Newspace parameters
Level: | \( N \) | \(=\) | \( 3072 = 2^{10} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 3072.p (of order \(8\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.53312771881\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{8})\) |
Coefficient field: | \(\Q(\zeta_{16})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{8} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{8}\) |
Projective field: | Galois closure of 8.0.57982058496.8 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times\).
\(n\) | \(1025\) | \(2047\) | \(2053\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(\zeta_{16}^{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
641.1 |
|
0 | −0.923880 | − | 0.382683i | 0 | 0 | 0 | 0.541196 | − | 0.541196i | 0 | 0.707107 | + | 0.707107i | 0 | ||||||||||||||||||||||||||||||||||||
641.2 | 0 | 0.923880 | + | 0.382683i | 0 | 0 | 0 | −0.541196 | + | 0.541196i | 0 | 0.707107 | + | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||||
1409.1 | 0 | −0.923880 | + | 0.382683i | 0 | 0 | 0 | 0.541196 | + | 0.541196i | 0 | 0.707107 | − | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||||
1409.2 | 0 | 0.923880 | − | 0.382683i | 0 | 0 | 0 | −0.541196 | − | 0.541196i | 0 | 0.707107 | − | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||||
2177.1 | 0 | −0.382683 | + | 0.923880i | 0 | 0 | 0 | 1.30656 | − | 1.30656i | 0 | −0.707107 | − | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||||
2177.2 | 0 | 0.382683 | − | 0.923880i | 0 | 0 | 0 | −1.30656 | + | 1.30656i | 0 | −0.707107 | − | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||||
2945.1 | 0 | −0.382683 | − | 0.923880i | 0 | 0 | 0 | 1.30656 | + | 1.30656i | 0 | −0.707107 | + | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||||
2945.2 | 0 | 0.382683 | + | 0.923880i | 0 | 0 | 0 | −1.30656 | − | 1.30656i | 0 | −0.707107 | + | 0.707107i | 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 |
32.g | even | 8 | 1 | inner |
32.h | odd | 8 | 1 | inner |
96.o | even | 8 | 1 | inner |
96.p | odd | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3072.1.p.c | yes | 8 |
3.b | odd | 2 | 1 | CM | 3072.1.p.c | yes | 8 |
4.b | odd | 2 | 1 | inner | 3072.1.p.c | yes | 8 |
8.b | even | 2 | 1 | 3072.1.p.b | yes | 8 | |
8.d | odd | 2 | 1 | 3072.1.p.b | yes | 8 | |
12.b | even | 2 | 1 | inner | 3072.1.p.c | yes | 8 |
16.e | even | 4 | 1 | 3072.1.p.a | ✓ | 8 | |
16.e | even | 4 | 1 | 3072.1.p.d | yes | 8 | |
16.f | odd | 4 | 1 | 3072.1.p.a | ✓ | 8 | |
16.f | odd | 4 | 1 | 3072.1.p.d | yes | 8 | |
24.f | even | 2 | 1 | 3072.1.p.b | yes | 8 | |
24.h | odd | 2 | 1 | 3072.1.p.b | yes | 8 | |
32.g | even | 8 | 1 | 3072.1.p.a | ✓ | 8 | |
32.g | even | 8 | 1 | 3072.1.p.b | yes | 8 | |
32.g | even | 8 | 1 | inner | 3072.1.p.c | yes | 8 |
32.g | even | 8 | 1 | 3072.1.p.d | yes | 8 | |
32.h | odd | 8 | 1 | 3072.1.p.a | ✓ | 8 | |
32.h | odd | 8 | 1 | 3072.1.p.b | yes | 8 | |
32.h | odd | 8 | 1 | inner | 3072.1.p.c | yes | 8 |
32.h | odd | 8 | 1 | 3072.1.p.d | yes | 8 | |
