Newspace parameters
Level: | \( N \) | \(=\) | \( 2496 = 2^{6} \cdot 3 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2496.bx (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.24566627153\) |
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
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Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{6}\) |
Projective field: | Galois closure of 6.2.1579309056.3 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times\).
\(n\) | \(703\) | \(769\) | \(833\) | \(1093\) |
\(\chi(n)\) | \(1\) | \(-\zeta_{12}^{2}\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
737.1 |
|
0 | −0.866025 | − | 0.500000i | 0 | 0 | 0 | −0.866025 | − | 1.50000i | 0 | 0.500000 | + | 0.866025i | 0 | ||||||||||||||||||||||||
737.2 | 0 | 0.866025 | + | 0.500000i | 0 | 0 | 0 | 0.866025 | + | 1.50000i | 0 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||
1121.1 | 0 | −0.866025 | + | 0.500000i | 0 | 0 | 0 | −0.866025 | + | 1.50000i | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||
1121.2 | 0 | 0.866025 | − | 0.500000i | 0 | 0 | 0 | 0.866025 | − | 1.50000i | 0 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
4.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
104.n | odd | 6 | 1 | inner |
104.r | even | 6 | 1 | inner |
312.bh | odd | 6 | 1 | inner |
312.bn | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2496.1.bx.a | ✓ | 4 |
3.b | odd | 2 | 1 | CM | 2496.1.bx.a | ✓ | 4 |
4.b | odd | 2 | 1 | inner | 2496.1.bx.a | ✓ | 4 |
8.b | even | 2 | 1 | 2496.1.bx.b | yes | 4 | |
8.d | odd | 2 | 1 | 2496.1.bx.b | yes | 4 | |
12.b | even | 2 | 1 | inner | 2496.1.bx.a | ✓ | 4 |
13.c | even | 3 | 1 | 2496.1.bx.b | yes | 4 | |
24.f | even | 2 | 1 | 2496.1.bx.b | yes | 4 | |
24.h | odd | 2 | 1 | 2496.1.bx.b | yes | 4 | |
39.i | odd | 6 | 1 | 2496.1.bx.b | yes | 4 | |
52.j | odd | 6 | 1 | 2496.1.bx.b | yes | 4 | |
104.n | odd | 6 | 1 | inner | 2496.1.bx.a | ✓ | 4 |
104.r | even | 6 | 1 | inner | 2496.1.bx.a | ✓ | 4 |
156.p | even | 6 | 1 | 2496.1.bx.b | yes | 4 | |
312.bh | odd | 6 | 1 | inner | 2496.1.bx.a | ✓ | 4 |
312.bn | even | 6 | 1 | inner | 2496.1.bx.a | ✓ | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2496.1.bx.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
2496.1.bx.a | ✓ | 4 | 3.b | odd | 2 | 1 | CM |
2496.1.bx.a | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
2496.1.bx.a | ✓ | 4 | 12.b | even | 2 | 1 | inner |
2496.1.bx.a | ✓ | 4 | 104.n | odd | 6 | 1 | inner |
2496.1.bx.a | ✓ | 4 | 104.r | even | 6 | 1 | inner |
2496.1.bx.a | ✓ | 4 | 312.bh | odd | 6 | 1 | inner |
2496.1.bx.a | ✓ | 4 | 312.bn | even | 6 | 1 | inner |
2496.1.bx.b | yes | 4 | 8.b | even | 2 | 1 | |
2496.1.bx.b | yes | 4 | 8.d | odd | 2 | 1 | |
2496.1.bx.b | yes | 4 | 13.c | even | 3 | 1 | |
2496.1.bx.b | yes | 4 | 24.f | even | 2 | 1 | |
2496.1.bx.b | yes | 4 | 24.h | odd | 2 | 1 | |
2496.1.bx.b | yes | 4 | 39.i | odd | 6 | 1 | |
2496.1.bx.b | yes | 4 | 52.j | odd | 6 | 1 | |
2496.1.bx.b | yes | 4 | 156.p | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{61}^{2} + 3T_{61} + 3 \)
acting on \(S_{1}^{\mathrm{new}}(2496, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} - T^{2} + 1 \)
$5$
\( T^{4} \)
$7$
\( T^{4} + 3T^{2} + 9 \)
$11$
\( T^{4} \)
$13$
\( (T^{2} + T + 1)^{2} \)
$17$
\( T^{4} \)
$19$
\( T^{4} - 4T^{2} + 16 \)
$23$
\( T^{4} \)
$29$
\( T^{4} \)
$31$
\( (T^{2} - 3)^{2} \)
$37$
\( T^{4} \)
$41$
\( T^{4} \)
$43$
\( T^{4} - T^{2} + 1 \)
$47$
\( T^{4} \)
$53$
\( T^{4} \)
$59$
\( T^{4} \)
$61$
\( (T^{2} + 3 T + 3)^{2} \)
$67$
\( T^{4} - T^{2} + 1 \)
$71$
\( T^{4} \)
$73$
\( (T + 1)^{4} \)
$79$
\( (T^{2} - 3)^{2} \)
$83$
\( T^{4} \)
$89$
\( T^{4} \)
$97$
\( (T^{2} + T + 1)^{2} \)
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