Newspace parameters
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 126.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.00611506547\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.309123.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
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
\( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} + 5\nu^{3} + \nu^{2} + 6 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} - 5\nu^{3} + 2\nu^{2} - 3\nu ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 6 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 33\nu - 9 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} - 6\beta _1 + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( -3\beta_{5} - 2\beta_{4} - 11\beta_{3} - 6\beta_{2} + 8\beta _1 + 7 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) |
| \(\chi(n)\) | \(-1 - \beta_{4}\) | \(-1 - \beta_{4}\) |
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
1.00000 | −1.29418 | + | 1.15113i | 1.00000 | −1.84981 | + | 3.20397i | −1.29418 | + | 1.15113i | 2.64400 | + | 0.0963576i | 1.00000 | 0.349814 | − | 2.97954i | −1.84981 | + | 3.20397i | ||||||||||||||||||||||||
| 25.2 | 1.00000 | −0.796790 | − | 1.53790i | 1.00000 | 0.230252 | − | 0.398809i | −0.796790 | − | 1.53790i | 0.0665372 | − | 2.64491i | 1.00000 | −1.73025 | + | 2.45076i | 0.230252 | − | 0.398809i | |||||||||||||||||||||||||
| 25.3 | 1.00000 | 1.09097 | − | 1.34528i | 1.00000 | −0.880438 | + | 1.52496i | 1.09097 | − | 1.34528i | −0.710533 | + | 2.54856i | 1.00000 | −0.619562 | − | 2.93533i | −0.880438 | + | 1.52496i | |||||||||||||||||||||||||
| 121.1 | 1.00000 | −1.29418 | − | 1.15113i | 1.00000 | −1.84981 | − | 3.20397i | −1.29418 | − | 1.15113i | 2.64400 | − | 0.0963576i | 1.00000 | 0.349814 | + | 2.97954i | −1.84981 | − | 3.20397i | |||||||||||||||||||||||||
| 121.2 | 1.00000 | −0.796790 | + | 1.53790i | 1.00000 | 0.230252 | + | 0.398809i | −0.796790 | + | 1.53790i | 0.0665372 | + | 2.64491i | 1.00000 | −1.73025 | − | 2.45076i | 0.230252 | + | 0.398809i | |||||||||||||||||||||||||
| 121.3 | 1.00000 | 1.09097 | + | 1.34528i | 1.00000 | −0.880438 | − | 1.52496i | 1.09097 | + | 1.34528i | −0.710533 | − | 2.54856i | 1.00000 | −0.619562 | + | 2.93533i | −0.880438 | − | 1.52496i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 63.h | even | 3 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 5T_{5}^{5} + 21T_{5}^{4} + 26T_{5}^{3} + 31T_{5}^{2} - 12T_{5} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T - 1)^{6} \)
$3$
\( T^{6} + 2 T^{5} + 4 T^{4} + 3 T^{3} + \cdots + 27 \)
$5$
\( T^{6} + 5 T^{5} + 21 T^{4} + 26 T^{3} + \cdots + 9 \)
$7$
\( T^{6} - 4 T^{5} + 14 T^{4} - 55 T^{3} + \cdots + 343 \)
$11$
\( T^{6} + T^{5} + 27 T^{4} - 92 T^{3} + \cdots + 1089 \)
$13$
\( T^{6} + 2 T^{5} + 7 T^{4} + 15 T^{2} + \cdots + 9 \)
$17$
\( T^{6} + 4 T^{5} + 60 T^{4} + \cdots + 28224 \)
$19$
\( T^{6} + 3 T^{5} + 15 T^{4} - 4 T^{3} + \cdots + 49 \)
$23$
\( T^{6} + 7 T^{5} + 45 T^{4} + 34 T^{3} + \cdots + 9 \)
$29$
\( T^{6} + 5 T^{5} + 57 T^{4} + \cdots + 1089 \)
$31$
\( (T^{3} - 14 T^{2} + 45 T + 27)^{2} \)
$37$
\( T^{6} + 9 T^{5} + 90 T^{4} + \cdots + 5329 \)
$41$
\( T^{6} + 12 T^{5} + 105 T^{4} + \cdots + 729 \)
$43$
\( T^{6} - 18 T^{5} + 243 T^{4} - 1456 T^{3} + \cdots + 1 \)
$47$
\( (T^{3} + 3 T^{2} - 24 T + 27)^{2} \)
$53$
\( T^{6} - 9 T^{5} + 123 T^{4} + 396 T^{3} + \cdots + 81 \)
$59$
\( (T^{3} + 4 T^{2} - 101 T + 177)^{2} \)
$61$
\( (T^{3} + 4 T^{2} - 135 T - 717)^{2} \)
$67$
\( (T^{3} + 5 T^{2} - 58 T - 149)^{2} \)
$71$
\( (T^{3} - 7 T^{2} - 50 T + 99)^{2} \)
$73$
\( T^{6} + 25 T^{5} + 473 T^{4} + \cdots + 2401 \)
$79$
\( (T^{3} + 7 T^{2} - 144 T - 771)^{2} \)
$83$
\( T^{6} - 8 T^{5} + 69 T^{4} + \cdots + 8649 \)
$89$
\( T^{6} + 9 T^{5} + 87 T^{4} + \cdots + 3969 \)
$97$
\( T^{6} + 28 T^{5} + 548 T^{4} + \cdots + 287296 \)
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