# Properties

 Label 560.2.bw.d Level $560$ Weight $2$ Character orbit 560.bw Analytic conductor $4.472$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 560.bw (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{5} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} -3 \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + ( \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{5} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{7} -3 \zeta_{12}^{2} q^{9} + ( 3 - 3 \zeta_{12}^{2} ) q^{11} -5 \zeta_{12}^{3} q^{13} -2 \zeta_{12} q^{17} + 5 \zeta_{12}^{2} q^{19} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{23} + ( -3 + 4 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{25} + 4 q^{29} + ( -2 + 2 \zeta_{12}^{2} ) q^{31} + ( -2 + 2 \zeta_{12} - \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{35} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{37} + 3 q^{41} -2 \zeta_{12}^{3} q^{43} + ( 6 - 3 \zeta_{12} - 6 \zeta_{12}^{2} ) q^{45} + ( -7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{47} + ( -5 + 8 \zeta_{12}^{2} ) q^{49} -9 \zeta_{12} q^{53} + ( 6 - 3 \zeta_{12}^{3} ) q^{55} + ( 4 - 4 \zeta_{12}^{2} ) q^{59} -6 \zeta_{12}^{2} q^{61} + ( -3 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{63} + ( 10 \zeta_{12} - 5 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{65} -2 \zeta_{12} q^{67} + 6 q^{71} + 16 \zeta_{12} q^{73} + ( -9 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{77} -14 \zeta_{12}^{2} q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + 6 \zeta_{12}^{3} q^{83} + ( -2 - 4 \zeta_{12}^{3} ) q^{85} + 2 \zeta_{12}^{2} q^{89} + ( -15 + 10 \zeta_{12}^{2} ) q^{91} + ( -10 + 5 \zeta_{12} + 10 \zeta_{12}^{2} ) q^{95} + 12 \zeta_{12}^{3} q^{97} -9 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{5} - 6 q^{9} + O(q^{10})$$ $$4 q + 4 q^{5} - 6 q^{9} + 6 q^{11} + 10 q^{19} - 6 q^{25} + 16 q^{29} - 4 q^{31} - 10 q^{35} + 12 q^{41} + 12 q^{45} - 4 q^{49} + 24 q^{55} + 8 q^{59} - 12 q^{61} - 10 q^{65} + 24 q^{71} - 28 q^{79} - 18 q^{81} - 8 q^{85} + 4 q^{89} - 40 q^{91} - 20 q^{95} - 36 q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
0 0 0 0.133975 2.23205i 0 1.73205 2.00000i 0 −1.50000 + 2.59808i 0
289.2 0 0 0 1.86603 1.23205i 0 −1.73205 + 2.00000i 0 −1.50000 + 2.59808i 0
529.1 0 0 0 0.133975 + 2.23205i 0 1.73205 + 2.00000i 0 −1.50000 2.59808i 0
529.2 0 0 0 1.86603 + 1.23205i 0 −1.73205 2.00000i 0 −1.50000 2.59808i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.bw.d 4
4.b odd 2 1 70.2.i.b 4
5.b even 2 1 inner 560.2.bw.d 4
7.c even 3 1 inner 560.2.bw.d 4
12.b even 2 1 630.2.u.a 4
20.d odd 2 1 70.2.i.b 4
20.e even 4 1 350.2.e.c 2
20.e even 4 1 350.2.e.j 2
28.d even 2 1 490.2.i.a 4
28.f even 6 1 490.2.c.d 2
28.f even 6 1 490.2.i.a 4
28.g odd 6 1 70.2.i.b 4
28.g odd 6 1 490.2.c.a 2
35.j even 6 1 inner 560.2.bw.d 4
60.h even 2 1 630.2.u.a 4
84.n even 6 1 630.2.u.a 4
140.c even 2 1 490.2.i.a 4
140.p odd 6 1 70.2.i.b 4
140.p odd 6 1 490.2.c.a 2
140.s even 6 1 490.2.c.d 2
140.s even 6 1 490.2.i.a 4
140.w even 12 1 350.2.e.c 2
140.w even 12 1 350.2.e.j 2
140.w even 12 1 2450.2.a.k 1
140.w even 12 1 2450.2.a.ba 1
140.x odd 12 1 2450.2.a.j 1
140.x odd 12 1 2450.2.a.bb 1
420.ba even 6 1 630.2.u.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.b 4 4.b odd 2 1
70.2.i.b 4 20.d odd 2 1
70.2.i.b 4 28.g odd 6 1
70.2.i.b 4 140.p odd 6 1
350.2.e.c 2 20.e even 4 1
350.2.e.c 2 140.w even 12 1
350.2.e.j 2 20.e even 4 1
350.2.e.j 2 140.w even 12 1
490.2.c.a 2 28.g odd 6 1
490.2.c.a 2 140.p odd 6 1
490.2.c.d 2 28.f even 6 1
490.2.c.d 2 140.s even 6 1
490.2.i.a 4 28.d even 2 1
490.2.i.a 4 28.f even 6 1
490.2.i.a 4 140.c even 2 1
490.2.i.a 4 140.s even 6 1
560.2.bw.d 4 1.a even 1 1 trivial
560.2.bw.d 4 5.b even 2 1 inner
560.2.bw.d 4 7.c even 3 1 inner
560.2.bw.d 4 35.j even 6 1 inner
630.2.u.a 4 12.b even 2 1
630.2.u.a 4 60.h even 2 1
630.2.u.a 4 84.n even 6 1
630.2.u.a 4 420.ba even 6 1
2450.2.a.j 1 140.x odd 12 1
2450.2.a.k 1 140.w even 12 1
2450.2.a.ba 1 140.w even 12 1
2450.2.a.bb 1 140.x odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$25 - 20 T + 11 T^{2} - 4 T^{3} + T^{4}$$
$7$ $$49 + 2 T^{2} + T^{4}$$
$11$ $$( 9 - 3 T + T^{2} )^{2}$$
$13$ $$( 25 + T^{2} )^{2}$$
$17$ $$16 - 4 T^{2} + T^{4}$$
$19$ $$( 25 - 5 T + T^{2} )^{2}$$
$23$ $$2401 - 49 T^{2} + T^{4}$$
$29$ $$( -4 + T )^{4}$$
$31$ $$( 4 + 2 T + T^{2} )^{2}$$
$37$ $$1 - T^{2} + T^{4}$$
$41$ $$( -3 + T )^{4}$$
$43$ $$( 4 + T^{2} )^{2}$$
$47$ $$2401 - 49 T^{2} + T^{4}$$
$53$ $$6561 - 81 T^{2} + T^{4}$$
$59$ $$( 16 - 4 T + T^{2} )^{2}$$
$61$ $$( 36 + 6 T + T^{2} )^{2}$$
$67$ $$16 - 4 T^{2} + T^{4}$$
$71$ $$( -6 + T )^{4}$$
$73$ $$65536 - 256 T^{2} + T^{4}$$
$79$ $$( 196 + 14 T + T^{2} )^{2}$$
$83$ $$( 36 + T^{2} )^{2}$$
$89$ $$( 4 - 2 T + T^{2} )^{2}$$
$97$ $$( 144 + T^{2} )^{2}$$