Newspace parameters
Level: | \( N \) | \(=\) | \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2016.dn (of order \(8\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.00611506547\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{8})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{8}\) |
Projective field: | Galois closure of 8.0.3758802906120192.31 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(577\) | \(1765\) | \(1793\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-\zeta_{8}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
251.1 |
|
1.00000 | 0 | 1.00000 | 0 | 0 | 0.707107 | − | 0.707107i | 1.00000 | 0 | 0 | ||||||||||||||||||||||||||||
755.1 | 1.00000 | 0 | 1.00000 | 0 | 0 | 0.707107 | + | 0.707107i | 1.00000 | 0 | 0 | |||||||||||||||||||||||||||||
1259.1 | 1.00000 | 0 | 1.00000 | 0 | 0 | −0.707107 | + | 0.707107i | 1.00000 | 0 | 0 | |||||||||||||||||||||||||||||
1763.1 | 1.00000 | 0 | 1.00000 | 0 | 0 | −0.707107 | − | 0.707107i | 1.00000 | 0 | 0 | |||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
96.o | even | 8 | 1 | inner |
672.br | odd | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2016.1.dn.d | yes | 4 |
3.b | odd | 2 | 1 | 2016.1.dn.a | ✓ | 4 | |
7.b | odd | 2 | 1 | CM | 2016.1.dn.d | yes | 4 |
21.c | even | 2 | 1 | 2016.1.dn.a | ✓ | 4 | |
32.h | odd | 8 | 1 | 2016.1.dn.a | ✓ | 4 | |
96.o | even | 8 | 1 | inner | 2016.1.dn.d | yes | 4 |
224.x | even | 8 | 1 | 2016.1.dn.a | ✓ | 4 | |
672.br | odd | 8 | 1 | inner | 2016.1.dn.d | yes | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2016.1.dn.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
2016.1.dn.a | ✓ | 4 | 21.c | even | 2 | 1 | |
2016.1.dn.a | ✓ | 4 | 32.h | odd | 8 | 1 | |
2016.1.dn.a | ✓ | 4 | 224.x | even | 8 | 1 | |
2016.1.dn.d | yes | 4 | 1.a | even | 1 | 1 | trivial |
2016.1.dn.d | yes | 4 | 7.b | odd | 2 | 1 | CM |
2016.1.dn.d | yes | 4 | 96.o | even | 8 | 1 | inner |
2016.1.dn.d | yes | 4 | 672.br | odd | 8 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{4} + 2T_{11}^{2} - 4T_{11} + 2 \)
acting on \(S_{1}^{\mathrm{new}}(2016, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T - 1)^{4} \)
$3$
\( T^{4} \)
$5$
\( T^{4} \)
$7$
\( T^{4} + 1 \)
$11$
\( T^{4} + 2 T^{2} - 4 T + 2 \)
$13$
\( T^{4} \)
$17$
\( T^{4} \)
$19$
\( T^{4} \)
$23$
\( T^{4} \)
$29$
\( T^{4} + 2 T^{2} - 4 T + 2 \)
$31$
\( T^{4} \)
$37$
\( T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2 \)
$41$
\( T^{4} \)
$43$
\( T^{4} + 2 T^{2} - 4 T + 2 \)
$47$
\( T^{4} \)
$53$
\( T^{4} + 2 T^{2} + 4 T + 2 \)
$59$
\( T^{4} \)
$61$
\( T^{4} \)
$67$
\( T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2 \)
$71$
\( (T^{2} + 2 T + 2)^{2} \)
$73$
\( T^{4} \)
$79$
\( (T^{2} - 2)^{2} \)
$83$
\( T^{4} \)
$89$
\( T^{4} \)
$97$
\( T^{4} \)
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