Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [931,2,Mod(1,931)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(931, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("931.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 931 = 7^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 931.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: | \(7.43407242818\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.5744.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 5x^{2} - 2x + 1 \)
|
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 5x^{2} - 2x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 5\nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + \nu^{2} + 4\nu - 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 5\beta _1 + 1 \)
|
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 |
|
−1.92022 | −1.52077 | 1.68725 | 2.00692 | 2.92022 | 0 | 0.600553 | −0.687248 | −3.85372 | ||||||||||||||||||||||||||||||
| 1.2 | −0.751024 | −2.33152 | −1.43596 | −4.26543 | 1.75102 | 0 | 2.58049 | 2.43596 | 3.20344 | |||||||||||||||||||||||||||||||
| 1.3 | 0.291367 | 2.43210 | −1.91511 | −2.06574 | 0.708633 | 0 | −1.14073 | 2.91511 | −0.601888 | |||||||||||||||||||||||||||||||
| 1.4 | 2.37988 | −0.579810 | 3.66382 | −3.67575 | −1.37988 | 0 | 3.95969 | −2.66382 | −8.74783 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 931.2.a.l | ✓ | 4 |
| 3.b | odd | 2 | 1 | 8379.2.a.bv | 4 | ||
| 7.b | odd | 2 | 1 | 931.2.a.m | yes | 4 | |
| 7.c | even | 3 | 2 | 931.2.f.o | 8 | ||
| 7.d | odd | 6 | 2 | 931.2.f.n | 8 | ||
| 21.c | even | 2 | 1 | 8379.2.a.bu | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 931.2.a.l | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 931.2.a.m | yes | 4 | 7.b | odd | 2 | 1 | |
| 931.2.f.n | 8 | 7.d | odd | 6 | 2 | ||
| 931.2.f.o | 8 | 7.c | even | 3 | 2 | ||
| 8379.2.a.bu | 4 | 21.c | even | 2 | 1 | ||
| 8379.2.a.bv | 4 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(931))\):
|
\( T_{2}^{4} - 5T_{2}^{2} - 2T_{2} + 1 \)
|
|
\( T_{3}^{4} + 2T_{3}^{3} - 5T_{3}^{2} - 12T_{3} - 5 \)
|
|
\( T_{5}^{4} + 8T_{5}^{3} + 12T_{5}^{2} - 32T_{5} - 65 \)
|
|
\( T_{13}^{4} + 2T_{13}^{3} - 20T_{13}^{2} + 26T_{13} - 5 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 5 T^{2} + \cdots + 1 \)
|
| $3$ |
\( T^{4} + 2 T^{3} + \cdots - 5 \)
|
| $5$ |
\( T^{4} + 8 T^{3} + \cdots - 65 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 6 T^{3} + \cdots - 59 \)
|
| $13$ |
\( T^{4} + 2 T^{3} + \cdots - 5 \)
|
| $17$ |
\( T^{4} + 8 T^{3} + \cdots - 125 \)
|
| $19$ |
\( (T - 1)^{4} \)
|
| $23$ |
\( T^{4} + 8 T^{3} + \cdots - 59 \)
|
| $29$ |
\( T^{4} - 2 T^{3} + \cdots - 191 \)
|
| $31$ |
\( T^{4} - 27 T^{2} + \cdots + 35 \)
|
| $37$ |
\( T^{4} - 10 T^{3} + \cdots - 691 \)
|
| $41$ |
\( T^{4} + 12 T^{3} + \cdots + 155 \)
|
| $43$ |
\( T^{4} - 4 T^{3} + \cdots - 448 \)
|
| $47$ |
\( T^{4} + 16 T^{3} + \cdots - 7945 \)
|
| $53$ |
\( T^{4} - 12 T^{3} + \cdots + 65 \)
|
| $59$ |
\( T^{4} + 14 T^{3} + \cdots - 305 \)
|
| $61$ |
\( T^{4} + 20 T^{3} + \cdots - 685 \)
|
| $67$ |
\( T^{4} - 2 T^{3} + \cdots + 2293 \)
|
| $71$ |
\( T^{4} + 2 T^{3} + \cdots + 3569 \)
|
| $73$ |
\( T^{4} - 16 T^{3} + \cdots - 305 \)
|
| $79$ |
\( T^{4} + 8 T^{3} + \cdots - 16 \)
|
| $83$ |
\( T^{4} + 20 T^{3} + \cdots + 4555 \)
|
| $89$ |
\( T^{4} + 16 T^{3} + \cdots - 2240 \)
|
| $97$ |
\( T^{4} + 2 T^{3} + \cdots + 815 \)
|
| show more | |
| show less |