48.i | odd | 4 | 1 | 3072.1.p.a | ✓ | 8 | |
48.i | odd | 4 | 1 | 3072.1.p.d | yes | 8 | |
48.k | even | 4 | 1 | 3072.1.p.a | ✓ | 8 | |
48.k | even | 4 | 1 | 3072.1.p.d | yes | 8 | |
96.o | even | 8 | 1 | 3072.1.p.a | ✓ | 8 | |
96.o | even | 8 | 1 | 3072.1.p.b | yes | 8 | |
96.o | even | 8 | 1 | inner | 3072.1.p.c | yes | 8 |
96.o | even | 8 | 1 | 3072.1.p.d | yes | 8 | |
96.p | odd | 8 | 1 | 3072.1.p.a | ✓ | 8 | |
96.p | odd | 8 | 1 | 3072.1.p.b | yes | 8 | |
96.p | odd | 8 | 1 | inner | 3072.1.p.c | yes | 8 |
96.p | odd | 8 | 1 | 3072.1.p.d | yes | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3072.1.p.a | ✓ | 8 | 16.e | even | 4 | 1 | |
3072.1.p.a | ✓ | 8 | 16.f | odd | 4 | 1 | |
3072.1.p.a | ✓ | 8 | 32.g | even | 8 | 1 | |
3072.1.p.a | ✓ | 8 | 32.h | odd | 8 | 1 | |
3072.1.p.a | ✓ | 8 | 48.i | odd | 4 | 1 | |
3072.1.p.a | ✓ | 8 | 48.k | even | 4 | 1 | |
3072.1.p.a | ✓ | 8 | 96.o | even | 8 | 1 | |
3072.1.p.a | ✓ | 8 | 96.p | odd | 8 | 1 | |
3072.1.p.b | yes | 8 | 8.b | even | 2 | 1 | |
3072.1.p.b | yes | 8 | 8.d | odd | 2 | 1 | |
3072.1.p.b | yes | 8 | 24.f | even | 2 | 1 | |
3072.1.p.b | yes | 8 | 24.h | odd | 2 | 1 | |
3072.1.p.b | yes | 8 | 32.g | even | 8 | 1 | |
3072.1.p.b | yes | 8 | 32.h | odd | 8 | 1 | |
3072.1.p.b | yes | 8 | 96.o | even | 8 | 1 | |
3072.1.p.b | yes | 8 | 96.p | odd | 8 | 1 | |
3072.1.p.c | yes | 8 | 1.a | even | 1 | 1 | trivial |
3072.1.p.c | yes | 8 | 3.b | odd | 2 | 1 | CM |
3072.1.p.c | yes | 8 | 4.b | odd | 2 | 1 | inner |
3072.1.p.c | yes | 8 | 12.b | even | 2 | 1 | inner |
3072.1.p.c | yes | 8 | 32.g | even | 8 | 1 | inner |
3072.1.p.c | yes | 8 | 32.h | odd | 8 | 1 | inner |
3072.1.p.c | yes | 8 | 96.o | even | 8 | 1 | inner |
3072.1.p.c | yes | 8 | 96.p | odd | 8 | 1 | inner |
3072.1.p.d | yes | 8 | 16.e | even | 4 | 1 | |
3072.1.p.d | yes | 8 | 16.f | odd | 4 | 1 | |
3072.1.p.d | yes | 8 | 32.g | even | 8 | 1 | |
3072.1.p.d | yes | 8 | 32.h | odd | 8 | 1 | |
3072.1.p.d | yes | 8 | 48.i | odd | 4 | 1 | |
3072.1.p.d | yes | 8 | 48.k | even | 4 | 1 | |
3072.1.p.d | yes | 8 | 96.o | even | 8 | 1 | |
3072.1.p.d | yes | 8 | 96.p | odd | 8 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{13}^{4} + 2T_{13}^{2} - 4T_{13} + 2 \)
acting on \(S_{1}^{\mathrm{new}}(3072, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} + 1 \)
$5$
\( T^{8} \)
$7$
\( T^{8} + 12T^{4} + 4 \)
$11$
\( T^{8} \)
$13$
\( (T^{4} + 2 T^{2} - 4 T + 2)^{2} \)
$17$
\( T^{8} \)
$19$
\( T^{8} + 16 \)
$23$
\( T^{8} \)
$29$
\( T^{8} \)
$31$
\( (T^{4} - 4 T^{2} + 2)^{2} \)
$37$
\( (T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2)^{2} \)
$41$
\( T^{8} \)
$43$
\( T^{8} + 16 \)
$47$
\( T^{8} \)
$53$
\( T^{8} \)
$59$
\( T^{8} \)
$61$
\( (T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2)^{2} \)
$67$
\( T^{8} \)
$71$
\( T^{8} \)
$73$
\( (T^{2} - 2 T + 2)^{4} \)
$79$
\( (T^{4} + 4 T^{2} + 2)^{2} \)
$83$
\( T^{8} \)
$89$
\( T^{8} \)
$97$
\( (T^{2} - 2)^{4} \)
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