Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [861,2,Mod(1,861)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(861, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("861.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 861 = 3 \cdot 7 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 861.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: | \(6.87511961403\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.148.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 3x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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
\( x^{3} - x^{2} - 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
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 |
|
−2.17009 | −1.00000 | 2.70928 | 2.70928 | 2.17009 | 1.00000 | −1.53919 | 1.00000 | −5.87936 | |||||||||||||||||||||||||||
| 1.2 | −0.311108 | −1.00000 | −1.90321 | −1.90321 | 0.311108 | 1.00000 | 1.21432 | 1.00000 | 0.592104 | ||||||||||||||||||||||||||||
| 1.3 | 1.48119 | −1.00000 | 0.193937 | 0.193937 | −1.48119 | 1.00000 | −2.67513 | 1.00000 | 0.287258 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(41\) | \(-1\) |
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 | 861.2.a.h | ✓ | 3 |
| 3.b | odd | 2 | 1 | 2583.2.a.n | 3 | ||
| 7.b | odd | 2 | 1 | 6027.2.a.o | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 861.2.a.h | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 2583.2.a.n | 3 | 3.b | odd | 2 | 1 | ||
| 6027.2.a.o | 3 | 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(861))\):
|
\( T_{2}^{3} + T_{2}^{2} - 3T_{2} - 1 \)
|
|
\( T_{5}^{3} - T_{5}^{2} - 5T_{5} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + T^{2} - 3T - 1 \)
|
| $3$ |
\( (T + 1)^{3} \)
|
| $5$ |
\( T^{3} - T^{2} - 5T + 1 \)
|
| $7$ |
\( (T - 1)^{3} \)
|
| $11$ |
\( T^{3} + 6 T^{2} + \cdots - 40 \)
|
| $13$ |
\( T^{3} + 7T^{2} + T - 43 \)
|
| $17$ |
\( T^{3} - 10 T^{2} + \cdots + 4 \)
|
| $19$ |
\( T^{3} + 16 T^{2} + \cdots + 92 \)
|
| $23$ |
\( T^{3} + 3 T^{2} + \cdots - 227 \)
|
| $29$ |
\( T^{3} + 7 T^{2} + \cdots - 37 \)
|
| $31$ |
\( T^{3} - 2 T^{2} + \cdots + 4 \)
|
| $37$ |
\( T^{3} + 17 T^{2} + \cdots + 151 \)
|
| $41$ |
\( (T - 1)^{3} \)
|
| $43$ |
\( T^{3} + 18 T^{2} + \cdots - 92 \)
|
| $47$ |
\( T^{3} - T^{2} - 9T + 13 \)
|
| $53$ |
\( T^{3} + T^{2} + \cdots - 25 \)
|
| $59$ |
\( T^{3} + 8 T^{2} + \cdots - 244 \)
|
| $61$ |
\( T^{3} - 8 T^{2} + \cdots - 4 \)
|
| $67$ |
\( T^{3} + 13 T^{2} + \cdots - 653 \)
|
| $71$ |
\( T^{3} + 8 T^{2} + \cdots - 1156 \)
|
| $73$ |
\( T^{3} + 6 T^{2} + \cdots - 740 \)
|
| $79$ |
\( T^{3} + 17 T^{2} + \cdots - 29 \)
|
| $83$ |
\( T^{3} - 10 T^{2} + \cdots + 856 \)
|
| $89$ |
\( T^{3} - 8 T^{2} + \cdots - 256 \)
|
| $97$ |
\( T^{3} + T^{2} + \cdots - 23 \)
|
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