Newspace parameters
| Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 378.h (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(3.01834519640\) |
| 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: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 126) |
| 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
\( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - \nu + 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(325\) |
| \(\chi(n)\) | \(-1 + \beta_{4}\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 289.1 |
|
−0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | −1.58836 | 0 | −2.64400 | + | 0.0963576i | 1.00000 | 0 | 0.794182 | + | 1.37556i | ||||||||||||||||||||||||||||
| 289.2 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.593579 | 0 | −0.0665372 | − | 2.64491i | 1.00000 | 0 | 0.296790 | + | 0.514055i | |||||||||||||||||||||||||||||
| 289.3 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | 3.18194 | 0 | 0.710533 | + | 2.54856i | 1.00000 | 0 | −1.59097 | − | 2.75564i | |||||||||||||||||||||||||||||
| 361.1 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −1.58836 | 0 | −2.64400 | − | 0.0963576i | 1.00000 | 0 | 0.794182 | − | 1.37556i | |||||||||||||||||||||||||||||
| 361.2 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −0.593579 | 0 | −0.0665372 | + | 2.64491i | 1.00000 | 0 | 0.296790 | − | 0.514055i | |||||||||||||||||||||||||||||
| 361.3 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 3.18194 | 0 | 0.710533 | − | 2.54856i | 1.00000 | 0 | −1.59097 | + | 2.75564i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 63.g | even | 3 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - T_{5}^{2} - 6T_{5} - 3 \)
acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + T + 1)^{3} \)
$3$
\( T^{6} \)
$5$
\( (T^{3} - T^{2} - 6 T - 3)^{2} \)
$7$
\( T^{6} + 4 T^{5} + 14 T^{4} + 55 T^{3} + \cdots + 343 \)
$11$
\( (T^{3} + T^{2} - 6 T + 3)^{2} \)
$13$
\( T^{6} - 8 T^{5} + 63 T^{4} + \cdots + 4761 \)
$17$
\( T^{6} - 4 T^{5} + 28 T^{4} + 240 T^{2} + \cdots + 576 \)
$19$
\( T^{6} + 3 T^{5} + 45 T^{4} + \cdots + 2401 \)
$23$
\( (T^{3} + 7 T^{2} + 12 T + 3)^{2} \)
$29$
\( T^{6} - 5 T^{5} + 91 T^{4} + \cdots + 131769 \)
$31$
\( T^{6} - 20 T^{5} + 279 T^{4} + \cdots + 40401 \)
$37$
\( (T^{2} - T + 1)^{3} \)
$41$
\( T^{6} + 33 T^{4} - 18 T^{3} + 1089 T^{2} + \cdots + 81 \)
$43$
\( T^{6} + 6 T^{5} + 105 T^{4} + \cdots + 16129 \)
$47$
\( T^{6} - 9 T^{5} + 135 T^{4} + \cdots + 35721 \)
$53$
\( T^{6} + 15 T^{5} + 159 T^{4} + \cdots + 6561 \)
$59$
\( T^{6} - 14 T^{5} + 157 T^{4} + \cdots + 3969 \)
$61$
\( T^{6} - 8 T^{5} + 69 T^{4} + \cdots + 8649 \)
$67$
\( T^{6} - T^{5} + 113 T^{4} + \cdots + 44521 \)
$71$
\( (T^{3} + 7 T^{2} - 198 T - 1593)^{2} \)
$73$
\( T^{6} - 19 T^{5} + 353 T^{4} + \cdots + 398161 \)
$79$
\( T^{6} - 5 T^{5} + 99 T^{4} + \cdots + 103041 \)
$83$
\( T^{6} + 2 T^{5} + 67 T^{4} + \cdots + 21609 \)
$89$
\( T^{6} - 9 T^{5} + 123 T^{4} + 396 T^{3} + \cdots + 81 \)
$97$
\( T^{6} - 28 T^{5} + 572 T^{4} + \cdots + 61504 \)
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