Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6566,2,Mod(1,6566)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6566, 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("6566.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6566 = 2 \cdot 7^{2} \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6566.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: | \(52.4297739672\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.98928184.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 12x^{4} + 13x^{3} + 31x^{2} - 25x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 938) |
| 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
\( x^{6} - 2x^{5} - 12x^{4} + 13x^{3} + 31x^{2} - 25x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 3\nu^{4} + 9\nu^{3} - 18\nu^{2} - 21\nu + 18 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 3\nu^{4} + 7\nu^{3} - 14\nu^{2} - 5\nu + 6 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{4} + 7\nu^{3} + 34\nu^{2} - 55\nu + 2 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{5} + 13\nu^{4} + 7\nu^{3} - 58\nu^{2} + 37\nu - 6 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} - \beta_{3} + 2\beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} + 2\beta_{4} - 3\beta_{3} + 2\beta_{2} + 12\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 12\beta_{5} + 11\beta_{4} - 16\beta_{3} + 7\beta_{2} + 37\beta _1 + 29 \)
|
| \(\nu^{5}\) | \(=\) |
\( 36\beta_{5} + 33\beta_{4} - 57\beta_{3} + 35\beta_{2} + 162\beta _1 + 51 \)
|
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 |
|
−1.00000 | −2.81447 | 1.00000 | 1.97031 | 2.81447 | 0 | −1.00000 | 4.92126 | −1.97031 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −0.791447 | 1.00000 | 0.635163 | 0.791447 | 0 | −1.00000 | −2.37361 | −0.635163 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0.165042 | 1.00000 | −2.99551 | −0.165042 | 0 | −1.00000 | −2.97276 | 2.99551 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.07352 | 1.00000 | 0.885600 | −1.07352 | 0 | −1.00000 | −1.84756 | −0.885600 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 3.16097 | 1.00000 | −3.97835 | −3.16097 | 0 | −1.00000 | 6.99172 | 3.97835 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 3.20639 | 1.00000 | 3.48279 | −3.20639 | 0 | −1.00000 | 7.28095 | −3.48279 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
| \(67\) | \(1\) |
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 | 6566.2.a.bk | 6 | |
| 7.b | odd | 2 | 1 | 938.2.a.i | ✓ | 6 | |
| 21.c | even | 2 | 1 | 8442.2.a.bg | 6 | ||
| 28.d | even | 2 | 1 | 7504.2.a.bo | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 938.2.a.i | ✓ | 6 | 7.b | odd | 2 | 1 | |
| 6566.2.a.bk | 6 | 1.a | even | 1 | 1 | trivial | |
| 7504.2.a.bo | 6 | 28.d | even | 2 | 1 | ||
| 8442.2.a.bg | 6 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6566))\):
|
\( T_{3}^{6} - 4T_{3}^{5} - 7T_{3}^{4} + 35T_{3}^{3} - 7T_{3}^{2} - 24T_{3} + 4 \)
|
|
\( T_{5}^{6} - 21T_{5}^{4} + 13T_{5}^{3} + 97T_{5}^{2} - 134T_{5} + 46 \)
|
|
\( T_{11}^{6} + 5T_{11}^{5} - 30T_{11}^{4} - 177T_{11}^{3} + 4T_{11}^{2} + 652T_{11} - 64 \)
|
|
\( T_{13}^{6} - 12T_{13}^{5} + 13T_{13}^{4} + 307T_{13}^{3} - 1189T_{13}^{2} + 910T_{13} + 794 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
|
| $3$ |
\( T^{6} - 4 T^{5} + \cdots + 4 \)
|
| $5$ |
\( T^{6} - 21 T^{4} + \cdots + 46 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + 5 T^{5} + \cdots - 64 \)
|
| $13$ |
\( T^{6} - 12 T^{5} + \cdots + 794 \)
|
| $17$ |
\( T^{6} + T^{5} + \cdots - 2176 \)
|
| $19$ |
\( T^{6} - 14 T^{5} + \cdots + 27904 \)
|
| $23$ |
\( T^{6} + 2 T^{5} + \cdots + 11384 \)
|
| $29$ |
\( T^{6} - 13 T^{5} + \cdots + 16064 \)
|
| $31$ |
\( T^{6} - 19 T^{5} + \cdots + 17600 \)
|
| $37$ |
\( T^{6} + 15 T^{5} + \cdots - 12256 \)
|
| $41$ |
\( T^{6} + 9 T^{5} + \cdots + 16 \)
|
| $43$ |
\( T^{6} + 21 T^{5} + \cdots + 2224 \)
|
| $47$ |
\( T^{6} - 3 T^{5} + \cdots + 13028 \)
|
| $53$ |
\( T^{6} - 21 T^{5} + \cdots + 266984 \)
|
| $59$ |
\( T^{6} - 10 T^{5} + \cdots - 14944 \)
|
| $61$ |
\( T^{6} - 25 T^{5} + \cdots + 216100 \)
|
| $67$ |
\( (T + 1)^{6} \)
|
| $71$ |
\( T^{6} + 14 T^{5} + \cdots - 25904 \)
|
| $73$ |
\( T^{6} - 29 T^{5} + \cdots - 8264 \)
|
| $79$ |
\( T^{6} + 18 T^{5} + \cdots - 214208 \)
|
| $83$ |
\( T^{6} + 6 T^{5} + \cdots + 49376 \)
|
| $89$ |
\( T^{6} - 7 T^{5} + \cdots + 53856 \)
|
| $97$ |
\( T^{6} - 34 T^{5} + \cdots - 29088 \)
|
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