# Properties

 Label 1045.1.br.b Level $1045$ Weight $1$ Character orbit 1045.br Analytic conductor $0.522$ Analytic rank $0$ Dimension $8$ Projective image $D_{20}$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1045.br (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.521522938201$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{20}^{3} q^{4} + \zeta_{20}^{8} q^{5} + ( \zeta_{20}^{5} - \zeta_{20}^{6} ) q^{7} + \zeta_{20}^{9} q^{9} +O(q^{10})$$ $$q -\zeta_{20}^{3} q^{4} + \zeta_{20}^{8} q^{5} + ( \zeta_{20}^{5} - \zeta_{20}^{6} ) q^{7} + \zeta_{20}^{9} q^{9} + \zeta_{20}^{7} q^{11} + \zeta_{20}^{6} q^{16} + ( -1 + \zeta_{20}^{7} ) q^{17} + \zeta_{20}^{2} q^{19} + \zeta_{20} q^{20} + ( \zeta_{20}^{6} - \zeta_{20}^{9} ) q^{23} -\zeta_{20}^{6} q^{25} + ( -\zeta_{20}^{8} + \zeta_{20}^{9} ) q^{28} + ( -\zeta_{20}^{3} + \zeta_{20}^{4} ) q^{35} + \zeta_{20}^{2} q^{36} + ( -\zeta_{20} + \zeta_{20}^{4} ) q^{43} + q^{44} -\zeta_{20}^{7} q^{45} + ( -\zeta_{20} + \zeta_{20}^{8} ) q^{47} + ( -1 + \zeta_{20} - \zeta_{20}^{2} ) q^{49} -\zeta_{20}^{5} q^{55} + ( \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{61} + ( -\zeta_{20}^{4} + \zeta_{20}^{5} ) q^{63} -\zeta_{20}^{9} q^{64} + ( 1 + \zeta_{20}^{3} ) q^{68} + ( \zeta_{20}^{3} + \zeta_{20}^{8} ) q^{73} -\zeta_{20}^{5} q^{76} + ( -\zeta_{20}^{2} + \zeta_{20}^{3} ) q^{77} -\zeta_{20}^{4} q^{80} -\zeta_{20}^{8} q^{81} + ( -\zeta_{20}^{3} - \zeta_{20}^{4} ) q^{83} + ( -\zeta_{20}^{5} - \zeta_{20}^{8} ) q^{85} + ( -\zeta_{20}^{2} - \zeta_{20}^{9} ) q^{92} - q^{95} -\zeta_{20}^{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{5} - 2q^{7} + O(q^{10})$$ $$8q - 2q^{5} - 2q^{7} + 2q^{16} - 8q^{17} + 2q^{19} + 2q^{23} - 2q^{25} + 2q^{28} - 2q^{35} + 2q^{36} - 2q^{43} + 8q^{44} - 2q^{47} - 10q^{49} + 2q^{63} + 8q^{68} - 2q^{73} - 2q^{77} + 2q^{80} + 2q^{81} + 2q^{83} + 2q^{85} - 2q^{92} - 8q^{95} - 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times$$.

 $$n$$ $$496$$ $$761$$ $$837$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{20}^{8}$$ $$-\zeta_{20}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
18.1
 −0.587785 − 0.809017i 0.587785 + 0.809017i 0.951057 + 0.309017i −0.951057 + 0.309017i −0.951057 − 0.309017i 0.951057 − 0.309017i 0.587785 − 0.809017i −0.587785 + 0.809017i
0 0 −0.951057 + 0.309017i 0.309017 + 0.951057i 0 −0.809017 + 1.58779i 0 0.587785 0.809017i 0
227.1 0 0 0.951057 0.309017i 0.309017 + 0.951057i 0 −0.809017 0.412215i 0 −0.587785 + 0.809017i 0
303.1 0 0 −0.587785 0.809017i −0.809017 + 0.587785i 0 0.309017 + 0.0489435i 0 −0.951057 + 0.309017i 0
398.1 0 0 0.587785 0.809017i −0.809017 0.587785i 0 0.309017 + 1.95106i 0 0.951057 + 0.309017i 0
512.1 0 0 0.587785 + 0.809017i −0.809017 + 0.587785i 0 0.309017 1.95106i 0 0.951057 0.309017i 0
607.1 0 0 −0.587785 + 0.809017i −0.809017 0.587785i 0 0.309017 0.0489435i 0 −0.951057 0.309017i 0
778.1 0 0 0.951057 + 0.309017i 0.309017 0.951057i 0 −0.809017 + 0.412215i 0 −0.587785 0.809017i 0
987.1 0 0 −0.951057 0.309017i 0.309017 0.951057i 0 −0.809017 1.58779i 0 0.587785 + 0.809017i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 987.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
55.l even 20 1 inner
1045.br odd 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.1.br.b yes 8
5.c odd 4 1 1045.1.br.a 8
11.d odd 10 1 1045.1.br.a 8
19.b odd 2 1 CM 1045.1.br.b yes 8
55.l even 20 1 inner 1045.1.br.b yes 8
95.g even 4 1 1045.1.br.a 8
209.k even 10 1 1045.1.br.a 8
1045.br odd 20 1 inner 1045.1.br.b yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.1.br.a 8 5.c odd 4 1
1045.1.br.a 8 11.d odd 10 1
1045.1.br.a 8 95.g even 4 1
1045.1.br.a 8 209.k even 10 1
1045.1.br.b yes 8 1.a even 1 1 trivial
1045.1.br.b yes 8 19.b odd 2 1 CM
1045.1.br.b yes 8 55.l even 20 1 inner
1045.1.br.b yes 8 1045.br odd 20 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + \cdots$$ acting on $$S_{1}^{\mathrm{new}}(1045, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$7$ $$1 - 4 T - 2 T^{2} + 10 T^{3} + 16 T^{4} + 10 T^{5} + 7 T^{6} + 2 T^{7} + T^{8}$$
$11$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$13$ $$T^{8}$$
$17$ $$1 + 4 T + 18 T^{2} + 40 T^{3} + 56 T^{4} + 50 T^{5} + 27 T^{6} + 8 T^{7} + T^{8}$$
$19$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$23$ $$1 - 6 T + 18 T^{2} - 20 T^{3} + 11 T^{4} + 2 T^{6} - 2 T^{7} + T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$1 + 6 T + 18 T^{2} + 20 T^{3} + 11 T^{4} + 2 T^{6} + 2 T^{7} + T^{8}$$
$47$ $$1 + 6 T + 13 T^{2} + 10 T^{3} + 16 T^{4} + 10 T^{5} + 2 T^{6} + 2 T^{7} + T^{8}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$1 - 4 T^{2} + 6 T^{4} + T^{6} + T^{8}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$16 + 16 T + 8 T^{2} - 4 T^{4} + 2 T^{6} + 2 T^{7} + T^{8}$$
$79$ $$T^{8}$$
$83$ $$1 - 6 T + 13 T^{2} - 10 T^{3} + 16 T^{4} - 10 T^{5} + 2 T^{6} - 2 T^{7} + T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$