Properties

Label 1792.1.bf.a
Level $1792$
Weight $1$
Character orbit 1792.bf
Analytic conductor $0.894$
Analytic rank $0$
Dimension $8$
Projective image $D_{16}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1792.bf (of order \(16\), degree \(8\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.894324502638\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 448)
Projective image: \(D_{16}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{16}^{7} q^{7} + \zeta_{16} q^{9} +O(q^{10})\) \( q -\zeta_{16}^{7} q^{7} + \zeta_{16} q^{9} + ( -\zeta_{16}^{2} - \zeta_{16}^{5} ) q^{11} + ( \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{23} -\zeta_{16}^{3} q^{25} + ( -\zeta_{16}^{2} - \zeta_{16}^{7} ) q^{29} + ( -\zeta_{16}^{5} + \zeta_{16}^{6} ) q^{37} + ( \zeta_{16}^{3} - \zeta_{16}^{4} ) q^{43} -\zeta_{16}^{6} q^{49} + ( \zeta_{16}^{3} + \zeta_{16}^{4} ) q^{53} + q^{63} + ( 1 + \zeta_{16} ) q^{67} + ( -\zeta_{16} + \zeta_{16}^{5} ) q^{71} + ( -\zeta_{16} - \zeta_{16}^{4} ) q^{77} + ( \zeta_{16} - \zeta_{16}^{3} ) q^{79} + \zeta_{16}^{2} q^{81} + ( -\zeta_{16}^{3} - \zeta_{16}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{63} + 8q^{67} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{16}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
0.923880 + 0.382683i
−0.923880 + 0.382683i
0.382683 0.923880i
0.382683 + 0.923880i
−0.923880 0.382683i
0.923880 0.382683i
−0.382683 + 0.923880i
−0.382683 0.923880i
0 0 0 0 0 0.923880 0.382683i 0 0.923880 + 0.382683i 0
433.1 0 0 0 0 0 −0.923880 0.382683i 0 −0.923880 + 0.382683i 0
657.1 0 0 0 0 0 0.382683 + 0.923880i 0 0.382683 0.923880i 0
881.1 0 0 0 0 0 0.382683 0.923880i 0 0.382683 + 0.923880i 0
1105.1 0 0 0 0 0 −0.923880 + 0.382683i 0 −0.923880 0.382683i 0
1329.1 0 0 0 0 0 0.923880 + 0.382683i 0 0.923880 0.382683i 0
1553.1 0 0 0 0 0 −0.382683 0.923880i 0 −0.382683 + 0.923880i 0
1777.1 0 0 0 0 0 −0.382683 + 0.923880i 0 −0.382683 0.923880i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1777.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
64.i even 16 1 inner
448.bf odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.1.bf.a 8
4.b odd 2 1 448.1.bf.a 8
7.b odd 2 1 CM 1792.1.bf.a 8
8.b even 2 1 3584.1.bf.b 8
8.d odd 2 1 3584.1.bf.a 8
28.d even 2 1 448.1.bf.a 8
28.f even 6 2 3136.1.ce.a 16
28.g odd 6 2 3136.1.ce.a 16
56.e even 2 1 3584.1.bf.a 8
56.h odd 2 1 3584.1.bf.b 8
64.i even 16 1 inner 1792.1.bf.a 8
64.i even 16 1 3584.1.bf.b 8
64.j odd 16 1 448.1.bf.a 8
64.j odd 16 1 3584.1.bf.a 8
448.bd even 16 1 448.1.bf.a 8
448.bd even 16 1 3584.1.bf.a 8
448.bf odd 16 1 inner 1792.1.bf.a 8
448.bf odd 16 1 3584.1.bf.b 8
448.bl odd 48 2 3136.1.ce.a 16
448.bm even 48 2 3136.1.ce.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.1.bf.a 8 4.b odd 2 1
448.1.bf.a 8 28.d even 2 1
448.1.bf.a 8 64.j odd 16 1
448.1.bf.a 8 448.bd even 16 1
1792.1.bf.a 8 1.a even 1 1 trivial
1792.1.bf.a 8 7.b odd 2 1 CM
1792.1.bf.a 8 64.i even 16 1 inner
1792.1.bf.a 8 448.bf odd 16 1 inner
3136.1.ce.a 16 28.f even 6 2
3136.1.ce.a 16 28.g odd 6 2
3136.1.ce.a 16 448.bl odd 48 2
3136.1.ce.a 16 448.bm even 48 2
3584.1.bf.a 8 8.d odd 2 1
3584.1.bf.a 8 56.e even 2 1
3584.1.bf.a 8 64.j odd 16 1
3584.1.bf.a 8 448.bd even 16 1
3584.1.bf.b 8 8.b even 2 1
3584.1.bf.b 8 56.h odd 2 1
3584.1.bf.b 8 64.i even 16 1
3584.1.bf.b 8 448.bf odd 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1792, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( 1 + T^{8} \)
$11$ \( 2 + 8 T + 20 T^{2} + 16 T^{3} + 2 T^{4} + T^{8} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( ( 2 - 4 T + 2 T^{2} + T^{4} )^{2} \)
$29$ \( 2 - 8 T + 12 T^{2} + 2 T^{4} + 8 T^{5} + T^{8} \)
$31$ \( T^{8} \)
$37$ \( 2 + 8 T + 12 T^{2} + 2 T^{4} - 8 T^{5} + T^{8} \)
$41$ \( T^{8} \)
$43$ \( 2 - 8 T + 4 T^{2} + 8 T^{3} + 6 T^{4} + 4 T^{6} + T^{8} \)
$47$ \( T^{8} \)
$53$ \( 2 + 8 T + 4 T^{2} - 8 T^{3} + 6 T^{4} + 4 T^{6} + T^{8} \)
$59$ \( T^{8} \)
$61$ \( T^{8} \)
$67$ \( 2 - 8 T + 28 T^{2} - 56 T^{3} + 70 T^{4} - 56 T^{5} + 28 T^{6} - 8 T^{7} + T^{8} \)
$71$ \( 16 + T^{8} \)
$73$ \( T^{8} \)
$79$ \( 4 + 12 T^{4} + T^{8} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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