Newspace parameters
Level: | \( N \) | \(=\) | \( 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 1428.bu (of order \(8\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.712664838040\) |
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 8.2.2918512474101504.2 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1428\mathbb{Z}\right)^\times\).
\(n\) | \(409\) | \(715\) | \(953\) | \(1261\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(\zeta_{16}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
83.1 |
|
−0.707107 | − | 0.707107i | −0.382683 | − | 0.923880i | 1.00000i | −1.30656 | + | 0.541196i | −0.382683 | + | 0.923880i | 0.923880 | + | 0.382683i | 0.707107 | − | 0.707107i | −0.707107 | + | 0.707107i | 1.30656 | + | 0.541196i | ||||||||||||||||||||||||||
83.2 | −0.707107 | − | 0.707107i | 0.382683 | + | 0.923880i | 1.00000i | 1.30656 | − | 0.541196i | 0.382683 | − | 0.923880i | −0.923880 | − | 0.382683i | 0.707107 | − | 0.707107i | −0.707107 | + | 0.707107i | −1.30656 | − | 0.541196i | |||||||||||||||||||||||||||
587.1 | 0.707107 | − | 0.707107i | −0.923880 | − | 0.382683i | − | 1.00000i | −0.541196 | + | 1.30656i | −0.923880 | + | 0.382683i | −0.382683 | − | 0.923880i | −0.707107 | − | 0.707107i | 0.707107 | + | 0.707107i | 0.541196 | + | 1.30656i | ||||||||||||||||||||||||||
587.2 | 0.707107 | − | 0.707107i | 0.923880 | + | 0.382683i | − | 1.00000i | 0.541196 | − | 1.30656i | 0.923880 | − | 0.382683i | 0.382683 | + | 0.923880i | −0.707107 | − | 0.707107i | 0.707107 | + | 0.707107i | −0.541196 | − | 1.30656i | ||||||||||||||||||||||||||
671.1 | −0.707107 | + | 0.707107i | −0.382683 | + | 0.923880i | − | 1.00000i | −1.30656 | − | 0.541196i | −0.382683 | − | 0.923880i | 0.923880 | − | 0.382683i | 0.707107 | + | 0.707107i | −0.707107 | − | 0.707107i | 1.30656 | − | 0.541196i | ||||||||||||||||||||||||||
671.2 | −0.707107 | + | 0.707107i | 0.382683 | − | 0.923880i | − | 1.00000i | 1.30656 | + | 0.541196i | 0.382683 | + | 0.923880i | −0.923880 | + | 0.382683i | 0.707107 | + | 0.707107i | −0.707107 | − | 0.707107i | −1.30656 | + | 0.541196i | ||||||||||||||||||||||||||
1175.1 | 0.707107 | + | 0.707107i | −0.923880 | + | 0.382683i | 1.00000i | −0.541196 | − | 1.30656i | −0.923880 | − | 0.382683i | −0.382683 | + | 0.923880i | −0.707107 | + | 0.707107i | 0.707107 | − | 0.707107i | 0.541196 | − | 1.30656i | |||||||||||||||||||||||||||
1175.2 | 0.707107 | + | 0.707107i | 0.923880 | − | 0.382683i | 1.00000i | 0.541196 | + | 1.30656i | 0.923880 | + | 0.382683i | 0.382683 | − | 0.923880i | −0.707107 | + | 0.707107i | 0.707107 | − | 0.707107i | −0.541196 | + | 1.30656i | |||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
84.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-21}) \) |
7.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
17.d | even | 8 | 1 | inner |
119.l | odd | 8 | 1 | inner |
204.p | even | 8 | 1 | inner |
1428.bu | odd | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1428.1.bu.a | ✓ | 8 |
3.b | odd | 2 | 1 | 1428.1.bu.b | yes | 8 | |
4.b | odd | 2 | 1 | 1428.1.bu.b | yes | 8 | |
7.b | odd | 2 | 1 | inner | 1428.1.bu.a | ✓ | 8 |
12.b | even | 2 | 1 | inner | 1428.1.bu.a | ✓ | 8 |
17.d | even | 8 | 1 | inner | 1428.1.bu.a | ✓ | 8 |
21.c | even | 2 | 1 | 1428.1.bu.b | yes | 8 | |
28.d | even | 2 | 1 | 1428.1.bu.b | yes | 8 | |
51.g | odd | 8 | 1 | 1428.1.bu.b | yes | 8 | |
68.g | odd | 8 | 1 | 1428.1.bu.b | yes | 8 | |
84.h | odd | 2 | 1 | CM | 1428.1.bu.a | ✓ | 8 |
119.l | odd | 8 | 1 | inner | 1428.1.bu.a | ✓ | 8 |
204.p | even | 8 | 1 | inner | 1428.1.bu.a | ✓ | 8 |
357.w | even | 8 | 1 | 1428.1.bu.b | yes | 8 | |
476.w | even | 8 | 1 | 1428.1.bu.b | yes | 8 | |
1428.bu | odd | 8 | 1 | inner | 1428.1.bu.a | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1428.1.bu.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
1428.1.bu.a | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
1428.1.bu.a | ✓ | 8 | 12.b | even | 2 | 1 | inner |
1428.1.bu.a | ✓ | 8 | 17.d | even | 8 | 1 | inner |
1428.1.bu.a | ✓ | 8 | 84.h | odd | 2 | 1 | CM |
1428.1.bu.a | ✓ | 8 | 119.l | odd | 8 | 1 | inner |
1428.1.bu.a | ✓ | 8 | 204.p | even | 8 | 1 | inner |
1428.1.bu.a | ✓ | 8 | 1428.bu | odd | 8 | 1 | inner |
1428.1.bu.b | yes | 8 | 3.b | odd | 2 | 1 | |
1428.1.bu.b | yes | 8 | 4.b | odd | 2 | 1 | |
1428.1.bu.b | yes | 8 | 21.c | even | 2 | 1 | |
1428.1.bu.b | yes | 8 | 28.d | even | 2 | 1 | |
1428.1.bu.b | yes | 8 | 51.g | odd | 8 | 1 | |
1428.1.bu.b | yes | 8 | 68.g | odd | 8 | 1 | |
1428.1.bu.b | yes | 8 | 357.w | even | 8 | 1 | |
1428.1.bu.b | yes | 8 | 476.w | even | 8 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{4} + 4T_{11}^{3} + 6T_{11}^{2} + 4T_{11} + 2 \)
acting on \(S_{1}^{\mathrm{new}}(1428, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} + 1)^{2} \)
$3$
\( T^{8} + 1 \)
$5$
\( T^{8} + 16 \)
$7$
\( T^{8} + 1 \)
$11$
\( (T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2)^{2} \)
$13$
\( T^{8} \)
$17$
\( T^{8} + 1 \)
$19$
\( T^{8} + 12T^{4} + 4 \)
$23$
\( (T^{4} + 2 T^{2} + 4 T + 2)^{2} \)
$29$
\( T^{8} \)
$31$
\( T^{8} + 16 \)
$37$
\( (T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2)^{2} \)
$41$
\( T^{8} \)
$43$
\( T^{8} \)
$47$
\( T^{8} \)
$53$
\( T^{8} \)
$59$
\( T^{8} \)
$61$
\( T^{8} \)
$67$
\( T^{8} \)
$71$
\( (T^{4} + 2 T^{2} - 4 T + 2)^{2} \)
$73$
\( T^{8} \)
$79$
\( T^{8} \)
$83$
\( T^{8} \)
$89$
\( (T^{4} + 4 T^{2} + 2)^{2} \)
$97$
\( T^{8} \)
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