Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [585,4,Mod(1,585)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(585, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("585.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 585 = 3^{2} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 585.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: | \(34.5161173534\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 23x^{2} + 18x + 48 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 195) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 23x^{2} + 18x + 48 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 19\nu + 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 21\beta _1 + 13 \)
|
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 |
|
−5.05160 | 0 | 17.5186 | −5.00000 | 0 | −5.94377 | −48.0842 | 0 | 25.2580 | ||||||||||||||||||||||||||||||
| 1.2 | −2.17930 | 0 | −3.25067 | −5.00000 | 0 | 19.3758 | 24.5185 | 0 | 10.8965 | |||||||||||||||||||||||||||||||
| 1.3 | 0.876058 | 0 | −7.23252 | −5.00000 | 0 | −32.5617 | −13.3446 | 0 | −4.38029 | |||||||||||||||||||||||||||||||
| 1.4 | 4.35483 | 0 | 10.9646 | −5.00000 | 0 | 23.1297 | 12.9102 | 0 | −21.7742 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(13\) | \(-1\) |
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 | 585.4.a.l | 4 | |
| 3.b | odd | 2 | 1 | 195.4.a.j | ✓ | 4 | |
| 15.d | odd | 2 | 1 | 975.4.a.p | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 195.4.a.j | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 585.4.a.l | 4 | 1.a | even | 1 | 1 | trivial | |
| 975.4.a.p | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(585))\):
|
\( T_{2}^{4} + 2T_{2}^{3} - 23T_{2}^{2} - 30T_{2} + 42 \)
|
|
\( T_{7}^{4} - 4T_{7}^{3} - 995T_{7}^{2} + 9030T_{7} + 86736 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 42 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T + 5)^{4} \)
|
| $7$ |
\( T^{4} - 4 T^{3} + \cdots + 86736 \)
|
| $11$ |
\( T^{4} + 10 T^{3} + \cdots + 641184 \)
|
| $13$ |
\( (T - 13)^{4} \)
|
| $17$ |
\( T^{4} + 212 T^{3} + \cdots - 2787888 \)
|
| $19$ |
\( T^{4} + 46 T^{3} + \cdots + 52740640 \)
|
| $23$ |
\( T^{4} + 104 T^{3} + \cdots - 43137120 \)
|
| $29$ |
\( T^{4} + 202 T^{3} + \cdots + 230383680 \)
|
| $31$ |
\( T^{4} + 60 T^{3} + \cdots - 28191488 \)
|
| $37$ |
\( T^{4} - 802 T^{3} + \cdots + 210462732 \)
|
| $41$ |
\( T^{4} - 258 T^{3} + \cdots + 33743484 \)
|
| $43$ |
\( T^{4} + \cdots - 16706398976 \)
|
| $47$ |
\( T^{4} + \cdots - 3396049920 \)
|
| $53$ |
\( T^{4} + 66 T^{3} + \cdots + 92689596 \)
|
| $59$ |
\( T^{4} + \cdots - 15822710784 \)
|
| $61$ |
\( T^{4} + \cdots + 59927430628 \)
|
| $67$ |
\( T^{4} + \cdots + 1649185024 \)
|
| $71$ |
\( T^{4} + \cdots + 89713589760 \)
|
| $73$ |
\( T^{4} + \cdots - 24626658912 \)
|
| $79$ |
\( T^{4} + \cdots - 12365170688 \)
|
| $83$ |
\( T^{4} + \cdots - 51146053632 \)
|
| $89$ |
\( T^{4} + \cdots + 1234501188 \)
|
| $97$ |
\( T^{4} + \cdots + 184770675720 \)
|
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