Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1785,2,Mod(1,1785)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1785, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("1785.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1785 = 3 \cdot 5 \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1785.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: | \(14.2532967608\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.674848.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 10x^{3} + 8x^{2} + 21x - 11 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{5} - x^{4} - 10x^{3} + 8x^{2} + 21x - 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 8\nu^{2} - \nu + 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 8\beta_{2} + \beta _1 + 23 \)
|
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 |
|
−2.76501 | 1.00000 | 5.64527 | −1.00000 | −2.76501 | 1.00000 | −10.0792 | 1.00000 | 2.76501 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.94778 | 1.00000 | 1.79385 | −1.00000 | −1.94778 | 1.00000 | 0.401533 | 1.00000 | 1.94778 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.489771 | 1.00000 | −1.76012 | −1.00000 | −0.489771 | 1.00000 | 1.84160 | 1.00000 | 0.489771 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.60618 | 1.00000 | 0.579823 | −1.00000 | 1.60618 | 1.00000 | −2.28106 | 1.00000 | −1.60618 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.59638 | 1.00000 | 4.74118 | −1.00000 | 2.59638 | 1.00000 | 7.11714 | 1.00000 | −2.59638 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(17\) | \( -1 \) |
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 | 1785.2.a.bc | ✓ | 5 |
| 3.b | odd | 2 | 1 | 5355.2.a.bs | 5 | ||
| 5.b | even | 2 | 1 | 8925.2.a.bz | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1785.2.a.bc | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 5355.2.a.bs | 5 | 3.b | odd | 2 | 1 | ||
| 8925.2.a.bz | 5 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1785))\):
|
\( T_{2}^{5} + T_{2}^{4} - 10T_{2}^{3} - 8T_{2}^{2} + 21T_{2} + 11 \)
|
|
\( T_{11}^{5} - 4T_{11}^{4} - 12T_{11}^{3} + 36T_{11}^{2} + 32T_{11} - 64 \)
|
|
\( T_{13}^{5} - 16T_{13}^{4} + 84T_{13}^{3} - 148T_{13}^{2} + 128 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + T^{4} + \cdots + 11 \)
|
| $3$ |
\( (T - 1)^{5} \)
|
| $5$ |
\( (T + 1)^{5} \)
|
| $7$ |
\( (T - 1)^{5} \)
|
| $11$ |
\( T^{5} - 4 T^{4} + \cdots - 64 \)
|
| $13$ |
\( T^{5} - 16 T^{4} + \cdots + 128 \)
|
| $17$ |
\( (T - 1)^{5} \)
|
| $19$ |
\( T^{5} + 2 T^{4} + \cdots + 800 \)
|
| $23$ |
\( T^{5} + 4 T^{4} + \cdots - 128 \)
|
| $29$ |
\( T^{5} - 2 T^{4} + \cdots + 704 \)
|
| $31$ |
\( T^{5} - 4 T^{4} + \cdots - 64 \)
|
| $37$ |
\( T^{5} - 2 T^{4} + \cdots - 1024 \)
|
| $41$ |
\( (T + 2)^{5} \)
|
| $43$ |
\( T^{5} - 6 T^{4} + \cdots - 6272 \)
|
| $47$ |
\( T^{5} - 22 T^{4} + \cdots - 1504 \)
|
| $53$ |
\( T^{5} + 8 T^{4} + \cdots - 13904 \)
|
| $59$ |
\( T^{5} - 12 T^{4} + \cdots - 2816 \)
|
| $61$ |
\( T^{5} - 14 T^{4} + \cdots + 176 \)
|
| $67$ |
\( T^{5} - 10 T^{4} + \cdots - 52864 \)
|
| $71$ |
\( T^{5} + 8 T^{4} + \cdots + 448 \)
|
| $73$ |
\( T^{5} - 24 T^{4} + \cdots + 6400 \)
|
| $79$ |
\( T^{5} - 6 T^{4} + \cdots - 35936 \)
|
| $83$ |
\( T^{5} - 14 T^{4} + \cdots - 6304 \)
|
| $89$ |
\( T^{5} - 6 T^{4} + \cdots - 7552 \)
|
| $97$ |
\( T^{5} - 268 T^{3} + \cdots - 3712 \)
|
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