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: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{15})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{15}\) |
Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{15} + \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1287\mathbb{Z}\right)^\times\).
\(n\) | \(496\) | \(937\) | \(1145\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(\zeta_{30}^{10}\) |
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 |
|
−0.913545 | + | 1.58231i | −0.104528 | − | 0.994522i | −1.16913 | − | 2.02499i | 0 | 1.66913 | + | 0.743145i | −0.309017 | + | 0.535233i | 2.44512 | −0.978148 | + | 0.207912i | 0 | ||||||||||||||||||||||||||||||
142.2 | −0.669131 | + | 1.15897i | −0.978148 | − | 0.207912i | −0.395472 | − | 0.684977i | 0 | 0.895472 | − | 0.994522i | 0.809017 | − | 1.40126i | −0.279773 | 0.913545 | + | 0.406737i | 0 | |||||||||||||||||||||||||||||||
142.3 | 0.104528 | − | 0.181049i | 0.913545 | − | 0.406737i | 0.478148 | + | 0.828176i | 0 | 0.0218524 | − | 0.207912i | −0.309017 | + | 0.535233i | 0.408977 | 0.669131 | − | 0.743145i | 0 | |||||||||||||||||||||||||||||||
142.4 | 0.978148 | − | 1.69420i | 0.669131 | + | 0.743145i | −1.41355 | − | 2.44833i | 0 | 1.91355 | − | 0.406737i | 0.809017 | − | 1.40126i | −3.57433 | −0.104528 | + | 0.994522i | 0 | |||||||||||||||||||||||||||||||
571.1 | −0.913545 | − | 1.58231i | −0.104528 | + | 0.994522i | −1.16913 | + | 2.02499i | 0 | 1.66913 | − | 0.743145i | −0.309017 | − | 0.535233i | 2.44512 | −0.978148 | − | 0.207912i | 0 | |||||||||||||||||||||||||||||||
571.2 | −0.669131 | − | 1.15897i | −0.978148 | + | 0.207912i | −0.395472 | + | 0.684977i | 0 | 0.895472 | + | 0.994522i | 0.809017 | + | 1.40126i | −0.279773 | 0.913545 | − | 0.406737i | 0 | |||||||||||||||||||||||||||||||
571.3 | 0.104528 | + | 0.181049i | 0.913545 | + | 0.406737i | 0.478148 | − | 0.828176i | 0 | 0.0218524 | + | 0.207912i | −0.309017 | − | 0.535233i | 0.408977 | 0.669131 | + | 0.743145i | 0 | |||||||||||||||||||||||||||||||
571.4 | 0.978148 | + | 1.69420i | 0.669131 | − | 0.743145i | −1.41355 | + | 2.44833i | 0 | 1.91355 | + | 0.406737i | 0.809017 | + | 1.40126i | −3.57433 | −0.104528 | − | 0.994522i | 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.c | ✓ | 8 |
3.b | odd | 2 | 1 | 3861.1.ba.d | 8 | ||
9.c | even | 3 | 1 | inner | 1287.1.ba.c | ✓ | 8 |
9.d | odd | 6 | 1 | 3861.1.ba.d | 8 | ||
11.b | odd | 2 | 1 | 1287.1.ba.d | yes | 8 | |
13.b | even | 2 | 1 | 1287.1.ba.d | yes | 8 | |
33.d | even | 2 | 1 | 3861.1.ba.c | 8 | ||
39.d | odd | 2 | 1 | 3861.1.ba.c | 8 | ||
99.g | even | 6 | 1 | 3861.1.ba.c | 8 | ||
99.h | odd | 6 | 1 | 1287.1.ba.d | yes | 8 | |
117.n | odd | 6 | 1 | 3861.1.ba.c | 8 | ||
117.t | even | 6 | 1 | 1287.1.ba.d | yes | 8 | |
143.d | odd | 2 | 1 | CM | 1287.1.ba.c | ✓ | 8 |
429.e | even | 2 | 1 | 3861.1.ba.d | 8 | ||
1287.ba | odd | 6 | 1 | inner | 1287.1.ba.c | ✓ | 8 |
1287.bh | even | 6 | 1 | 3861.1.ba.d | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1287.1.ba.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
1287.1.ba.c | ✓ | 8 | 9.c | even | 3 | 1 | inner |
1287.1.ba.c | ✓ | 8 | 143.d | odd | 2 | 1 | CM |
1287.1.ba.c | ✓ | 8 | 1287.ba | odd | 6 | 1 | inner |
1287.1.ba.d | yes | 8 | 11.b | odd | 2 | 1 | |
1287.1.ba.d | yes | 8 | 13.b | even | 2 | 1 | |
1287.1.ba.d | yes | 8 | 99.h | odd | 6 | 1 | |
1287.1.ba.d | yes | 8 | 117.t | even | 6 | 1 | |
3861.1.ba.c | 8 | 33.d | even | 2 | 1 | ||
3861.1.ba.c | 8 | 39.d | odd | 2 | 1 | ||
3861.1.ba.c | 8 | 99.g | even | 6 | 1 | ||
3861.1.ba.c | 8 | 117.n | odd | 6 | 1 | ||
3861.1.ba.d | 8 | 3.b | odd | 2 | 1 | ||
3861.1.ba.d | 8 | 9.d | odd | 6 | 1 | ||
3861.1.ba.d | 8 | 429.e | even | 2 | 1 | ||
3861.1.ba.d | 8 | 1287.bh | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + T_{2}^{7} + 5T_{2}^{6} + 4T_{2}^{5} + 19T_{2}^{4} + 14T_{2}^{3} + 20T_{2}^{2} - 4T_{2} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(1287, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + T^{7} + 5 T^{6} + 4 T^{5} + 19 T^{4} + \cdots + 1 \)
$3$
\( T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1 \)
$5$
\( T^{8} \)
$7$
\( (T^{4} - T^{3} + 2 T^{2} + T + 1)^{2} \)
$11$
\( (T^{2} + T + 1)^{4} \)
$13$
\( (T^{2} + T + 1)^{4} \)
$17$
\( T^{8} \)
$19$
\( (T^{4} - T^{3} - 4 T^{2} + 4 T + 1)^{2} \)
$23$
\( T^{8} + T^{7} + 5 T^{6} + 4 T^{5} + 19 T^{4} + \cdots + 1 \)
$29$
\( T^{8} \)
$31$
\( T^{8} \)
$37$
\( T^{8} \)
$41$
\( T^{8} + T^{7} + 5 T^{6} + 4 T^{5} + 19 T^{4} + \cdots + 1 \)
$43$
\( T^{8} \)
$47$
\( T^{8} \)
$53$
\( (T^{2} + T - 1)^{4} \)
$59$
\( T^{8} \)
$61$
\( T^{8} \)
$67$
\( T^{8} \)
$71$
\( T^{8} \)
$73$
\( (T^{2} + T - 1)^{4} \)
$79$
\( T^{8} \)
$83$
\( T^{8} + T^{7} + 5 T^{6} + 4 T^{5} + 19 T^{4} + \cdots + 1 \)
$89$
\( T^{8} \)
$97$
\( T^{8} \)
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