# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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