Newspace parameters
Level: | \( N \) | \(=\) | \( 1287 = 3^{2} \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 1287.ba (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.642296671259\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\zeta_{6})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{3}\) |
Projective field: | Galois closure of 3.1.11583.1 |
Artin image: | $C_3\times S_3$ |
Artin field: | Galois closure of 6.0.236860767.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1287\mathbb{Z}\right)^\times\).
\(n\) | \(496\) | \(937\) | \(1145\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
142.1 |
|
0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | 0 | 0 | 0.500000 | + | 0.866025i | −1.00000 | + | 1.73205i | 1.00000 | −0.500000 | − | 0.866025i | 0 | ||||||||||||||
571.1 | 0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | 0 | 0 | 0.500000 | − | 0.866025i | −1.00000 | − | 1.73205i | 1.00000 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
143.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-143}) \) |
9.c | even | 3 | 1 | inner |
1287.ba | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1287.1.ba.b | yes | 2 |
3.b | odd | 2 | 1 | 3861.1.ba.a | 2 | ||
9.c | even | 3 | 1 | inner | 1287.1.ba.b | yes | 2 |
9.d | odd | 6 | 1 | 3861.1.ba.a | 2 | ||
11.b | odd | 2 | 1 | 1287.1.ba.a | ✓ | 2 | |
13.b | even | 2 | 1 | 1287.1.ba.a | ✓ | 2 | |
33.d | even | 2 | 1 | 3861.1.ba.b | 2 | ||
39.d | odd | 2 | 1 | 3861.1.ba.b | 2 | ||
99.g | even | 6 | 1 | 3861.1.ba.b | 2 | ||
99.h | odd | 6 | 1 | 1287.1.ba.a | ✓ | 2 | |
117.n | odd | 6 | 1 | 3861.1.ba.b | 2 | ||
117.t | even | 6 | 1 | 1287.1.ba.a | ✓ | 2 | |
143.d | odd | 2 | 1 | CM | 1287.1.ba.b | yes | 2 |
429.e | even | 2 | 1 | 3861.1.ba.a | 2 | ||
1287.ba | odd | 6 | 1 | inner | 1287.1.ba.b | yes | 2 |
1287.bh | even | 6 | 1 | 3861.1.ba.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1287.1.ba.a | ✓ | 2 | 11.b | odd | 2 | 1 | |
1287.1.ba.a | ✓ | 2 | 13.b | even | 2 | 1 | |
1287.1.ba.a | ✓ | 2 | 99.h | odd | 6 | 1 | |
1287.1.ba.a | ✓ | 2 | 117.t | even | 6 | 1 | |
1287.1.ba.b | yes | 2 | 1.a | even | 1 | 1 | trivial |
1287.1.ba.b | yes | 2 | 9.c | even | 3 | 1 | inner |
1287.1.ba.b | yes | 2 | 143.d | odd | 2 | 1 | CM |
1287.1.ba.b | yes | 2 | 1287.ba | odd | 6 | 1 | inner |
3861.1.ba.a | 2 | 3.b | odd | 2 | 1 | ||
3861.1.ba.a | 2 | 9.d | odd | 6 | 1 | ||
3861.1.ba.a | 2 | 429.e | even | 2 | 1 | ||
3861.1.ba.a | 2 | 1287.bh | even | 6 | 1 | ||
3861.1.ba.b | 2 | 33.d | even | 2 | 1 | ||
3861.1.ba.b | 2 | 39.d | odd | 2 | 1 | ||
3861.1.ba.b | 2 | 99.g | even | 6 | 1 | ||
3861.1.ba.b | 2 | 117.n | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(1287, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - T + 1 \)
$3$
\( T^{2} + T + 1 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 2T + 4 \)
$11$
\( T^{2} + T + 1 \)
$13$
\( T^{2} + T + 1 \)
$17$
\( T^{2} \)
$19$
\( (T + 1)^{2} \)
$23$
\( T^{2} - T + 1 \)
$29$
\( T^{2} \)
$31$
\( T^{2} \)
$37$
\( T^{2} \)
$41$
\( T^{2} - T + 1 \)
$43$
\( T^{2} \)
$47$
\( T^{2} \)
$53$
\( (T - 2)^{2} \)
$59$
\( T^{2} \)
$61$
\( T^{2} \)
$67$
\( T^{2} \)
$71$
\( T^{2} \)
$73$
\( (T - 2)^{2} \)
$79$
\( T^{2} \)
$83$
\( T^{2} - T + 1 \)
$89$
\( T^{2} \)
$97$
\( T^{2} \)
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