# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 560.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 3 - 3 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 1 - 3 \zeta_{6} ) q^{7} -6 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 3 - 3 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 1 - 3 \zeta_{6} ) q^{7} -6 \zeta_{6} q^{9} + ( -2 + 2 \zeta_{6} ) q^{11} + 3 q^{15} + ( 4 - 4 \zeta_{6} ) q^{17} -6 \zeta_{6} q^{19} + ( -6 - 3 \zeta_{6} ) q^{21} + 3 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} -9 q^{27} + 9 q^{29} + ( -4 + 4 \zeta_{6} ) q^{31} + 6 \zeta_{6} q^{33} + ( 3 - 2 \zeta_{6} ) q^{35} + 4 \zeta_{6} q^{37} -7 q^{41} + 5 q^{43} + ( 6 - 6 \zeta_{6} ) q^{45} + 8 \zeta_{6} q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} -12 \zeta_{6} q^{51} + ( 2 - 2 \zeta_{6} ) q^{53} -2 q^{55} -18 q^{57} + ( 10 - 10 \zeta_{6} ) q^{59} -\zeta_{6} q^{61} + ( -18 + 12 \zeta_{6} ) q^{63} + ( -9 + 9 \zeta_{6} ) q^{67} + 9 q^{69} -2 q^{71} + ( 4 - 4 \zeta_{6} ) q^{73} + 3 \zeta_{6} q^{75} + ( 4 + 2 \zeta_{6} ) q^{77} + 10 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + 7 q^{83} + 4 q^{85} + ( 27 - 27 \zeta_{6} ) q^{87} -\zeta_{6} q^{89} + 12 \zeta_{6} q^{93} + ( 6 - 6 \zeta_{6} ) q^{95} + 14 q^{97} + 12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} + q^{5} - q^{7} - 6q^{9} + O(q^{10})$$ $$2q + 3q^{3} + q^{5} - q^{7} - 6q^{9} - 2q^{11} + 6q^{15} + 4q^{17} - 6q^{19} - 15q^{21} + 3q^{23} - q^{25} - 18q^{27} + 18q^{29} - 4q^{31} + 6q^{33} + 4q^{35} + 4q^{37} - 14q^{41} + 10q^{43} + 6q^{45} + 8q^{47} - 13q^{49} - 12q^{51} + 2q^{53} - 4q^{55} - 36q^{57} + 10q^{59} - q^{61} - 24q^{63} - 9q^{67} + 18q^{69} - 4q^{71} + 4q^{73} + 3q^{75} + 10q^{77} + 10q^{79} - 9q^{81} + 14q^{83} + 8q^{85} + 27q^{87} - q^{89} + 12q^{93} + 6q^{95} + 28q^{97} + 24q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$241$$ $$337$$ $$351$$ $$421$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 2.59808i 0 0.500000 + 0.866025i 0 −0.500000 2.59808i 0 −3.00000 5.19615i 0
401.1 0 1.50000 + 2.59808i 0 0.500000 0.866025i 0 −0.500000 + 2.59808i 0 −3.00000 + 5.19615i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.q.i 2
4.b odd 2 1 70.2.e.a 2
7.c even 3 1 inner 560.2.q.i 2
7.c even 3 1 3920.2.a.b 1
7.d odd 6 1 3920.2.a.bk 1
12.b even 2 1 630.2.k.f 2
20.d odd 2 1 350.2.e.l 2
20.e even 4 2 350.2.j.f 4
28.d even 2 1 490.2.e.f 2
28.f even 6 1 490.2.a.e 1
28.f even 6 1 490.2.e.f 2
28.g odd 6 1 70.2.e.a 2
28.g odd 6 1 490.2.a.k 1
84.j odd 6 1 4410.2.a.h 1
84.n even 6 1 630.2.k.f 2
84.n even 6 1 4410.2.a.r 1
140.p odd 6 1 350.2.e.l 2
140.p odd 6 1 2450.2.a.b 1
140.s even 6 1 2450.2.a.q 1
140.w even 12 2 350.2.j.f 4
140.w even 12 2 2450.2.c.s 2
140.x odd 12 2 2450.2.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.a 2 4.b odd 2 1
70.2.e.a 2 28.g odd 6 1
350.2.e.l 2 20.d odd 2 1
350.2.e.l 2 140.p odd 6 1
350.2.j.f 4 20.e even 4 2
350.2.j.f 4 140.w even 12 2
490.2.a.e 1 28.f even 6 1
490.2.a.k 1 28.g odd 6 1
490.2.e.f 2 28.d even 2 1
490.2.e.f 2 28.f even 6 1
560.2.q.i 2 1.a even 1 1 trivial
560.2.q.i 2 7.c even 3 1 inner
630.2.k.f 2 12.b even 2 1
630.2.k.f 2 84.n even 6 1
2450.2.a.b 1 140.p odd 6 1
2450.2.a.q 1 140.s even 6 1
2450.2.c.a 2 140.x odd 12 2
2450.2.c.s 2 140.w even 12 2
3920.2.a.b 1 7.c even 3 1
3920.2.a.bk 1 7.d odd 6 1
4410.2.a.h 1 84.j odd 6 1
4410.2.a.r 1 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$:

 $$T_{3}^{2} - 3 T_{3} + 9$$ $$T_{11}^{2} + 2 T_{11} + 4$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 - 3 T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$7 + T + T^{2}$$
$11$ $$4 + 2 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$16 - 4 T + T^{2}$$
$19$ $$36 + 6 T + T^{2}$$
$23$ $$9 - 3 T + T^{2}$$
$29$ $$( -9 + T )^{2}$$
$31$ $$16 + 4 T + T^{2}$$
$37$ $$16 - 4 T + T^{2}$$
$41$ $$( 7 + T )^{2}$$
$43$ $$( -5 + T )^{2}$$
$47$ $$64 - 8 T + T^{2}$$
$53$ $$4 - 2 T + T^{2}$$
$59$ $$100 - 10 T + T^{2}$$
$61$ $$1 + T + T^{2}$$
$67$ $$81 + 9 T + T^{2}$$
$71$ $$( 2 + T )^{2}$$
$73$ $$16 - 4 T + T^{2}$$
$79$ $$100 - 10 T + T^{2}$$
$83$ $$( -7 + T )^{2}$$
$89$ $$1 + T + T^{2}$$
$97$ $$( -14 + T )^{2}$$