# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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