Properties

Label 4000.1.cq.a
Level 4000
Weight 1
Character orbit 4000.cq
Analytic conductor 1.996
Analytic rank 0
Dimension 40
Projective image \(D_{100}\)
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.cq (of order \(100\), degree \(40\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.99626005053\)
Analytic rank: \(0\)
Dimension: \(40\)
Coefficient field: \(\Q(\zeta_{100})\)
Defining polynomial: \(x^{40} - x^{30} + x^{20} - x^{10} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{100}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{100} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{100}^{13} q^{5} -\zeta_{100}^{49} q^{9} +O(q^{10})\) \( q + \zeta_{100}^{13} q^{5} -\zeta_{100}^{49} q^{9} + ( \zeta_{100}^{2} - \zeta_{100}^{21} ) q^{13} + ( \zeta_{100}^{17} - \zeta_{100}^{44} ) q^{17} + \zeta_{100}^{26} q^{25} + ( -\zeta_{100}^{6} - \zeta_{100}^{28} ) q^{29} + ( -\zeta_{100}^{18} + \zeta_{100}^{35} ) q^{37} + ( -\zeta_{100}^{11} + \zeta_{100}^{47} ) q^{41} + \zeta_{100}^{12} q^{45} + \zeta_{100}^{45} q^{49} + ( \zeta_{100}^{5} - \zeta_{100}^{14} ) q^{53} + ( \zeta_{100}^{9} - \zeta_{100}^{33} ) q^{61} + ( \zeta_{100}^{15} - \zeta_{100}^{34} ) q^{65} + ( \zeta_{100}^{8} + \zeta_{100}^{29} ) q^{73} -\zeta_{100}^{48} q^{81} + ( \zeta_{100}^{7} + \zeta_{100}^{30} ) q^{85} + ( -\zeta_{100}^{22} - \zeta_{100}^{40} ) q^{89} + ( \zeta_{100}^{23} + \zeta_{100}^{36} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q + O(q^{10}) \) \( 40q + 10q^{85} + 10q^{89} + O(q^{100}) \)

Character values

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

\(n\) \(1377\) \(2501\) \(2751\)
\(\chi(n)\) \(\zeta_{100}^{7}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
−0.368125 0.929776i
0.844328 0.535827i
−0.770513 0.637424i
0.998027 0.0627905i
0.125333 0.992115i
0.125333 + 0.992115i
−0.125333 0.992115i
−0.125333 + 0.992115i
−0.998027 + 0.0627905i
0.770513 + 0.637424i
−0.844328 + 0.535827i
0.368125 + 0.929776i
−0.904827 0.425779i
−0.904827 + 0.425779i
−0.982287 0.187381i
−0.982287 + 0.187381i
−0.248690 + 0.968583i
−0.368125 + 0.929776i
0.248690 + 0.968583i
−0.770513 + 0.637424i
0 0 0 0.982287 0.187381i 0 0 0 −0.368125 + 0.929776i 0
97.1 0 0 0 0.481754 0.876307i 0 0 0 0.844328 + 0.535827i 0
353.1 0 0 0 0.904827 0.425779i 0 0 0 −0.770513 + 0.637424i 0
417.1 0 0 0 0.684547 0.728969i 0 0 0 0.998027 + 0.0627905i 0
513.1 0 0 0 0.998027 + 0.0627905i 0 0 0 0.125333 + 0.992115i 0
577.1 0 0 0 0.998027 0.0627905i 0 0 0 0.125333 0.992115i 0
673.1 0 0 0 −0.998027 + 0.0627905i 0 0 0 −0.125333 + 0.992115i 0
737.1 0 0 0 −0.998027 0.0627905i 0 0 0 −0.125333 0.992115i 0
833.1 0 0 0 −0.684547 + 0.728969i 0 0 0 −0.998027 0.0627905i 0
897.1 0 0 0 −0.904827 + 0.425779i 0 0 0 0.770513 0.637424i 0
1153.1 0 0 0 −0.481754 + 0.876307i 0 0 0 −0.844328 0.535827i 0
1217.1 0 0 0 −0.982287 + 0.187381i 0 0 0 0.368125 0.929776i 0
1313.1 0 0 0 −0.844328 + 0.535827i 0 0 0 −0.904827 + 0.425779i 0
1377.1 0 0 0 −0.844328 0.535827i 0 0 0 −0.904827 0.425779i 0
1473.1 0 0 0 0.770513 0.637424i 0 0 0 −0.982287 + 0.187381i 0
1537.1 0 0 0 0.770513 + 0.637424i 0 0 0 −0.982287 0.187381i 0
1633.1 0 0 0 0.125333 0.992115i 0 0 0 −0.248690 0.968583i 0
1697.1 0 0 0 0.982287 + 0.187381i 0 0 0 −0.368125 0.929776i 0
1953.1 0 0 0 −0.125333 0.992115i 0 0 0 0.248690 0.968583i 0
2017.1 0 0 0 0.904827 + 0.425779i 0 0 0 −0.770513 0.637424i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3937.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
125.i odd 100 1 inner
500.r even 100 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4000.1.cq.a 40
4.b odd 2 1 CM 4000.1.cq.a 40
125.i odd 100 1 inner 4000.1.cq.a 40
500.r even 100 1 inner 4000.1.cq.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4000.1.cq.a 40 1.a even 1 1 trivial
4000.1.cq.a 40 4.b odd 2 1 CM
4000.1.cq.a 40 125.i odd 100 1 inner
4000.1.cq.a 40 500.r even 100 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(4000, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T^{20} + T^{40} - T^{60} + T^{80} \)
$5$ \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \)
$7$ \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{5} \)
$11$ \( ( 1 - T^{10} + T^{20} - T^{30} + T^{40} )^{2} \)
$13$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 - T^{10} + T^{20} - T^{30} + T^{40} ) \)
$17$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 - T^{10} + T^{20} - T^{30} + T^{40} ) \)
$19$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
$23$ \( 1 - T^{20} + T^{40} - T^{60} + T^{80} \)
$29$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
$31$ \( ( 1 - T^{10} + T^{20} - T^{30} + T^{40} )^{2} \)
$37$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
$41$ \( ( 1 - T^{10} + T^{20} - T^{30} + T^{40} )^{2} \)
$43$ \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{5} \)
$47$ \( 1 - T^{20} + T^{40} - T^{60} + T^{80} \)
$53$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
$59$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
$61$ \( ( 1 - T^{10} + T^{20} - T^{30} + T^{40} )^{2} \)
$67$ \( 1 - T^{20} + T^{40} - T^{60} + T^{80} \)
$71$ \( ( 1 - T^{10} + T^{20} - T^{30} + T^{40} )^{2} \)
$73$ \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2}( 1 - T^{10} + T^{20} - T^{30} + T^{40} ) \)
$79$ \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
$83$ \( 1 - T^{20} + T^{40} - T^{60} + T^{80} \)
$89$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{10}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
$97$ \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2}( 1 - T^{10} + T^{20} - T^{30} + T^{40} ) \)
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