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: | 6.6.29995216.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 8x^{4} + 7x^{3} + 12x^{2} - 3x - 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 8x^{4} + 7x^{3} + 12x^{2} - 3x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{4} + 6\nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} + \nu^{3} - 5\nu^{2} - 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + \nu^{3} - 6\nu^{2} - 4\nu + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 7\nu^{3} + \nu^{2} + 8\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6\beta_{4} + 6\beta_{3} - \beta_{2} - \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 8\beta_{4} - \beta_{3} + 7\beta_{2} + 27\beta _1 - 3 \)
|
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 |
|
−2.26764 | −1.00000 | 3.14217 | −3.32110 | 2.26764 | −0.826648 | −2.59003 | 1.00000 | 7.53105 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.07755 | −1.00000 | 2.31621 | 2.29444 | 2.07755 | −0.596213 | −0.656942 | 1.00000 | −4.76681 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.405427 | −1.00000 | −1.83563 | 2.34991 | 0.405427 | 3.06111 | 1.55507 | 1.00000 | −0.952715 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.198272 | −1.00000 | −1.96069 | −0.590280 | −0.198272 | −3.84531 | −0.785293 | 1.00000 | −0.117036 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.05910 | −1.00000 | −0.878305 | 1.30186 | −1.05910 | 1.11490 | −3.04842 | 1.00000 | 1.37880 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.49324 | −1.00000 | 4.21624 | −2.03482 | −2.49324 | 3.09215 | 5.52561 | 1.00000 | −5.07330 | |||||||||||||||||||||||||||||||||||||
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.x | ✓ | 6 |
| 11.b | odd | 2 | 1 | 6171.2.a.z | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6171.2.a.x | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 6171.2.a.z | yes | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6171))\):
|
\( T_{2}^{6} + T_{2}^{5} - 8T_{2}^{4} - 7T_{2}^{3} + 12T_{2}^{2} + 3T_{2} - 1 \)
|
|
\( T_{5}^{6} - 14T_{5}^{4} + 6T_{5}^{3} + 48T_{5}^{2} - 24T_{5} - 28 \)
|
|
\( T_{7}^{6} - 2T_{7}^{5} - 16T_{7}^{4} + 34T_{7}^{3} + 28T_{7}^{2} - 32T_{7} - 20 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 8 T^{4} + \cdots - 1 \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( T^{6} - 14 T^{4} + \cdots - 28 \)
|
| $7$ |
\( T^{6} - 2 T^{5} + \cdots - 20 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} - 6 T^{5} + \cdots + 1889 \)
|
| $17$ |
\( (T + 1)^{6} \)
|
| $19$ |
\( T^{6} - 8 T^{5} + \cdots + 127 \)
|
| $23$ |
\( T^{6} - 6 T^{5} + \cdots + 320 \)
|
| $29$ |
\( T^{6} + 2 T^{5} + \cdots - 44 \)
|
| $31$ |
\( T^{6} + 2 T^{5} + \cdots - 28 \)
|
| $37$ |
\( T^{6} - 2 T^{5} + \cdots - 704 \)
|
| $41$ |
\( T^{6} - 8 T^{5} + \cdots - 256 \)
|
| $43$ |
\( T^{6} - 12 T^{5} + \cdots + 81191 \)
|
| $47$ |
\( T^{6} + 12 T^{5} + \cdots + 3467 \)
|
| $53$ |
\( T^{6} + 14 T^{5} + \cdots - 7792 \)
|
| $59$ |
\( T^{6} + 4 T^{5} + \cdots - 1079552 \)
|
| $61$ |
\( T^{6} - 12 T^{5} + \cdots - 108 \)
|
| $67$ |
\( T^{6} - 16 T^{5} + \cdots + 2663 \)
|
| $71$ |
\( T^{6} - 20 T^{5} + \cdots + 592900 \)
|
| $73$ |
\( T^{6} + 4 T^{5} + \cdots + 19712 \)
|
| $79$ |
\( T^{6} - 24 T^{5} + \cdots + 32960 \)
|
| $83$ |
\( T^{6} + 12 T^{5} + \cdots - 96845 \)
|
| $89$ |
\( T^{6} - 18 T^{5} + \cdots + 72149 \)
|
| $97$ |
\( T^{6} - 296 T^{4} + \cdots - 140308 \)
|
| show more | |
| show less |