# Properties

 Label 560.2.g.d Level $560$ Weight $2$ Character orbit 560.g 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.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} + (i + 2) q^{5} + i q^{7} + 2 q^{9}+O(q^{10})$$ q + i * q^3 + (i + 2) * q^5 + i * q^7 + 2 * q^9 $$q + i q^{3} + (i + 2) q^{5} + i q^{7} + 2 q^{9} + q^{11} + i q^{13} + (2 i - 1) q^{15} - 3 i q^{17} - 4 q^{19} - q^{21} + 2 i q^{23} + (4 i + 3) q^{25} + 5 i q^{27} + q^{29} + 6 q^{31} + i q^{33} + (2 i - 1) q^{35} + 2 i q^{37} - q^{39} - 10 q^{41} + (2 i + 4) q^{45} - 9 i q^{47} - q^{49} + 3 q^{51} + 14 i q^{53} + (i + 2) q^{55} - 4 i q^{57} + 6 q^{59} - 4 q^{61} + 2 i q^{63} + (2 i - 1) q^{65} - 10 i q^{67} - 2 q^{69} + 16 q^{71} - 10 i q^{73} + (3 i - 4) q^{75} + i q^{77} - 11 q^{79} + q^{81} + 4 i q^{83} + ( - 6 i + 3) q^{85} + i q^{87} - 12 q^{89} - q^{91} + 6 i q^{93} + ( - 4 i - 8) q^{95} - 19 i q^{97} + 2 q^{99} +O(q^{100})$$ q + i * q^3 + (i + 2) * q^5 + i * q^7 + 2 * q^9 + q^11 + i * q^13 + (2*i - 1) * q^15 - 3*i * q^17 - 4 * q^19 - q^21 + 2*i * q^23 + (4*i + 3) * q^25 + 5*i * q^27 + q^29 + 6 * q^31 + i * q^33 + (2*i - 1) * q^35 + 2*i * q^37 - q^39 - 10 * q^41 + (2*i + 4) * q^45 - 9*i * q^47 - q^49 + 3 * q^51 + 14*i * q^53 + (i + 2) * q^55 - 4*i * q^57 + 6 * q^59 - 4 * q^61 + 2*i * q^63 + (2*i - 1) * q^65 - 10*i * q^67 - 2 * q^69 + 16 * q^71 - 10*i * q^73 + (3*i - 4) * q^75 + i * q^77 - 11 * q^79 + q^81 + 4*i * q^83 + (-6*i + 3) * q^85 + i * q^87 - 12 * q^89 - q^91 + 6*i * q^93 + (-4*i - 8) * q^95 - 19*i * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^5 + 4 * q^9 $$2 q + 4 q^{5} + 4 q^{9} + 2 q^{11} - 2 q^{15} - 8 q^{19} - 2 q^{21} + 6 q^{25} + 2 q^{29} + 12 q^{31} - 2 q^{35} - 2 q^{39} - 20 q^{41} + 8 q^{45} - 2 q^{49} + 6 q^{51} + 4 q^{55} + 12 q^{59} - 8 q^{61} - 2 q^{65} - 4 q^{69} + 32 q^{71} - 8 q^{75} - 22 q^{79} + 2 q^{81} + 6 q^{85} - 24 q^{89} - 2 q^{91} - 16 q^{95} + 4 q^{99}+O(q^{100})$$ 2 * q + 4 * q^5 + 4 * q^9 + 2 * q^11 - 2 * q^15 - 8 * q^19 - 2 * q^21 + 6 * q^25 + 2 * q^29 + 12 * q^31 - 2 * q^35 - 2 * q^39 - 20 * q^41 + 8 * q^45 - 2 * q^49 + 6 * q^51 + 4 * q^55 + 12 * q^59 - 8 * q^61 - 2 * q^65 - 4 * q^69 + 32 * q^71 - 8 * q^75 - 22 * q^79 + 2 * q^81 + 6 * q^85 - 24 * q^89 - 2 * q^91 - 16 * q^95 + 4 * q^99

## 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 1.00000i 0 2.00000 1.00000i 0 1.00000i 0 2.00000 0
449.2 0 1.00000i 0 2.00000 + 1.00000i 0 1.00000i 0 2.00000 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.d 2
3.b odd 2 1 5040.2.t.a 2
4.b odd 2 1 280.2.g.a 2
5.b even 2 1 inner 560.2.g.d 2
5.c odd 4 1 2800.2.a.k 1
5.c odd 4 1 2800.2.a.u 1
8.b even 2 1 2240.2.g.a 2
8.d odd 2 1 2240.2.g.b 2
12.b even 2 1 2520.2.t.a 2
15.d odd 2 1 5040.2.t.a 2
20.d odd 2 1 280.2.g.a 2
20.e even 4 1 1400.2.a.d 1
20.e even 4 1 1400.2.a.j 1
28.d even 2 1 1960.2.g.a 2
40.e odd 2 1 2240.2.g.b 2
40.f even 2 1 2240.2.g.a 2
60.h even 2 1 2520.2.t.a 2
140.c even 2 1 1960.2.g.a 2
140.j odd 4 1 9800.2.a.p 1
140.j odd 4 1 9800.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.g.a 2 4.b odd 2 1
280.2.g.a 2 20.d odd 2 1
560.2.g.d 2 1.a even 1 1 trivial
560.2.g.d 2 5.b even 2 1 inner
1400.2.a.d 1 20.e even 4 1
1400.2.a.j 1 20.e even 4 1
1960.2.g.a 2 28.d even 2 1
1960.2.g.a 2 140.c even 2 1
2240.2.g.a 2 8.b even 2 1
2240.2.g.a 2 40.f even 2 1
2240.2.g.b 2 8.d odd 2 1
2240.2.g.b 2 40.e odd 2 1
2520.2.t.a 2 12.b even 2 1
2520.2.t.a 2 60.h even 2 1
2800.2.a.k 1 5.c odd 4 1
2800.2.a.u 1 5.c odd 4 1
5040.2.t.a 2 3.b odd 2 1
5040.2.t.a 2 15.d odd 2 1
9800.2.a.p 1 140.j odd 4 1
9800.2.a.bb 1 140.j odd 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{11} - 1$$ T11 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2} + 1$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 9$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 4$$
$29$ $$(T - 1)^{2}$$
$31$ $$(T - 6)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 10)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 81$$
$53$ $$T^{2} + 196$$
$59$ $$(T - 6)^{2}$$
$61$ $$(T + 4)^{2}$$
$67$ $$T^{2} + 100$$
$71$ $$(T - 16)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T + 11)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T + 12)^{2}$$
$97$ $$T^{2} + 361$$