# Properties

 Label 560.2.q.f 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: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} + ( 6 - 6 \zeta_{6} ) q^{11} + 2 q^{13} + q^{15} + ( 6 - 6 \zeta_{6} ) q^{17} + 8 \zeta_{6} q^{19} + ( -2 + 3 \zeta_{6} ) q^{21} + 3 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 5 q^{27} + 3 q^{29} + ( 2 - 2 \zeta_{6} ) q^{31} -6 \zeta_{6} q^{33} + ( -1 - 2 \zeta_{6} ) q^{35} -8 \zeta_{6} q^{37} + ( 2 - 2 \zeta_{6} ) q^{39} -3 q^{41} -5 q^{43} + ( -2 + 2 \zeta_{6} ) q^{45} + ( 8 - 5 \zeta_{6} ) q^{49} -6 \zeta_{6} q^{51} + ( -12 + 12 \zeta_{6} ) q^{53} + 6 q^{55} + 8 q^{57} + \zeta_{6} q^{61} + ( -2 - 4 \zeta_{6} ) q^{63} + 2 \zeta_{6} q^{65} + ( -7 + 7 \zeta_{6} ) q^{67} + 3 q^{69} + ( 10 - 10 \zeta_{6} ) q^{73} + \zeta_{6} q^{75} + ( -12 + 18 \zeta_{6} ) q^{77} -4 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -3 q^{83} + 6 q^{85} + ( 3 - 3 \zeta_{6} ) q^{87} + 3 \zeta_{6} q^{89} + ( -6 + 2 \zeta_{6} ) q^{91} -2 \zeta_{6} q^{93} + ( -8 + 8 \zeta_{6} ) q^{95} -10 q^{97} + 12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} + q^{5} - 5q^{7} + 2q^{9} + O(q^{10})$$ $$2q + q^{3} + q^{5} - 5q^{7} + 2q^{9} + 6q^{11} + 4q^{13} + 2q^{15} + 6q^{17} + 8q^{19} - q^{21} + 3q^{23} - q^{25} + 10q^{27} + 6q^{29} + 2q^{31} - 6q^{33} - 4q^{35} - 8q^{37} + 2q^{39} - 6q^{41} - 10q^{43} - 2q^{45} + 11q^{49} - 6q^{51} - 12q^{53} + 12q^{55} + 16q^{57} + q^{61} - 8q^{63} + 2q^{65} - 7q^{67} + 6q^{69} + 10q^{73} + q^{75} - 6q^{77} - 4q^{79} - q^{81} - 6q^{83} + 12q^{85} + 3q^{87} + 3q^{89} - 10q^{91} - 2q^{93} - 8q^{95} - 20q^{97} + 24q^{99} + O(q^{100})$$

## 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 0.500000 0.866025i 0 0.500000 + 0.866025i 0 −2.50000 + 0.866025i 0 1.00000 + 1.73205i 0
401.1 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 −2.50000 0.866025i 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
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.f 2
4.b odd 2 1 140.2.i.a 2
7.c even 3 1 inner 560.2.q.f 2
7.c even 3 1 3920.2.a.k 1
7.d odd 6 1 3920.2.a.w 1
12.b even 2 1 1260.2.s.c 2
20.d odd 2 1 700.2.i.b 2
20.e even 4 2 700.2.r.a 4
28.d even 2 1 980.2.i.f 2
28.f even 6 1 980.2.a.e 1
28.f even 6 1 980.2.i.f 2
28.g odd 6 1 140.2.i.a 2
28.g odd 6 1 980.2.a.g 1
84.j odd 6 1 8820.2.a.a 1
84.n even 6 1 1260.2.s.c 2
84.n even 6 1 8820.2.a.p 1
140.p odd 6 1 700.2.i.b 2
140.p odd 6 1 4900.2.a.i 1
140.s even 6 1 4900.2.a.q 1
140.w even 12 2 700.2.r.a 4
140.w even 12 2 4900.2.e.m 2
140.x odd 12 2 4900.2.e.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.a 2 4.b odd 2 1
140.2.i.a 2 28.g odd 6 1
560.2.q.f 2 1.a even 1 1 trivial
560.2.q.f 2 7.c even 3 1 inner
700.2.i.b 2 20.d odd 2 1
700.2.i.b 2 140.p odd 6 1
700.2.r.a 4 20.e even 4 2
700.2.r.a 4 140.w even 12 2
980.2.a.e 1 28.f even 6 1
980.2.a.g 1 28.g odd 6 1
980.2.i.f 2 28.d even 2 1
980.2.i.f 2 28.f even 6 1
1260.2.s.c 2 12.b even 2 1
1260.2.s.c 2 84.n even 6 1
3920.2.a.k 1 7.c even 3 1
3920.2.a.w 1 7.d odd 6 1
4900.2.a.i 1 140.p odd 6 1
4900.2.a.q 1 140.s even 6 1
4900.2.e.m 2 140.w even 12 2
4900.2.e.n 2 140.x odd 12 2
8820.2.a.a 1 84.j odd 6 1
8820.2.a.p 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 $$S_{2}^{\mathrm{new}}(560, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{11}^{2} - 6 T_{11} + 36$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}$$
$5$ $$1 - T + T^{2}$$
$7$ $$1 + 5 T + 7 T^{2}$$
$11$ $$1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 2 T + 13 T^{2} )^{2}$$
$17$ $$1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4}$$
$19$ $$( 1 - 7 T + 19 T^{2} )( 1 - T + 19 T^{2} )$$
$23$ $$1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 3 T + 29 T^{2} )^{2}$$
$31$ $$1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4}$$
$37$ $$1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 3 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + 5 T + 43 T^{2} )^{2}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$
$67$ $$1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} )$$
$79$ $$( 1 - 13 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} )$$
$83$ $$( 1 + 3 T + 83 T^{2} )^{2}$$
$89$ $$1 - 3 T - 80 T^{2} - 267 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 10 T + 97 T^{2} )^{2}$$