} + \cdots + 48599 \)
|
| $97$ |
\( T^{6} + 22 T^{5} + \cdots - 142255 \)
|
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