Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1210,4,Mod(1,1210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1210, 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("1210.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1210 = 2 \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1210.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: | \(71.3923111069\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.300520.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 67x - 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 67x - 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 45 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 45 \)
|
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 | −9.22977 | 4.00000 | 5.00000 | 18.4595 | 5.22977 | −8.00000 | 58.1886 | −10.0000 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | −0.910437 | 4.00000 | 5.00000 | 1.82087 | −3.08956 | −8.00000 | −26.1711 | −10.0000 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 7.14020 | 4.00000 | 5.00000 | −14.2804 | −11.1402 | −8.00000 | 23.9825 | −10.0000 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( +1 \) |
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 | 1210.4.a.t | ✓ | 3 |
| 11.b | odd | 2 | 1 | 1210.4.a.w | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1210.4.a.t | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1210.4.a.w | yes | 3 | 11.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1210))\):
|
\( T_{3}^{3} + 3T_{3}^{2} - 64T_{3} - 60 \)
|
|
\( T_{7}^{3} + 9T_{7}^{2} - 40T_{7} - 180 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( T^{3} + 3 T^{2} + \cdots - 60 \)
|
| $5$ |
\( (T - 5)^{3} \)
|
| $7$ |
\( T^{3} + 9 T^{2} + \cdots - 180 \)
|
| $11$ |
\( T^{3} \)
|
| $13$ |
\( T^{3} + 8 T^{2} + \cdots - 2112 \)
|
| $17$ |
\( T^{3} - 25 T^{2} + \cdots - 49896 \)
|
| $19$ |
\( T^{3} + 105 T^{2} + \cdots - 915600 \)
|
| $23$ |
\( T^{3} - 44 T^{2} + \cdots - 116160 \)
|
| $29$ |
\( T^{3} + 345 T^{2} + \cdots - 10344000 \)
|
| $31$ |
\( T^{3} - 73 T^{2} + \cdots - 3835280 \)
|
| $37$ |
\( T^{3} - 197 T^{2} + \cdots + 2948220 \)
|
| $41$ |
\( T^{3} - 226 T^{2} + \cdots + 17404320 \)
|
| $43$ |
\( T^{3} - 80 T^{2} + \cdots - 1688688 \)
|
| $47$ |
\( T^{3} + 146 T^{2} + \cdots + 22683240 \)
|
| $53$ |
\( T^{3} - 421 T^{2} + \cdots + 30296700 \)
|
| $59$ |
\( T^{3} - 516 T^{2} + \cdots + 8726400 \)
|
| $61$ |
\( T^{3} + 1415 T^{2} + \cdots + 51519600 \)
|
| $67$ |
\( T^{3} + 136 T^{2} + \cdots + 8191760 \)
|
| $71$ |
\( T^{3} + 1457 T^{2} + \cdots + 39605760 \)
|
| $73$ |
\( T^{3} + 578 T^{2} + \cdots - 39298032 \)
|
| $79$ |
\( T^{3} + 364 T^{2} + \cdots - 39787200 \)
|
| $83$ |
\( T^{3} + 896 T^{2} + \cdots - 424970352 \)
|
| $89$ |
\( T^{3} + 107 T^{2} + \cdots - 14720580 \)
|
| $97$ |
\( T^{3} - 2290 T^{2} + \cdots + 831400 \)
|
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