Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 4000.p (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.99626005053\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Projective image \(D_{20}\)
Projective field Galois closure of 20.2.400000000000000000000000000.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \zeta_{20}^{6} + \zeta_{20}^{9} ) q^{3} + ( -\zeta_{20}^{2} - \zeta_{20}^{3} ) q^{7} + ( -\zeta_{20}^{2} - \zeta_{20}^{5} - \zeta_{20}^{8} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{20}^{6} + \zeta_{20}^{9} ) q^{3} + ( -\zeta_{20}^{2} - \zeta_{20}^{3} ) q^{7} + ( -\zeta_{20}^{2} - \zeta_{20}^{5} - \zeta_{20}^{8} ) q^{9} + ( \zeta_{20} + \zeta_{20}^{2} - \zeta_{20}^{8} - \zeta_{20}^{9} ) q^{21} + ( \zeta_{20}^{2} - \zeta_{20}^{3} ) q^{23} + ( \zeta_{20} + \zeta_{20}^{4} + \zeta_{20}^{7} - \zeta_{20}^{8} ) q^{27} + ( \zeta_{20} + \zeta_{20}^{9} ) q^{29} + ( -\zeta_{20} + \zeta_{20}^{9} ) q^{41} + ( -\zeta_{20} + \zeta_{20}^{4} ) q^{43} + ( \zeta_{20}^{6} - \zeta_{20}^{9} ) q^{47} + ( \zeta_{20}^{4} + \zeta_{20}^{5} + \zeta_{20}^{6} ) q^{49} + ( \zeta_{20}^{3} - \zeta_{20}^{7} ) q^{61} + ( -1 - \zeta_{20} + \zeta_{20}^{4} + \zeta_{20}^{5} + \zeta_{20}^{7} + \zeta_{20}^{8} ) q^{63} + ( 1 + \zeta_{20}^{5} ) q^{67} + ( -\zeta_{20} + \zeta_{20}^{2} + \zeta_{20}^{8} - \zeta_{20}^{9} ) q^{69} + ( -1 - \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{6} + \zeta_{20}^{7} ) q^{81} + ( \zeta_{20}^{7} + \zeta_{20}^{8} ) q^{83} + ( -1 - \zeta_{20}^{5} + \zeta_{20}^{7} - \zeta_{20}^{8} ) q^{87} + ( \zeta_{20}^{3} + \zeta_{20}^{7} ) q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{3} - 2q^{7} + O(q^{10}) \) \( 8q + 2q^{3} - 2q^{7} + 4q^{21} + 2q^{23} - 2q^{43} + 2q^{47} - 12q^{63} + 8q^{67} - 12q^{81} - 2q^{83} - 6q^{87} + O(q^{100}) \)

Character values

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

\(n\) \(1377\) \(2501\) \(2751\)
\(\chi(n)\) \(\zeta_{20}^{5}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.951057 0.309017i
0.587785 + 0.809017i
−0.951057 0.309017i
−0.587785 + 0.809017i
0.951057 + 0.309017i
0.587785 0.809017i
−0.951057 + 0.309017i
−0.587785 0.809017i
0 −1.26007 1.26007i 0 0 0 −1.39680 + 1.39680i 0 2.17557i 0
193.2 0 0.221232 + 0.221232i 0 0 0 1.26007 1.26007i 0 0.902113i 0
193.3 0 0.642040 + 0.642040i 0 0 0 −0.221232 + 0.221232i 0 0.175571i 0
193.4 0 1.39680 + 1.39680i 0 0 0 −0.642040 + 0.642040i 0 2.90211i 0
1057.1 0 −1.26007 + 1.26007i 0 0 0 −1.39680 1.39680i 0 2.17557i 0
1057.2 0 0.221232 0.221232i 0 0 0 1.26007 + 1.26007i 0 0.902113i 0
1057.3 0 0.642040 0.642040i 0 0 0 −0.221232 0.221232i 0 0.175571i 0
1057.4 0 1.39680 1.39680i 0 0 0 −0.642040 0.642040i 0 2.90211i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1057.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4000.1.p.b yes 8
4.b odd 2 1 4000.1.p.a 8
5.b even 2 1 4000.1.p.a 8
5.c odd 4 1 4000.1.p.a 8
5.c odd 4 1 inner 4000.1.p.b yes 8
20.d odd 2 1 CM 4000.1.p.b yes 8
20.e even 4 1 4000.1.p.a 8
20.e even 4 1 inner 4000.1.p.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4000.1.p.a 8 4.b odd 2 1
4000.1.p.a 8 5.b even 2 1
4000.1.p.a 8 5.c odd 4 1
4000.1.p.a 8 20.e even 4 1
4000.1.p.b yes 8 1.a even 1 1 trivial
4000.1.p.b yes 8 5.c odd 4 1 inner
4000.1.p.b yes 8 20.d odd 2 1 CM
4000.1.p.b yes 8 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 2 T_{3}^{7} + 2 T_{3}^{6} + 11 T_{3}^{4} - 20 T_{3}^{3} + 18 T_{3}^{2} - 6 T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(4000, [\chi])\).

Hecke characteristic polynomials

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