# Properties

 Label 560.1.bt.a Level 560 Weight 1 Character orbit 560.bt Analytic conductor 0.279 Analytic rank 0 Dimension 4 Projective image $$D_{6}$$ CM discriminant -20 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 560.bt (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.279476407074$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{6}$$ Projective field Galois closure of 6.0.3841600.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{3} + \zeta_{12}^{2} q^{5} -\zeta_{12} q^{7} + ( -1 - \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{3} + \zeta_{12}^{2} q^{5} -\zeta_{12} q^{7} + ( -1 - \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{9} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{15} + ( -1 + \zeta_{12}^{4} ) q^{21} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{23} + \zeta_{12}^{4} q^{25} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{27} - q^{29} -\zeta_{12}^{3} q^{35} + q^{41} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{43} + ( 1 - \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{45} + \zeta_{12}^{2} q^{49} -\zeta_{12}^{2} q^{61} + ( \zeta_{12} + \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{63} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{67} + ( 2 + \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{69} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{75} + \zeta_{12}^{4} q^{81} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{83} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{87} + \zeta_{12}^{2} q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{5} - 4q^{9} + O(q^{10})$$ $$4q + 2q^{5} - 4q^{9} - 6q^{21} - 2q^{25} - 4q^{29} + 4q^{41} + 4q^{45} + 2q^{49} - 2q^{61} + 12q^{69} - 2q^{81} + 2q^{89} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$351$$ $$421$$ $$\chi(n)$$ $$-\zeta_{12}^{2}$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
0 −0.866025 1.50000i 0 0.500000 0.866025i 0 0.866025 0.500000i 0 −1.00000 + 1.73205i 0
79.2 0 0.866025 + 1.50000i 0 0.500000 0.866025i 0 −0.866025 + 0.500000i 0 −1.00000 + 1.73205i 0
319.1 0 −0.866025 + 1.50000i 0 0.500000 + 0.866025i 0 0.866025 + 0.500000i 0 −1.00000 1.73205i 0
319.2 0 0.866025 1.50000i 0 0.500000 + 0.866025i 0 −0.866025 0.500000i 0 −1.00000 1.73205i 0
 $$n$$: e.g. 2-40 or 990-1000 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 $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner
35.j even 6 1 inner
140.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.1.bt.a 4
4.b odd 2 1 inner 560.1.bt.a 4
5.b even 2 1 inner 560.1.bt.a 4
5.c odd 4 1 2800.1.ce.a 2
5.c odd 4 1 2800.1.ce.b 2
7.b odd 2 1 3920.1.bt.c 4
7.c even 3 1 inner 560.1.bt.a 4
7.c even 3 1 3920.1.j.b 2
7.d odd 6 1 3920.1.j.d 2
7.d odd 6 1 3920.1.bt.c 4
8.b even 2 1 2240.1.bt.c 4
8.d odd 2 1 2240.1.bt.c 4
20.d odd 2 1 CM 560.1.bt.a 4
20.e even 4 1 2800.1.ce.a 2
20.e even 4 1 2800.1.ce.b 2
28.d even 2 1 3920.1.bt.c 4
28.f even 6 1 3920.1.j.d 2
28.f even 6 1 3920.1.bt.c 4
28.g odd 6 1 inner 560.1.bt.a 4
28.g odd 6 1 3920.1.j.b 2
35.c odd 2 1 3920.1.bt.c 4
35.i odd 6 1 3920.1.j.d 2
35.i odd 6 1 3920.1.bt.c 4
35.j even 6 1 inner 560.1.bt.a 4
35.j even 6 1 3920.1.j.b 2
35.l odd 12 1 2800.1.ce.a 2
35.l odd 12 1 2800.1.ce.b 2
40.e odd 2 1 2240.1.bt.c 4
40.f even 2 1 2240.1.bt.c 4
56.k odd 6 1 2240.1.bt.c 4
56.p even 6 1 2240.1.bt.c 4
140.c even 2 1 3920.1.bt.c 4
140.p odd 6 1 inner 560.1.bt.a 4
140.p odd 6 1 3920.1.j.b 2
140.s even 6 1 3920.1.j.d 2
140.s even 6 1 3920.1.bt.c 4
140.w even 12 1 2800.1.ce.a 2
140.w even 12 1 2800.1.ce.b 2
280.bf even 6 1 2240.1.bt.c 4
280.bi odd 6 1 2240.1.bt.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.1.bt.a 4 1.a even 1 1 trivial
560.1.bt.a 4 4.b odd 2 1 inner
560.1.bt.a 4 5.b even 2 1 inner
560.1.bt.a 4 7.c even 3 1 inner
560.1.bt.a 4 20.d odd 2 1 CM
560.1.bt.a 4 28.g odd 6 1 inner
560.1.bt.a 4 35.j even 6 1 inner
560.1.bt.a 4 140.p odd 6 1 inner
2240.1.bt.c 4 8.b even 2 1
2240.1.bt.c 4 8.d odd 2 1
2240.1.bt.c 4 40.e odd 2 1
2240.1.bt.c 4 40.f even 2 1
2240.1.bt.c 4 56.k odd 6 1
2240.1.bt.c 4 56.p even 6 1
2240.1.bt.c 4 280.bf even 6 1
2240.1.bt.c 4 280.bi odd 6 1
2800.1.ce.a 2 5.c odd 4 1
2800.1.ce.a 2 20.e even 4 1
2800.1.ce.a 2 35.l odd 12 1
2800.1.ce.a 2 140.w even 12 1
2800.1.ce.b 2 5.c odd 4 1
2800.1.ce.b 2 20.e even 4 1
2800.1.ce.b 2 35.l odd 12 1
2800.1.ce.b 2 140.w even 12 1
3920.1.j.b 2 7.c even 3 1
3920.1.j.b 2 28.g odd 6 1
3920.1.j.b 2 35.j even 6 1
3920.1.j.b 2 140.p odd 6 1
3920.1.j.d 2 7.d odd 6 1
3920.1.j.d 2 28.f even 6 1
3920.1.j.d 2 35.i odd 6 1
3920.1.j.d 2 140.s even 6 1
3920.1.bt.c 4 7.b odd 2 1
3920.1.bt.c 4 7.d odd 6 1
3920.1.bt.c 4 28.d even 2 1
3920.1.bt.c 4 28.f even 6 1
3920.1.bt.c 4 35.c odd 2 1
3920.1.bt.c 4 35.i odd 6 1
3920.1.bt.c 4 140.c even 2 1
3920.1.bt.c 4 140.s even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(560, [\chi])$$.

## Hecke characteristic polynomials

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