Properties

Label 3200.1.bi.a
Level $3200$
Weight $1$
Character orbit 3200.bi
Analytic conductor $1.597$
Analytic rank $0$
Dimension $4$
Projective image $D_{10}$
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.bi (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.59700804043\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.400000000000000000.16

$q$-expansion

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

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

Character values

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

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{10}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
−0.309017 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 + 0.951057i
0 0 0 −0.809017 0.587785i 0 0 0 0.309017 0.951057i 0
959.1 0 0 0 0.309017 0.951057i 0 0 0 −0.809017 + 0.587785i 0
2239.1 0 0 0 0.309017 + 0.951057i 0 0 0 −0.809017 0.587785i 0
2879.1 0 0 0 −0.809017 + 0.587785i 0 0 0 0.309017 + 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
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.o even 10 1 inner
200.s odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.1.bi.a 4
4.b odd 2 1 CM 3200.1.bi.a 4
8.b even 2 1 3200.1.bi.b yes 4
8.d odd 2 1 3200.1.bi.b yes 4
25.e even 10 1 3200.1.bi.b yes 4
100.h odd 10 1 3200.1.bi.b yes 4
200.o even 10 1 inner 3200.1.bi.a 4
200.s odd 10 1 inner 3200.1.bi.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.1.bi.a 4 1.a even 1 1 trivial
3200.1.bi.a 4 4.b odd 2 1 CM
3200.1.bi.a 4 200.o even 10 1 inner
3200.1.bi.a 4 200.s odd 10 1 inner
3200.1.bi.b yes 4 8.b even 2 1
3200.1.bi.b yes 4 8.d odd 2 1
3200.1.bi.b yes 4 25.e even 10 1
3200.1.bi.b yes 4 100.h odd 10 1

Hecke kernels

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

Hecke characteristic polynomials

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