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: | 3.3.515892.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 55x - 59 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{3} - x^{2} - 55x - 59 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3\nu - 36 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3\beta _1 + 36 \)
|
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 | −6.27060 | 4.00000 | −5.00000 | 12.5412 | −29.1322 | −8.00000 | 12.3204 | 10.0000 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | −1.12121 | 4.00000 | −5.00000 | 2.24242 | 24.3793 | −8.00000 | −25.7429 | 10.0000 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 8.39181 | 4.00000 | −5.00000 | −16.7836 | −16.2470 | −8.00000 | 43.4225 | 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.u | ✓ | 3 |
| 11.b | odd | 2 | 1 | 1210.4.a.x | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1210.4.a.u | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1210.4.a.x | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1210))\):
|
\( T_{3}^{3} - T_{3}^{2} - 55T_{3} - 59 \)
|
|
\( T_{7}^{3} + 21T_{7}^{2} - 633T_{7} - 11539 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( T^{3} - T^{2} + \cdots - 59 \)
|
| $5$ |
\( (T + 5)^{3} \)
|
| $7$ |
\( T^{3} + 21 T^{2} + \cdots - 11539 \)
|
| $11$ |
\( T^{3} \)
|
| $13$ |
\( T^{3} - 34 T^{2} + \cdots + 16540 \)
|
| $17$ |
\( T^{3} - 44 T^{2} + \cdots + 328848 \)
|
| $19$ |
\( T^{3} + 200 T^{2} + \cdots + 181588 \)
|
| $23$ |
\( T^{3} + 54 T^{2} + \cdots - 18540 \)
|
| $29$ |
\( T^{3} + 14 T^{2} + \cdots + 3874728 \)
|
| $31$ |
\( T^{3} + 10 T^{2} + \cdots + 109364 \)
|
| $37$ |
\( T^{3} - 276 T^{2} + \cdots - 503120 \)
|
| $41$ |
\( T^{3} + 659 T^{2} + \cdots + 9861105 \)
|
| $43$ |
\( T^{3} + 169 T^{2} + \cdots - 23484747 \)
|
| $47$ |
\( T^{3} - 105 T^{2} + \cdots + 33293025 \)
|
| $53$ |
\( T^{3} - 182 T^{2} + \cdots + 2107116 \)
|
| $59$ |
\( T^{3} - 710 T^{2} + \cdots + 18346020 \)
|
| $61$ |
\( T^{3} - 99 T^{2} + \cdots + 155427107 \)
|
| $67$ |
\( T^{3} + 393 T^{2} + \cdots - 116811617 \)
|
| $71$ |
\( T^{3} - 506 T^{2} + \cdots + 666754308 \)
|
| $73$ |
\( T^{3} - 1252 T^{2} + \cdots - 51343136 \)
|
| $79$ |
\( T^{3} + 238 T^{2} + \cdots - 172684200 \)
|
| $83$ |
\( T^{3} + 1774 T^{2} + \cdots + 178650132 \)
|
| $89$ |
\( T^{3} - 627 T^{2} + \cdots - 20910177 \)
|
| $97$ |
\( T^{3} - 266 T^{2} + \cdots - 151931332 \)
|
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