Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6171,2,Mod(1,6171)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6171, 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("6171.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6171 = 3 \cdot 11^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6171.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: | \(49.2756830873\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.78067472.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 11x^{4} + 9x^{3} + 32x^{2} - 20x - 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 561) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 11x^{4} + 9x^{3} + 32x^{2} - 20x - 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 9\nu^{3} - 2\nu^{2} + 16\nu + 6 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 7\nu^{2} + 5\nu + 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{3} + 8\beta_{2} + 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{4} + 9\beta_{3} + 11\beta_{2} + 29\beta _1 + 2 \)
|
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 |
|
−2.46313 | 1.00000 | 4.06699 | −0.856910 | −2.46313 | 1.06699 | −5.09126 | 1.00000 | 2.11068 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.95170 | 1.00000 | 1.80914 | 2.87246 | −1.95170 | −1.19086 | 0.372510 | 1.00000 | −5.60618 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.543722 | 1.00000 | −1.70437 | −0.945829 | −0.543722 | −4.70437 | 2.01415 | 1.00000 | 0.514268 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.32296 | 1.00000 | −0.249770 | 3.44009 | 1.32296 | −3.24977 | −2.97636 | 1.00000 | 4.55111 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.90934 | 1.00000 | 1.64556 | −4.00593 | 1.90934 | −1.35444 | −0.676741 | 1.00000 | −7.64867 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.72625 | 1.00000 | 5.43245 | 1.49612 | 2.72625 | 2.43245 | 9.35771 | 1.00000 | 4.07879 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(11\) | \(-1\) |
| \(17\) | \(-1\) |
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 | 6171.2.a.bc | 6 | |
| 11.b | odd | 2 | 1 | 561.2.a.k | ✓ | 6 | |
| 33.d | even | 2 | 1 | 1683.2.a.bc | 6 | ||
| 44.c | even | 2 | 1 | 8976.2.a.cm | 6 | ||
| 187.b | odd | 2 | 1 | 9537.2.a.bg | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 561.2.a.k | ✓ | 6 | 11.b | odd | 2 | 1 | |
| 1683.2.a.bc | 6 | 33.d | even | 2 | 1 | ||
| 6171.2.a.bc | 6 | 1.a | even | 1 | 1 | trivial | |
| 8976.2.a.cm | 6 | 44.c | even | 2 | 1 | ||
| 9537.2.a.bg | 6 | 187.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6171))\):
|
\( T_{2}^{6} - T_{2}^{5} - 11T_{2}^{4} + 9T_{2}^{3} + 32T_{2}^{2} - 20T_{2} - 18 \)
|
|
\( T_{5}^{6} - 2T_{5}^{5} - 18T_{5}^{4} + 38T_{5}^{3} + 44T_{5}^{2} - 56T_{5} - 48 \)
|
|
\( T_{7}^{6} + 7T_{7}^{5} + 3T_{7}^{4} - 51T_{7}^{3} - 60T_{7}^{2} + 48T_{7} + 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - T^{5} + \cdots - 18 \)
|
| $3$ |
\( (T - 1)^{6} \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots - 48 \)
|
| $7$ |
\( T^{6} + 7 T^{5} + \cdots + 64 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} + 8 T^{5} + \cdots + 2608 \)
|
| $17$ |
\( (T - 1)^{6} \)
|
| $19$ |
\( T^{6} + 12 T^{5} + \cdots - 256 \)
|
| $23$ |
\( T^{6} \)
|
| $29$ |
\( T^{6} - 7 T^{5} + \cdots + 4536 \)
|
| $31$ |
\( T^{6} - 14 T^{5} + \cdots + 1024 \)
|
| $37$ |
\( T^{6} - 16 T^{5} + \cdots + 16 \)
|
| $41$ |
\( T^{6} + 3 T^{5} + \cdots - 101688 \)
|
| $43$ |
\( T^{6} + 8 T^{5} + \cdots + 32 \)
|
| $47$ |
\( T^{6} - 13 T^{5} + \cdots + 136176 \)
|
| $53$ |
\( T^{6} - 5 T^{5} + \cdots - 1896 \)
|
| $59$ |
\( T^{6} - 13 T^{5} + \cdots - 16176 \)
|
| $61$ |
\( T^{6} - 16 T^{5} + \cdots + 34976 \)
|
| $67$ |
\( T^{6} - 21 T^{5} + \cdots + 28736 \)
|
| $71$ |
\( T^{6} - 8 T^{5} + \cdots + 768 \)
|
| $73$ |
\( T^{6} + 3 T^{5} + \cdots + 10328 \)
|
| $79$ |
\( T^{6} - 4 T^{5} + \cdots + 432896 \)
|
| $83$ |
\( T^{6} - 6 T^{5} + \cdots - 64128 \)
|
| $89$ |
\( T^{6} - 5 T^{5} + \cdots + 155304 \)
|
| $97$ |
\( T^{6} - 14 T^{5} + \cdots + 9008 \)
|
| show more | |
| show less |