Properties

Label 3200.1.bz.b
Level $3200$
Weight $1$
Character orbit 3200.bz
Analytic conductor $1.597$
Analytic rank $0$
Dimension $8$
Projective image $D_{20}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3200.bz (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.59700804043\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{20}^{9} q^{5} -\zeta_{20}^{7} q^{9} +O(q^{10})\) \( q + \zeta_{20}^{9} q^{5} -\zeta_{20}^{7} q^{9} + ( -\zeta_{20}^{3} + \zeta_{20}^{6} ) q^{13} + ( -\zeta_{20} - \zeta_{20}^{2} ) q^{17} -\zeta_{20}^{8} q^{25} + ( \zeta_{20}^{4} + \zeta_{20}^{8} ) q^{29} + ( -\zeta_{20}^{4} - \zeta_{20}^{5} ) q^{37} + ( -\zeta_{20} - \zeta_{20}^{3} ) q^{41} + \zeta_{20}^{6} q^{45} + \zeta_{20}^{5} q^{49} + ( -\zeta_{20}^{2} - \zeta_{20}^{5} ) q^{53} + ( \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{61} + ( \zeta_{20}^{2} - \zeta_{20}^{5} ) q^{65} + ( -\zeta_{20}^{4} + \zeta_{20}^{7} ) q^{73} -\zeta_{20}^{4} q^{81} + ( 1 + \zeta_{20} ) q^{85} + ( 1 + \zeta_{20}^{6} ) q^{89} + ( -\zeta_{20}^{8} + \zeta_{20}^{9} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q + 2 q^{13} - 2 q^{17} + 2 q^{25} - 4 q^{29} + 2 q^{37} + 2 q^{45} - 2 q^{53} + 2 q^{65} + 2 q^{73} + 2 q^{81} + 8 q^{85} + 10 q^{89} + 2 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{20}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
−0.951057 0.309017i
−0.587785 0.809017i
0.587785 + 0.809017i
0.951057 + 0.309017i
−0.951057 + 0.309017i
−0.587785 + 0.809017i
0.587785 0.809017i
0.951057 0.309017i
0 0 0 0.951057 0.309017i 0 0 0 −0.587785 + 0.809017i 0
833.1 0 0 0 0.587785 0.809017i 0 0 0 0.951057 + 0.309017i 0
1217.1 0 0 0 −0.587785 + 0.809017i 0 0 0 −0.951057 0.309017i 0
1473.1 0 0 0 −0.951057 + 0.309017i 0 0 0 0.587785 0.809017i 0
2113.1 0 0 0 0.951057 + 0.309017i 0 0 0 −0.587785 0.809017i 0
2497.1 0 0 0 0.587785 + 0.809017i 0 0 0 0.951057 0.309017i 0
2753.1 0 0 0 −0.587785 0.809017i 0 0 0 −0.951057 + 0.309017i 0
3137.1 0 0 0 −0.951057 0.309017i 0 0 0 0.587785 + 0.809017i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3137.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
200.v even 20 1 inner
200.x odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.1.bz.b yes 8
4.b odd 2 1 CM 3200.1.bz.b yes 8
8.b even 2 1 3200.1.bz.a 8
8.d odd 2 1 3200.1.bz.a 8
25.f odd 20 1 3200.1.bz.a 8
100.l even 20 1 3200.1.bz.a 8
200.v even 20 1 inner 3200.1.bz.b yes 8
200.x odd 20 1 inner 3200.1.bz.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.1.bz.a 8 8.b even 2 1
3200.1.bz.a 8 8.d odd 2 1
3200.1.bz.a 8 25.f odd 20 1
3200.1.bz.a 8 100.l even 20 1
3200.1.bz.b yes 8 1.a even 1 1 trivial
3200.1.bz.b yes 8 4.b odd 2 1 CM
3200.1.bz.b yes 8 200.v even 20 1 inner
3200.1.bz.b yes 8 200.x odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{8} - 2 T_{13}^{7} + 2 T_{13}^{6} - 4 T_{13}^{4} + 10 T_{13}^{3} + 13 T_{13}^{2} + 4 T_{13} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3200, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( T^{8} \)
$13$ \( 1 + 4 T + 13 T^{2} + 10 T^{3} - 4 T^{4} + 2 T^{6} - 2 T^{7} + T^{8} \)
$17$ \( 1 - 4 T + 13 T^{2} - 10 T^{3} - 4 T^{4} + 2 T^{6} + 2 T^{7} + T^{8} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( ( 1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$31$ \( T^{8} \)
$37$ \( 1 + 4 T - 2 T^{2} - 10 T^{3} + 16 T^{4} - 10 T^{5} + 7 T^{6} - 2 T^{7} + T^{8} \)
$41$ \( 25 + 25 T^{2} + 10 T^{4} + T^{8} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( 1 - 4 T - 2 T^{2} + 10 T^{3} + 16 T^{4} + 10 T^{5} + 7 T^{6} + 2 T^{7} + T^{8} \)
$59$ \( T^{8} \)
$61$ \( 1 + T^{2} + 6 T^{4} - 4 T^{6} + T^{8} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( 1 + 4 T + 13 T^{2} + 10 T^{3} - 4 T^{4} + 2 T^{6} - 2 T^{7} + T^{8} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( ( 5 - 10 T + 10 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$97$ \( 1 + 4 T + 13 T^{2} + 10 T^{3} - 4 T^{4} + 2 T^{6} - 2 T^{7} + T^{8} \)
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