Newspace parameters
| Level: | \( N \) | \(=\) | \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1260.de (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.628821915918\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{6}\) |
| Projective field: | Galois closure of 6.2.661624362000.11 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).
| \(n\) | \(281\) | \(631\) | \(757\) | \(1081\) |
| \(\chi(n)\) | \(\zeta_{12}^{2}\) | \(-1\) | \(-1\) | \(-\zeta_{12}^{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 59.1 |
|
−0.866025 | − | 0.500000i | −0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | −1.00000 | 0.500000 | + | 0.866025i | 0.866025 | − | 0.500000i | − | 1.00000i | 0.500000 | + | 0.866025i | 0.866025 | + | 0.500000i | |||||||||||||||
| 59.2 | 0.866025 | + | 0.500000i | 0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | −1.00000 | 0.500000 | + | 0.866025i | −0.866025 | + | 0.500000i | 1.00000i | 0.500000 | + | 0.866025i | −0.866025 | − | 0.500000i | |||||||||||||||||
| 299.1 | −0.866025 | + | 0.500000i | −0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | −1.00000 | 0.500000 | − | 0.866025i | 0.866025 | + | 0.500000i | 1.00000i | 0.500000 | − | 0.866025i | 0.866025 | − | 0.500000i | |||||||||||||||||
| 299.2 | 0.866025 | − | 0.500000i | 0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | −1.00000 | 0.500000 | − | 0.866025i | −0.866025 | − | 0.500000i | − | 1.00000i | 0.500000 | − | 0.866025i | −0.866025 | + | 0.500000i | ||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 20.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-5}) \) |
| 4.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 63.s | even | 6 | 1 | inner |
| 252.bn | odd | 6 | 1 | inner |
| 315.u | even | 6 | 1 | inner |
| 1260.de | odd | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{23}^{2} + 4 \)
acting on \(S_{1}^{\mathrm{new}}(1260, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - T^{2} + 1 \)
$3$
\( T^{4} - T^{2} + 1 \)
$5$
\( (T + 1)^{4} \)
$7$
\( T^{4} - T^{2} + 1 \)
$11$
\( T^{4} \)
$13$
\( T^{4} \)
$17$
\( T^{4} \)
$19$
\( T^{4} \)
$23$
\( (T^{2} + 4)^{2} \)
$29$
\( (T^{2} - 3 T + 3)^{2} \)
$31$
\( T^{4} \)
$37$
\( T^{4} \)
$41$
\( (T^{2} - T + 1)^{2} \)
$43$
\( T^{4} + 3T^{2} + 9 \)
$47$
\( T^{4} + 3T^{2} + 9 \)
$53$
\( T^{4} \)
$59$
\( T^{4} \)
$61$
\( T^{4} \)
$67$
\( T^{4} \)
$71$
\( T^{4} \)
$73$
\( T^{4} \)
$79$
\( T^{4} \)
$83$
\( T^{4} + 3T^{2} + 9 \)
$89$
\( (T^{2} + 2 T + 4)^{2} \)
$97$
\( T^{4} \)
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