# Properties

 Label 560.2.g.a Level $560$ Weight $2$ Character orbit 560.g Analytic conductor $4.472$ Analytic rank $1$ 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.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + ( -2 - i ) q^{5} -i q^{7} -6 q^{9} +O(q^{10})$$ $$q + 3 i q^{3} + ( -2 - i ) q^{5} -i q^{7} -6 q^{9} -3 q^{11} -i q^{13} + ( 3 - 6 i ) q^{15} -5 i q^{17} -8 q^{19} + 3 q^{21} + 2 i q^{23} + ( 3 + 4 i ) q^{25} -9 i q^{27} + q^{29} + 2 q^{31} -9 i q^{33} + ( -1 + 2 i ) q^{35} + 10 i q^{37} + 3 q^{39} -6 q^{41} -4 i q^{43} + ( 12 + 6 i ) q^{45} -11 i q^{47} - q^{49} + 15 q^{51} -6 i q^{53} + ( 6 + 3 i ) q^{55} -24 i q^{57} -10 q^{59} + 6 i q^{63} + ( -1 + 2 i ) q^{65} + 10 i q^{67} -6 q^{69} + 10 i q^{73} + ( -12 + 9 i ) q^{75} + 3 i q^{77} -7 q^{79} + 9 q^{81} + 12 i q^{83} + ( -5 + 10 i ) q^{85} + 3 i q^{87} -8 q^{89} - q^{91} + 6 i q^{93} + ( 16 + 8 i ) q^{95} + 3 i q^{97} + 18 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} - 12q^{9} + O(q^{10})$$ $$2q - 4q^{5} - 12q^{9} - 6q^{11} + 6q^{15} - 16q^{19} + 6q^{21} + 6q^{25} + 2q^{29} + 4q^{31} - 2q^{35} + 6q^{39} - 12q^{41} + 24q^{45} - 2q^{49} + 30q^{51} + 12q^{55} - 20q^{59} - 2q^{65} - 12q^{69} - 24q^{75} - 14q^{79} + 18q^{81} - 10q^{85} - 16q^{89} - 2q^{91} + 32q^{95} + 36q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$241$$ $$337$$ $$351$$ $$421$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 1.00000i 1.00000i
0 3.00000i 0 −2.00000 + 1.00000i 0 1.00000i 0 −6.00000 0
449.2 0 3.00000i 0 −2.00000 1.00000i 0 1.00000i 0 −6.00000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.g.a 2
3.b odd 2 1 5040.2.t.s 2
4.b odd 2 1 140.2.e.a 2
5.b even 2 1 inner 560.2.g.a 2
5.c odd 4 1 2800.2.a.a 1
5.c odd 4 1 2800.2.a.bf 1
8.b even 2 1 2240.2.g.f 2
8.d odd 2 1 2240.2.g.e 2
12.b even 2 1 1260.2.k.c 2
15.d odd 2 1 5040.2.t.s 2
20.d odd 2 1 140.2.e.a 2
20.e even 4 1 700.2.a.a 1
20.e even 4 1 700.2.a.j 1
28.d even 2 1 980.2.e.b 2
28.f even 6 2 980.2.q.c 4
28.g odd 6 2 980.2.q.f 4
40.e odd 2 1 2240.2.g.e 2
40.f even 2 1 2240.2.g.f 2
60.h even 2 1 1260.2.k.c 2
60.l odd 4 1 6300.2.a.c 1
60.l odd 4 1 6300.2.a.t 1
140.c even 2 1 980.2.e.b 2
140.j odd 4 1 4900.2.a.b 1
140.j odd 4 1 4900.2.a.w 1
140.p odd 6 2 980.2.q.f 4
140.s even 6 2 980.2.q.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.a 2 4.b odd 2 1
140.2.e.a 2 20.d odd 2 1
560.2.g.a 2 1.a even 1 1 trivial
560.2.g.a 2 5.b even 2 1 inner
700.2.a.a 1 20.e even 4 1
700.2.a.j 1 20.e even 4 1
980.2.e.b 2 28.d even 2 1
980.2.e.b 2 140.c even 2 1
980.2.q.c 4 28.f even 6 2
980.2.q.c 4 140.s even 6 2
980.2.q.f 4 28.g odd 6 2
980.2.q.f 4 140.p odd 6 2
1260.2.k.c 2 12.b even 2 1
1260.2.k.c 2 60.h even 2 1
2240.2.g.e 2 8.d odd 2 1
2240.2.g.e 2 40.e odd 2 1
2240.2.g.f 2 8.b even 2 1
2240.2.g.f 2 40.f even 2 1
2800.2.a.a 1 5.c odd 4 1
2800.2.a.bf 1 5.c odd 4 1
4900.2.a.b 1 140.j odd 4 1
4900.2.a.w 1 140.j odd 4 1
5040.2.t.s 2 3.b odd 2 1
5040.2.t.s 2 15.d odd 2 1
6300.2.a.c 1 60.l odd 4 1
6300.2.a.t 1 60.l odd 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 9$$ $$T_{11} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$5 + 4 T + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$25 + T^{2}$$
$19$ $$( 8 + T )^{2}$$
$23$ $$4 + T^{2}$$
$29$ $$( -1 + T )^{2}$$
$31$ $$( -2 + T )^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$121 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( 10 + T )^{2}$$
$61$ $$T^{2}$$
$67$ $$100 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( 7 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( 8 + T )^{2}$$
$97$ $$9 + T^{2}$$