# Properties

 Label 4000.1.bo.b 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

# Related objects

## 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.

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

## 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: 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 $$\Q(\sqrt{-1})$$
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.b 8
4.b odd 2 1 CM 4000.1.bo.b 8
5.b even 2 1 800.1.bo.a 8
5.c odd 4 1 4000.1.bo.a 8
5.c odd 4 1 4000.1.bo.c 8
20.d odd 2 1 800.1.bo.a 8
20.e even 4 1 4000.1.bo.a 8
20.e even 4 1 4000.1.bo.c 8
25.d even 5 1 4000.1.bo.a 8
25.e even 10 1 4000.1.bo.c 8
25.f odd 20 1 800.1.bo.a 8
25.f odd 20 1 inner 4000.1.bo.b 8
40.e odd 2 1 1600.1.bw.a 8
40.f even 2 1 1600.1.bw.a 8
100.h odd 10 1 4000.1.bo.c 8
100.j odd 10 1 4000.1.bo.a 8
100.l even 20 1 800.1.bo.a 8
100.l even 20 1 inner 4000.1.bo.b 8
200.v even 20 1 1600.1.bw.a 8
200.x odd 20 1 1600.1.bw.a 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{8} + 2 T_{13}^{7} + 2 T_{13}^{6} - 4 T_{13}^{4} - 10 T_{13}^{3} + 13 T_{13}^{2} - 4 T_{13} + 1$$ acting on $$S_{1}^{\mathrm{new}}(4000, [\chi])$$.

## Hecke Characteristic Polynomials

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