# Properties

 Label 3200.1.bz.a Level $3200$ Weight $1$ Character orbit 3200.bz Analytic conductor $1.597$ Analytic rank $0$ Dimension $8$ Projective image $D_{20}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3200.bz (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.59700804043$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{20}^{9} q^{5} -\zeta_{20}^{7} q^{9} +O(q^{10})$$ $$q -\zeta_{20}^{9} q^{5} -\zeta_{20}^{7} q^{9} + ( \zeta_{20}^{3} - \zeta_{20}^{6} ) q^{13} + ( -\zeta_{20} - \zeta_{20}^{2} ) q^{17} -\zeta_{20}^{8} q^{25} + ( -\zeta_{20}^{4} - \zeta_{20}^{8} ) q^{29} + ( \zeta_{20}^{4} + \zeta_{20}^{5} ) q^{37} + ( -\zeta_{20} - \zeta_{20}^{3} ) q^{41} -\zeta_{20}^{6} q^{45} + \zeta_{20}^{5} q^{49} + ( \zeta_{20}^{2} + \zeta_{20}^{5} ) q^{53} + ( -\zeta_{20}^{7} + \zeta_{20}^{9} ) q^{61} + ( \zeta_{20}^{2} - \zeta_{20}^{5} ) q^{65} + ( -\zeta_{20}^{4} + \zeta_{20}^{7} ) q^{73} -\zeta_{20}^{4} q^{81} + ( -1 - \zeta_{20} ) q^{85} + ( 1 + \zeta_{20}^{6} ) q^{89} + ( -\zeta_{20}^{8} + \zeta_{20}^{9} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q - 2 q^{13} - 2 q^{17} + 2 q^{25} + 4 q^{29} - 2 q^{37} - 2 q^{45} + 2 q^{53} + 2 q^{65} + 2 q^{73} + 2 q^{81} - 8 q^{85} + 10 q^{89} + 2 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1151$$ $$2177$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{20}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 −0.951057 − 0.309017i −0.587785 − 0.809017i 0.587785 + 0.809017i 0.951057 + 0.309017i −0.951057 + 0.309017i −0.587785 + 0.809017i 0.587785 − 0.809017i 0.951057 − 0.309017i
0 0 0 −0.951057 + 0.309017i 0 0 0 −0.587785 + 0.809017i 0
833.1 0 0 0 −0.587785 + 0.809017i 0 0 0 0.951057 + 0.309017i 0
1217.1 0 0 0 0.587785 0.809017i 0 0 0 −0.951057 0.309017i 0
1473.1 0 0 0 0.951057 0.309017i 0 0 0 0.587785 0.809017i 0
2113.1 0 0 0 −0.951057 0.309017i 0 0 0 −0.587785 0.809017i 0
2497.1 0 0 0 −0.587785 0.809017i 0 0 0 0.951057 0.309017i 0
2753.1 0 0 0 0.587785 + 0.809017i 0 0 0 −0.951057 + 0.309017i 0
3137.1 0 0 0 0.951057 + 0.309017i 0 0 0 0.587785 + 0.809017i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3137.1 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 $$\Q(\sqrt{-1})$$
200.v even 20 1 inner
200.x odd 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.1.bz.a 8
4.b odd 2 1 CM 3200.1.bz.a 8
8.b even 2 1 3200.1.bz.b yes 8
8.d odd 2 1 3200.1.bz.b yes 8
25.f odd 20 1 3200.1.bz.b yes 8
100.l even 20 1 3200.1.bz.b yes 8
200.v even 20 1 inner 3200.1.bz.a 8
200.x odd 20 1 inner 3200.1.bz.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.1.bz.a 8 1.a even 1 1 trivial
3200.1.bz.a 8 4.b odd 2 1 CM
3200.1.bz.a 8 200.v even 20 1 inner
3200.1.bz.a 8 200.x odd 20 1 inner
3200.1.bz.b yes 8 8.b even 2 1
3200.1.bz.b yes 8 8.d odd 2 1
3200.1.bz.b yes 8 25.f odd 20 1
3200.1.bz.b yes 8 100.l even 20 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 $$S_{1}^{\mathrm{new}}(3200, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$T^{8}$$
$13$ $$1 - 4 T + 13 T^{2} - 10 T^{3} - 4 T^{4} + 2 T^{6} + 2 T^{7} + T^{8}$$
$17$ $$1 - 4 T + 13 T^{2} - 10 T^{3} - 4 T^{4} + 2 T^{6} + 2 T^{7} + T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$( 1 - 3 T + 4 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$31$ $$T^{8}$$
$37$ $$1 - 4 T - 2 T^{2} + 10 T^{3} + 16 T^{4} + 10 T^{5} + 7 T^{6} + 2 T^{7} + T^{8}$$
$41$ $$25 + 25 T^{2} + 10 T^{4} + T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$1 + 4 T - 2 T^{2} - 10 T^{3} + 16 T^{4} - 10 T^{5} + 7 T^{6} - 2 T^{7} + T^{8}$$
$59$ $$T^{8}$$
$61$ $$1 + T^{2} + 6 T^{4} - 4 T^{6} + T^{8}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$1 + 4 T + 13 T^{2} + 10 T^{3} - 4 T^{4} + 2 T^{6} - 2 T^{7} + T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$( 5 - 10 T + 10 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$97$ $$1 + 4 T + 13 T^{2} + 10 T^{3} - 4 T^{4} + 2 T^{6} - 2 T^{7} + T^{8}$$