Newspace parameters
Level: | \( N \) | \(=\) | \( 2808 = 2^{3} \cdot 3^{3} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2808.dp (of order \(12\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.40137455547\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
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: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 936) |
Projective image: | \(S_{4}\) |
Projective field: | Galois closure of 4.0.2847312.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2808\mathbb{Z}\right)^\times\).
\(n\) | \(703\) | \(1081\) | \(1405\) | \(2081\) |
\(\chi(n)\) | \(1\) | \(\zeta_{12}^{3}\) | \(1\) | \(-\zeta_{12}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
73.1 |
|
0 | 0 | 0 | 0.366025 | − | 1.36603i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
577.1 | 0 | 0 | 0 | 0.366025 | + | 1.36603i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
1009.1 | 0 | 0 | 0 | −1.36603 | + | 0.366025i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
2449.1 | 0 | 0 | 0 | −1.36603 | − | 0.366025i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
13.d | odd | 4 | 1 | inner |
117.y | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2808.1.dp.a | 4 | |
3.b | odd | 2 | 1 | 936.1.dm.b | ✓ | 4 | |
9.c | even | 3 | 1 | inner | 2808.1.dp.a | 4 | |
9.d | odd | 6 | 1 | 936.1.dm.b | ✓ | 4 | |
12.b | even | 2 | 1 | 1872.1.fi.a | 4 | ||
13.d | odd | 4 | 1 | inner | 2808.1.dp.a | 4 | |
36.h | even | 6 | 1 | 1872.1.fi.a | 4 | ||
39.f | even | 4 | 1 | 936.1.dm.b | ✓ | 4 | |
117.y | odd | 12 | 1 | inner | 2808.1.dp.a | 4 | |
117.z | even | 12 | 1 | 936.1.dm.b | ✓ | 4 | |
156.l | odd | 4 | 1 | 1872.1.fi.a | 4 | ||
468.ch | odd | 12 | 1 | 1872.1.fi.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
936.1.dm.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
936.1.dm.b | ✓ | 4 | 9.d | odd | 6 | 1 | |
936.1.dm.b | ✓ | 4 | 39.f | even | 4 | 1 | |
936.1.dm.b | ✓ | 4 | 117.z | even | 12 | 1 | |
1872.1.fi.a | 4 | 12.b | even | 2 | 1 | ||
1872.1.fi.a | 4 | 36.h | even | 6 | 1 | ||
1872.1.fi.a | 4 | 156.l | odd | 4 | 1 | ||
1872.1.fi.a | 4 | 468.ch | odd | 12 | 1 | ||
2808.1.dp.a | 4 | 1.a | even | 1 | 1 | trivial | |
2808.1.dp.a | 4 | 9.c | even | 3 | 1 | inner | |
2808.1.dp.a | 4 | 13.d | odd | 4 | 1 | inner | |
2808.1.dp.a | 4 | 117.y | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 2T_{5}^{3} + 2T_{5}^{2} + 4T_{5} + 4 \)
acting on \(S_{1}^{\mathrm{new}}(2808, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} \)
$5$
\( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \)
$7$
\( T^{4} \)
$11$
\( T^{4} \)
$13$
\( (T^{2} + T + 1)^{2} \)
$17$
\( (T^{2} + 1)^{2} \)
$19$
\( T^{4} \)
$23$
\( T^{4} - T^{2} + 1 \)
$29$
\( T^{4} \)
$31$
\( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \)
$37$
\( (T^{2} + 2 T + 2)^{2} \)
$41$
\( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \)
$43$
\( T^{4} - T^{2} + 1 \)
$47$
\( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \)
$53$
\( (T + 1)^{4} \)
$59$
\( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \)
$61$
\( (T^{2} + T + 1)^{2} \)
$67$
\( T^{4} \)
$71$
\( T^{4} \)
$73$
\( (T^{2} - 2 T + 2)^{2} \)
$79$
\( (T^{2} - T + 1)^{2} \)
$83$
\( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \)
$89$
\( T^{4} \)
$97$
\( T^{4} \)
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