# Properties

 Label 560.1.p.a Level $560$ Weight $1$ Character orbit 560.p Self dual yes Analytic conductor $0.279$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -35 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 560.p (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.279476407074$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.140.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.1568000.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} + q^{7}+O(q^{10})$$ q - q^3 - q^5 + q^7 $$q - q^{3} - q^{5} + q^{7} + q^{11} + q^{13} + q^{15} + q^{17} - q^{21} + q^{25} + q^{27} - q^{29} - q^{33} - q^{35} - q^{39} - q^{47} + q^{49} - q^{51} - q^{55} - q^{65} - 2 q^{71} - 2 q^{73} - q^{75} + q^{77} + q^{79} - q^{81} + 2 q^{83} - q^{85} + q^{87} + q^{91} + q^{97}+O(q^{100})$$ q - q^3 - q^5 + q^7 + q^11 + q^13 + q^15 + q^17 - q^21 + q^25 + q^27 - q^29 - q^33 - q^35 - q^39 - q^47 + q^49 - q^51 - q^55 - q^65 - 2 * q^71 - 2 * q^73 - q^75 + q^77 + q^79 - q^81 + 2 * q^83 - q^85 + q^87 + q^91 + q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 0
0 −1.00000 0 −1.00000 0 1.00000 0 0 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
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.1.p.a 1
4.b odd 2 1 140.1.h.b yes 1
5.b even 2 1 560.1.p.b 1
5.c odd 4 2 2800.1.f.c 2
7.b odd 2 1 560.1.p.b 1
7.c even 3 2 3920.1.br.b 2
7.d odd 6 2 3920.1.br.a 2
8.b even 2 1 2240.1.p.d 1
8.d odd 2 1 2240.1.p.b 1
12.b even 2 1 1260.1.p.b 1
20.d odd 2 1 140.1.h.a 1
20.e even 4 2 700.1.d.a 2
28.d even 2 1 140.1.h.a 1
28.f even 6 2 980.1.n.b 2
28.g odd 6 2 980.1.n.a 2
35.c odd 2 1 CM 560.1.p.a 1
35.f even 4 2 2800.1.f.c 2
35.i odd 6 2 3920.1.br.b 2
35.j even 6 2 3920.1.br.a 2
40.e odd 2 1 2240.1.p.c 1
40.f even 2 1 2240.1.p.a 1
56.e even 2 1 2240.1.p.c 1
56.h odd 2 1 2240.1.p.a 1
60.h even 2 1 1260.1.p.a 1
84.h odd 2 1 1260.1.p.a 1
140.c even 2 1 140.1.h.b yes 1
140.j odd 4 2 700.1.d.a 2
140.p odd 6 2 980.1.n.b 2
140.s even 6 2 980.1.n.a 2
280.c odd 2 1 2240.1.p.d 1
280.n even 2 1 2240.1.p.b 1
420.o odd 2 1 1260.1.p.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.h.a 1 20.d odd 2 1
140.1.h.a 1 28.d even 2 1
140.1.h.b yes 1 4.b odd 2 1
140.1.h.b yes 1 140.c even 2 1
560.1.p.a 1 1.a even 1 1 trivial
560.1.p.a 1 35.c odd 2 1 CM
560.1.p.b 1 5.b even 2 1
560.1.p.b 1 7.b odd 2 1
700.1.d.a 2 20.e even 4 2
700.1.d.a 2 140.j odd 4 2
980.1.n.a 2 28.g odd 6 2
980.1.n.a 2 140.s even 6 2
980.1.n.b 2 28.f even 6 2
980.1.n.b 2 140.p odd 6 2
1260.1.p.a 1 60.h even 2 1
1260.1.p.a 1 84.h odd 2 1
1260.1.p.b 1 12.b even 2 1
1260.1.p.b 1 420.o odd 2 1
2240.1.p.a 1 40.f even 2 1
2240.1.p.a 1 56.h odd 2 1
2240.1.p.b 1 8.d odd 2 1
2240.1.p.b 1 280.n even 2 1
2240.1.p.c 1 40.e odd 2 1
2240.1.p.c 1 56.e even 2 1
2240.1.p.d 1 8.b even 2 1
2240.1.p.d 1 280.c odd 2 1
2800.1.f.c 2 5.c odd 4 2
2800.1.f.c 2 35.f even 4 2
3920.1.br.a 2 7.d odd 6 2
3920.1.br.a 2 35.j even 6 2
3920.1.br.b 2 7.c even 3 2
3920.1.br.b 2 35.i odd 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(560, [\chi])$$.

## Hecke characteristic polynomials

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