# Properties

 Label 560.2.a.h Level 560 Weight 2 Character orbit 560.a Self dual yes Analytic conductor 4.472 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 560.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - q^{5} + q^{7} + ( 5 + \beta ) q^{9} +O(q^{10})$$ $$q + \beta q^{3} - q^{5} + q^{7} + ( 5 + \beta ) q^{9} + ( -4 + \beta ) q^{11} + ( 2 - \beta ) q^{13} -\beta q^{15} + ( 2 + \beta ) q^{17} -2 \beta q^{19} + \beta q^{21} -2 \beta q^{23} + q^{25} + ( 8 + 3 \beta ) q^{27} + ( -2 + \beta ) q^{29} + 8 q^{31} + ( 8 - 3 \beta ) q^{33} - q^{35} -2 q^{37} + ( -8 + \beta ) q^{39} + ( 2 - 2 \beta ) q^{41} + ( 4 - 2 \beta ) q^{43} + ( -5 - \beta ) q^{45} -3 \beta q^{47} + q^{49} + ( 8 + 3 \beta ) q^{51} + ( 6 - 2 \beta ) q^{53} + ( 4 - \beta ) q^{55} + ( -16 - 2 \beta ) q^{57} -8 q^{59} + ( 2 + 2 \beta ) q^{61} + ( 5 + \beta ) q^{63} + ( -2 + \beta ) q^{65} + 4 q^{67} + ( -16 - 2 \beta ) q^{69} -8 q^{71} -6 q^{73} + \beta q^{75} + ( -4 + \beta ) q^{77} + ( -8 + 3 \beta ) q^{79} + ( 9 + 8 \beta ) q^{81} -4 \beta q^{83} + ( -2 - \beta ) q^{85} + ( 8 - \beta ) q^{87} + ( 10 - 2 \beta ) q^{89} + ( 2 - \beta ) q^{91} + 8 \beta q^{93} + 2 \beta q^{95} + ( 2 + 5 \beta ) q^{97} + ( -12 + 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - 2q^{5} + 2q^{7} + 11q^{9} + O(q^{10})$$ $$2q + q^{3} - 2q^{5} + 2q^{7} + 11q^{9} - 7q^{11} + 3q^{13} - q^{15} + 5q^{17} - 2q^{19} + q^{21} - 2q^{23} + 2q^{25} + 19q^{27} - 3q^{29} + 16q^{31} + 13q^{33} - 2q^{35} - 4q^{37} - 15q^{39} + 2q^{41} + 6q^{43} - 11q^{45} - 3q^{47} + 2q^{49} + 19q^{51} + 10q^{53} + 7q^{55} - 34q^{57} - 16q^{59} + 6q^{61} + 11q^{63} - 3q^{65} + 8q^{67} - 34q^{69} - 16q^{71} - 12q^{73} + q^{75} - 7q^{77} - 13q^{79} + 26q^{81} - 4q^{83} - 5q^{85} + 15q^{87} + 18q^{89} + 3q^{91} + 8q^{93} + 2q^{95} + 9q^{97} - 22q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.37228 3.37228
0 −2.37228 0 −1.00000 0 1.00000 0 2.62772 0
1.2 0 3.37228 0 −1.00000 0 1.00000 0 8.37228 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.a.h 2
3.b odd 2 1 5040.2.a.by 2
4.b odd 2 1 280.2.a.c 2
5.b even 2 1 2800.2.a.bk 2
5.c odd 4 2 2800.2.g.r 4
7.b odd 2 1 3920.2.a.bt 2
8.b even 2 1 2240.2.a.bg 2
8.d odd 2 1 2240.2.a.bk 2
12.b even 2 1 2520.2.a.x 2
20.d odd 2 1 1400.2.a.r 2
20.e even 4 2 1400.2.g.i 4
28.d even 2 1 1960.2.a.s 2
28.f even 6 2 1960.2.q.r 4
28.g odd 6 2 1960.2.q.t 4
140.c even 2 1 9800.2.a.bu 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.c 2 4.b odd 2 1
560.2.a.h 2 1.a even 1 1 trivial
1400.2.a.r 2 20.d odd 2 1
1400.2.g.i 4 20.e even 4 2
1960.2.a.s 2 28.d even 2 1
1960.2.q.r 4 28.f even 6 2
1960.2.q.t 4 28.g odd 6 2
2240.2.a.bg 2 8.b even 2 1
2240.2.a.bk 2 8.d odd 2 1
2520.2.a.x 2 12.b even 2 1
2800.2.a.bk 2 5.b even 2 1
2800.2.g.r 4 5.c odd 4 2
3920.2.a.bt 2 7.b odd 2 1
5040.2.a.by 2 3.b odd 2 1
9800.2.a.bu 2 140.c even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(560))$$:

 $$T_{3}^{2} - T_{3} - 8$$ $$T_{11}^{2} + 7 T_{11} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( 1 - T )^{2}$$
$11$ $$1 + 7 T + 26 T^{2} + 77 T^{3} + 121 T^{4}$$
$13$ $$1 - 3 T + 20 T^{2} - 39 T^{3} + 169 T^{4}$$
$17$ $$1 - 5 T + 32 T^{2} - 85 T^{3} + 289 T^{4}$$
$19$ $$1 + 2 T + 6 T^{2} + 38 T^{3} + 361 T^{4}$$
$23$ $$1 + 2 T + 14 T^{2} + 46 T^{3} + 529 T^{4}$$
$29$ $$1 + 3 T + 52 T^{2} + 87 T^{3} + 841 T^{4}$$
$31$ $$( 1 - 8 T + 31 T^{2} )^{2}$$
$37$ $$( 1 + 2 T + 37 T^{2} )^{2}$$
$41$ $$1 - 2 T + 50 T^{2} - 82 T^{3} + 1681 T^{4}$$
$43$ $$1 - 6 T + 62 T^{2} - 258 T^{3} + 1849 T^{4}$$
$47$ $$1 + 3 T + 22 T^{2} + 141 T^{3} + 2209 T^{4}$$
$53$ $$1 - 10 T + 98 T^{2} - 530 T^{3} + 2809 T^{4}$$
$59$ $$( 1 + 8 T + 59 T^{2} )^{2}$$
$61$ $$1 - 6 T + 98 T^{2} - 366 T^{3} + 3721 T^{4}$$
$67$ $$( 1 - 4 T + 67 T^{2} )^{2}$$
$71$ $$( 1 + 8 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 6 T + 73 T^{2} )^{2}$$
$79$ $$1 + 13 T + 126 T^{2} + 1027 T^{3} + 6241 T^{4}$$
$83$ $$1 + 4 T + 38 T^{2} + 332 T^{3} + 6889 T^{4}$$
$89$ $$1 - 18 T + 226 T^{2} - 1602 T^{3} + 7921 T^{4}$$
$97$ $$1 - 9 T + 8 T^{2} - 873 T^{3} + 9409 T^{4}$$