Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [810,4,Mod(1,810)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(810, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("810.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 810 = 2 \cdot 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 810.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: | \(47.7915471046\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.3732.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 13x + 19 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 90) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 13x + 19 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 3\nu^{2} + 3\nu - 28 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - \beta _1 + 27 ) / 3 \)
|
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 |
|
−2.00000 | 0 | 4.00000 | 5.00000 | 0 | −22.3190 | −8.00000 | 0 | −10.0000 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | 0 | 4.00000 | 5.00000 | 0 | −3.77733 | −8.00000 | 0 | −10.0000 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0 | 4.00000 | 5.00000 | 0 | 29.0963 | −8.00000 | 0 | −10.0000 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
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 | 810.4.a.q | 3 | |
| 3.b | odd | 2 | 1 | 810.4.a.r | 3 | ||
| 9.c | even | 3 | 2 | 90.4.e.d | ✓ | 6 | |
| 9.d | odd | 6 | 2 | 270.4.e.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 90.4.e.d | ✓ | 6 | 9.c | even | 3 | 2 | |
| 270.4.e.d | 6 | 9.d | odd | 6 | 2 | ||
| 810.4.a.q | 3 | 1.a | even | 1 | 1 | trivial | |
| 810.4.a.r | 3 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(810))\):
|
\( T_{7}^{3} - 3T_{7}^{2} - 675T_{7} - 2453 \)
|
|
\( T_{11}^{3} + 18T_{11}^{2} - 216T_{11} - 540 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( (T - 5)^{3} \)
|
| $7$ |
\( T^{3} - 3 T^{2} + \cdots - 2453 \)
|
| $11$ |
\( T^{3} + 18 T^{2} + \cdots - 540 \)
|
| $13$ |
\( T^{3} - 36 T^{2} + \cdots - 11324 \)
|
| $17$ |
\( T^{3} + 66 T^{2} + \cdots - 67716 \)
|
| $19$ |
\( T^{3} + 84 T^{2} + \cdots - 473444 \)
|
| $23$ |
\( T^{3} + 159 T^{2} + \cdots - 64395 \)
|
| $29$ |
\( T^{3} + 351 T^{2} + \cdots - 11864259 \)
|
| $31$ |
\( T^{3} - 48 T^{2} + \cdots + 7183660 \)
|
| $37$ |
\( T^{3} - 30 T^{2} + \cdots + 16733596 \)
|
| $41$ |
\( T^{3} + 777 T^{2} + \cdots + 10883619 \)
|
| $43$ |
\( T^{3} + 18 T^{2} + \cdots - 4382228 \)
|
| $47$ |
\( T^{3} + 465 T^{2} + \cdots - 26373519 \)
|
| $53$ |
\( T^{3} + 354 T^{2} + \cdots + 903528 \)
|
| $59$ |
\( T^{3} + 894 T^{2} + \cdots - 135998136 \)
|
| $61$ |
\( T^{3} - 837 T^{2} + \cdots + 17030329 \)
|
| $67$ |
\( T^{3} - 1293 T^{2} + \cdots + 108584659 \)
|
| $71$ |
\( T^{3} + 1206 T^{2} + \cdots - 103343796 \)
|
| $73$ |
\( T^{3} - 840 T^{2} + \cdots + 215989888 \)
|
| $79$ |
\( T^{3} - 354 T^{2} + \cdots - 74791832 \)
|
| $83$ |
\( T^{3} + 1245 T^{2} + \cdots + 63872631 \)
|
| $89$ |
\( T^{3} + 465 T^{2} + \cdots - 373653405 \)
|
| $97$ |
\( T^{3} - 294 T^{2} + \cdots + 126830152 \)
|
| show more | |
| show less |