Newspace parameters
Level: | \( N \) | \(=\) | \( 2304 = 2^{8} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2304.t (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.14984578911\) |
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: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 288) |
Projective image: | \(A_{4}\) |
Projective field: | Galois closure of 4.0.5184.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).
\(n\) | \(1279\) | \(1793\) | \(2053\) |
\(\chi(n)\) | \(-1\) | \(-\zeta_{12}^{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
895.1 |
|
0 | 1.00000 | 0 | −0.866025 | − | 0.500000i | 0 | 0.866025 | − | 0.500000i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||
895.2 | 0 | 1.00000 | 0 | 0.866025 | + | 0.500000i | 0 | −0.866025 | + | 0.500000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||
1663.1 | 0 | 1.00000 | 0 | −0.866025 | + | 0.500000i | 0 | 0.866025 | + | 0.500000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||
1663.2 | 0 | 1.00000 | 0 | 0.866025 | − | 0.500000i | 0 | −0.866025 | − | 0.500000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
9.c | even | 3 | 1 | inner |
72.p | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2304.1.t.b | 4 | |
4.b | odd | 2 | 1 | 2304.1.t.a | 4 | ||
8.b | even | 2 | 1 | 2304.1.t.a | 4 | ||
8.d | odd | 2 | 1 | inner | 2304.1.t.b | 4 | |
9.c | even | 3 | 1 | inner | 2304.1.t.b | 4 | |
16.e | even | 4 | 1 | 288.1.o.a | ✓ | 4 | |
16.e | even | 4 | 1 | 576.1.o.a | 4 | ||
16.f | odd | 4 | 1 | 288.1.o.a | ✓ | 4 | |
16.f | odd | 4 | 1 | 576.1.o.a | 4 | ||
36.f | odd | 6 | 1 | 2304.1.t.a | 4 | ||
48.i | odd | 4 | 1 | 864.1.o.a | 4 | ||
48.i | odd | 4 | 1 | 1728.1.o.a | 4 | ||
48.k | even | 4 | 1 | 864.1.o.a | 4 | ||
48.k | even | 4 | 1 | 1728.1.o.a | 4 | ||
72.n | even | 6 | 1 | 2304.1.t.a | 4 | ||
72.p | odd | 6 | 1 | inner | 2304.1.t.b | 4 | |
144.u | even | 12 | 1 | 864.1.o.a | 4 | ||
144.u | even | 12 | 1 | 1728.1.o.a | 4 | ||
144.u | even | 12 | 1 | 2592.1.g.b | 2 | ||
144.v | odd | 12 | 1 | 288.1.o.a | ✓ | 4 | |
144.v | odd | 12 | 1 | 576.1.o.a | 4 | ||
144.v | odd | 12 | 1 | 2592.1.g.a | 2 | ||
144.w | odd | 12 | 1 | 864.1.o.a | 4 | ||
144.w | odd | 12 | 1 | 1728.1.o.a | 4 | ||
144.w | odd | 12 | 1 | 2592.1.g.b | 2 | ||
144.x | even | 12 | 1 | 288.1.o.a | ✓ | 4 | |
144.x | even | 12 | 1 | 576.1.o.a | 4 | ||
144.x | even | 12 | 1 | 2592.1.g.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
288.1.o.a | ✓ | 4 | 16.e | even | 4 | 1 | |
288.1.o.a | ✓ | 4 | 16.f | odd | 4 | 1 | |
288.1.o.a | ✓ | 4 | 144.v | odd | 12 | 1 | |
288.1.o.a | ✓ | 4 | 144.x | even | 12 | 1 | |
576.1.o.a | 4 | 16.e | even | 4 | 1 | ||
576.1.o.a | 4 | 16.f | odd | 4 | 1 | ||
576.1.o.a | 4 | 144.v | odd | 12 | 1 | ||
576.1.o.a | 4 | 144.x | even | 12 | 1 | ||
864.1.o.a | 4 | 48.i | odd | 4 | 1 | ||
864.1.o.a | 4 | 48.k | even | 4 | 1 | ||
864.1.o.a | 4 | 144.u | even | 12 | 1 | ||
864.1.o.a | 4 | 144.w | odd | 12 | 1 | ||
1728.1.o.a | 4 | 48.i | odd | 4 | 1 | ||
1728.1.o.a | 4 | 48.k | even | 4 | 1 | ||
1728.1.o.a | 4 | 144.u | even | 12 | 1 | ||
1728.1.o.a | 4 | 144.w | odd | 12 | 1 | ||
2304.1.t.a | 4 | 4.b | odd | 2 | 1 | ||
2304.1.t.a | 4 | 8.b | even | 2 | 1 | ||
2304.1.t.a | 4 | 36.f | odd | 6 | 1 | ||
2304.1.t.a | 4 | 72.n | even | 6 | 1 | ||
2304.1.t.b | 4 | 1.a | even | 1 | 1 | trivial | |
2304.1.t.b | 4 | 8.d | odd | 2 | 1 | inner | |
2304.1.t.b | 4 | 9.c | even | 3 | 1 | inner | |
2304.1.t.b | 4 | 72.p | odd | 6 | 1 | inner | |
2592.1.g.a | 2 | 144.v | odd | 12 | 1 | ||
2592.1.g.a | 2 | 144.x | even | 12 | 1 | ||
2592.1.g.b | 2 | 144.u | even | 12 | 1 | ||
2592.1.g.b | 2 | 144.w | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{2} + T_{11} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2304, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( (T - 1)^{4} \)
$5$
\( T^{4} - T^{2} + 1 \)
$7$
\( T^{4} - T^{2} + 1 \)
$11$
\( (T^{2} + T + 1)^{2} \)
$13$
\( T^{4} - T^{2} + 1 \)
$17$
\( T^{4} \)
$19$
\( T^{4} \)
$23$
\( T^{4} - T^{2} + 1 \)
$29$
\( T^{4} - T^{2} + 1 \)
$31$
\( T^{4} - T^{2} + 1 \)
$37$
\( T^{4} \)
$41$
\( (T^{2} - T + 1)^{2} \)
$43$
\( (T^{2} - T + 1)^{2} \)
$47$
\( T^{4} - T^{2} + 1 \)
$53$
\( T^{4} \)
$59$
\( (T^{2} + T + 1)^{2} \)
$61$
\( T^{4} - T^{2} + 1 \)
$67$
\( (T^{2} + T + 1)^{2} \)
$71$
\( (T^{2} + 4)^{2} \)
$73$
\( T^{4} \)
$79$
\( T^{4} - T^{2} + 1 \)
$83$
\( (T^{2} + T + 1)^{2} \)
$89$
\( T^{4} \)
$97$
\( (T^{2} + T + 1)^{2} \)
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