Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [585,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("585.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 585 = 3^{2} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(93.8245345906\) |
| 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: |
\( x^{4} - 73x^{2} - 104x + 368 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{4} - 73x^{2} - 104x + 368 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 59\nu - 4 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{2} - 3\nu - 36 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{3} + 3\beta _1 + 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{3} + 4\beta_{2} + 65\beta _1 + 76 \)
|
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 |
|
−5.94579 | 0 | 3.35241 | −25.0000 | 0 | −85.2494 | 170.333 | 0 | 148.645 | ||||||||||||||||||||||||||||||
| 1.2 | 1.33454 | 0 | −30.2190 | −25.0000 | 0 | 21.2190 | −83.0335 | 0 | −33.3634 | |||||||||||||||||||||||||||||||
| 1.3 | 6.44825 | 0 | 9.57987 | −25.0000 | 0 | −186.275 | −144.570 | 0 | −161.206 | |||||||||||||||||||||||||||||||
| 1.4 | 10.1630 | 0 | 71.2867 | −25.0000 | 0 | 177.305 | 399.271 | 0 | −254.075 | |||||||||||||||||||||||||||||||
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 | 585.6.a.h | 4 | |
| 3.b | odd | 2 | 1 | 195.6.a.b | ✓ | 4 | |
| 15.d | odd | 2 | 1 | 975.6.a.i | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 195.6.a.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 585.6.a.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 975.6.a.i | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 12T_{2}^{3} - 19T_{2}^{2} + 434T_{2} - 520 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(585))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 12 T^{3} + \cdots - 520 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T + 25)^{4} \)
|
| $7$ |
\( T^{4} + 73 T^{3} + \cdots + 59743472 \)
|
| $11$ |
\( T^{4} + \cdots - 25061186176 \)
|
| $13$ |
\( (T + 169)^{4} \)
|
| $17$ |
\( T^{4} + \cdots - 2802645020656 \)
|
| $19$ |
\( T^{4} + \cdots + 1689434978400 \)
|
| $23$ |
\( T^{4} + \cdots + 230304798000 \)
|
| $29$ |
\( T^{4} + \cdots - 469988477120 \)
|
| $31$ |
\( T^{4} + \cdots + 35599249544832 \)
|
| $37$ |
\( T^{4} + \cdots + 535784181287112 \)
|
| $41$ |
\( T^{4} + \cdots - 58\!\cdots\!84 \)
|
| $43$ |
\( T^{4} + \cdots + 19\!\cdots\!76 \)
|
| $47$ |
\( T^{4} + \cdots - 81\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 51\!\cdots\!44 \)
|
| $59$ |
\( T^{4} + \cdots + 30\!\cdots\!16 \)
|
| $61$ |
\( T^{4} + \cdots + 46\!\cdots\!52 \)
|
| $67$ |
\( T^{4} + \cdots + 30\!\cdots\!64 \)
|
| $71$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots - 39\!\cdots\!52 \)
|
| $79$ |
\( T^{4} + \cdots + 31\!\cdots\!96 \)
|
| $83$ |
\( T^{4} + \cdots - 10\!\cdots\!08 \)
|
| $89$ |
\( T^{4} + \cdots - 30\!\cdots\!40 \)
|
| $97$ |
\( T^{4} + \cdots - 82\!\cdots\!00 \)
|
| show more | |
| show less |