Properties

Label 4000.1.bo.c
Level 4000
Weight 1
Character orbit 4000.bo
Analytic conductor 1.996
Analytic rank 0
Dimension 8
Projective image \(D_{20}\)
CM discriminant -4
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 4000.bo (of order \(20\), degree \(8\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.99626005053\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 800)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times\).

\(n\) \(1377\) \(2501\) \(2751\)
\(\chi(n)\) \(\zeta_{20}^{7}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
−0.951057 0.309017i
0.951057 + 0.309017i
0.587785 0.809017i
0.951057 0.309017i
−0.587785 0.809017i
0.587785 + 0.809017i
−0.951057 + 0.309017i
−0.587785 + 0.809017i
0 0 0 0 0 0 0 −0.951057 + 0.309017i 0
993.1 0 0 0 0 0 0 0 0.951057 0.309017i 0
1793.1 0 0 0 0 0 0 0 0.587785 + 0.809017i 0
1857.1 0 0 0 0 0 0 0 0.951057 + 0.309017i 0
2593.1 0 0 0 0 0 0 0 −0.587785 + 0.809017i 0
2657.1 0 0 0 0 0 0 0 0.587785 0.809017i 0
3393.1 0 0 0 0 0 0 0 −0.951057 0.309017i 0
3457.1 0 0 0 0 0 0 0 −0.587785 0.809017i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3457.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
25.f odd 20 1 inner
100.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4000.1.bo.c 8
4.b odd 2 1 CM 4000.1.bo.c 8
5.b even 2 1 4000.1.bo.a 8
5.c odd 4 1 800.1.bo.a 8
5.c odd 4 1 4000.1.bo.b 8
20.d odd 2 1 4000.1.bo.a 8
20.e even 4 1 800.1.bo.a 8
20.e even 4 1 4000.1.bo.b 8
25.d even 5 1 800.1.bo.a 8
25.e even 10 1 4000.1.bo.b 8
25.f odd 20 1 4000.1.bo.a 8
25.f odd 20 1 inner 4000.1.bo.c 8
40.i odd 4 1 1600.1.bw.a 8
40.k even 4 1 1600.1.bw.a 8
100.h odd 10 1 4000.1.bo.b 8
100.j odd 10 1 800.1.bo.a 8
100.l even 20 1 4000.1.bo.a 8
100.l even 20 1 inner 4000.1.bo.c 8
200.n odd 10 1 1600.1.bw.a 8
200.t even 10 1 1600.1.bw.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.1.bo.a 8 5.c odd 4 1
800.1.bo.a 8 20.e even 4 1
800.1.bo.a 8 25.d even 5 1
800.1.bo.a 8 100.j odd 10 1
1600.1.bw.a 8 40.i odd 4 1
1600.1.bw.a 8 40.k even 4 1
1600.1.bw.a 8 200.n odd 10 1
1600.1.bw.a 8 200.t even 10 1
4000.1.bo.a 8 5.b even 2 1
4000.1.bo.a 8 20.d odd 2 1
4000.1.bo.a 8 25.f odd 20 1
4000.1.bo.a 8 100.l even 20 1
4000.1.bo.b 8 5.c odd 4 1
4000.1.bo.b 8 20.e even 4 1
4000.1.bo.b 8 25.e even 10 1
4000.1.bo.b 8 100.h odd 10 1
4000.1.bo.c 8 1.a even 1 1 trivial
4000.1.bo.c 8 4.b odd 2 1 CM
4000.1.bo.c 8 25.f odd 20 1 inner
4000.1.bo.c 8 100.l even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{13}^{8} - \cdots\) acting on \(S_{1}^{\mathrm{new}}(4000, [\chi])\).

Hecke Characteristic Polynomials

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