# Properties

 Label 560.2.bb.a Level 560 Weight 2 Character orbit 560.bb 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.bb (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 - i ) q^{2} + ( -1 - i ) q^{3} + 2 i q^{4} + ( -1 + 2 i ) q^{5} + 2 i q^{6} + q^{7} + ( 2 - 2 i ) q^{8} -i q^{9} +O(q^{10})$$ $$q + ( -1 - i ) q^{2} + ( -1 - i ) q^{3} + 2 i q^{4} + ( -1 + 2 i ) q^{5} + 2 i q^{6} + q^{7} + ( 2 - 2 i ) q^{8} -i q^{9} + ( 3 - i ) q^{10} + ( 1 + i ) q^{11} + ( 2 - 2 i ) q^{12} + ( -3 - 3 i ) q^{13} + ( -1 - i ) q^{14} + ( 3 - i ) q^{15} -4 q^{16} + ( -1 + i ) q^{18} + ( -3 + 3 i ) q^{19} + ( -4 - 2 i ) q^{20} + ( -1 - i ) q^{21} -2 i q^{22} -4 q^{24} + ( -3 - 4 i ) q^{25} + 6 i q^{26} + ( -4 + 4 i ) q^{27} + 2 i q^{28} + ( -1 + i ) q^{29} + ( -4 - 2 i ) q^{30} -8 q^{31} + ( 4 + 4 i ) q^{32} -2 i q^{33} + ( -1 + 2 i ) q^{35} + 2 q^{36} + ( -5 + 5 i ) q^{37} + 6 q^{38} + 6 i q^{39} + ( 2 + 6 i ) q^{40} -12 i q^{41} + 2 i q^{42} + ( -1 + i ) q^{43} + ( -2 + 2 i ) q^{44} + ( 2 + i ) q^{45} + 10 i q^{47} + ( 4 + 4 i ) q^{48} + q^{49} + ( -1 + 7 i ) q^{50} + ( 6 - 6 i ) q^{52} + ( 3 - 3 i ) q^{53} + 8 q^{54} + ( -3 + i ) q^{55} + ( 2 - 2 i ) q^{56} + 6 q^{57} + 2 q^{58} + ( -1 - i ) q^{59} + ( 2 + 6 i ) q^{60} + ( -7 + 7 i ) q^{61} + ( 8 + 8 i ) q^{62} -i q^{63} -8 i q^{64} + ( 9 - 3 i ) q^{65} + ( -2 + 2 i ) q^{66} + ( -3 - 3 i ) q^{67} + ( 3 - i ) q^{70} + 6 i q^{71} + ( -2 - 2 i ) q^{72} -10 q^{73} + 10 q^{74} + ( -1 + 7 i ) q^{75} + ( -6 - 6 i ) q^{76} + ( 1 + i ) q^{77} + ( 6 - 6 i ) q^{78} -8 q^{79} + ( 4 - 8 i ) q^{80} + 5 q^{81} + ( -12 + 12 i ) q^{82} + ( -5 - 5 i ) q^{83} + ( 2 - 2 i ) q^{84} + 2 q^{86} + 2 q^{87} + 4 q^{88} + ( -1 - 3 i ) q^{90} + ( -3 - 3 i ) q^{91} + ( 8 + 8 i ) q^{93} + ( 10 - 10 i ) q^{94} + ( -3 - 9 i ) q^{95} -8 i q^{96} + 8 i q^{97} + ( -1 - i ) q^{98} + ( 1 - i ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} - 2q^{5} + 2q^{7} + 4q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} - 2q^{5} + 2q^{7} + 4q^{8} + 6q^{10} + 2q^{11} + 4q^{12} - 6q^{13} - 2q^{14} + 6q^{15} - 8q^{16} - 2q^{18} - 6q^{19} - 8q^{20} - 2q^{21} - 8q^{24} - 6q^{25} - 8q^{27} - 2q^{29} - 8q^{30} - 16q^{31} + 8q^{32} - 2q^{35} + 4q^{36} - 10q^{37} + 12q^{38} + 4q^{40} - 2q^{43} - 4q^{44} + 4q^{45} + 8q^{48} + 2q^{49} - 2q^{50} + 12q^{52} + 6q^{53} + 16q^{54} - 6q^{55} + 4q^{56} + 12q^{57} + 4q^{58} - 2q^{59} + 4q^{60} - 14q^{61} + 16q^{62} + 18q^{65} - 4q^{66} - 6q^{67} + 6q^{70} - 4q^{72} - 20q^{73} + 20q^{74} - 2q^{75} - 12q^{76} + 2q^{77} + 12q^{78} - 16q^{79} + 8q^{80} + 10q^{81} - 24q^{82} - 10q^{83} + 4q^{84} + 4q^{86} + 4q^{87} + 8q^{88} - 2q^{90} - 6q^{91} + 16q^{93} + 20q^{94} - 6q^{95} - 2q^{98} + 2q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 − 1.00000i 1.00000i
−1.00000 + 1.00000i −1.00000 + 1.00000i 2.00000i −1.00000 2.00000i 2.00000i 1.00000 2.00000 + 2.00000i 1.00000i 3.00000 + 1.00000i
309.1 −1.00000 1.00000i −1.00000 1.00000i 2.00000i −1.00000 + 2.00000i 2.00000i 1.00000 2.00000 2.00000i 1.00000i 3.00000 1.00000i
 $$n$$: 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
80.q even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.bb.a 2
5.b even 2 1 560.2.bb.b yes 2
16.e even 4 1 560.2.bb.b yes 2
80.q even 4 1 inner 560.2.bb.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.bb.a 2 1.a even 1 1 trivial
560.2.bb.a 2 80.q even 4 1 inner
560.2.bb.b yes 2 5.b even 2 1
560.2.bb.b yes 2 16.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 2 T_{3} + 2$$ acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2}$$
$3$ $$1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}$$
$5$ $$1 + 2 T + 5 T^{2}$$
$7$ $$( 1 - T )^{2}$$
$11$ $$1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4}$$
$13$ $$1 + 6 T + 18 T^{2} + 78 T^{3} + 169 T^{4}$$
$17$ $$( 1 - 17 T^{2} )^{2}$$
$19$ $$1 + 6 T + 18 T^{2} + 114 T^{3} + 361 T^{4}$$
$23$ $$( 1 + 23 T^{2} )^{2}$$
$29$ $$1 + 2 T + 2 T^{2} + 58 T^{3} + 841 T^{4}$$
$31$ $$( 1 + 8 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 2 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$1 + 62 T^{2} + 1681 T^{4}$$
$43$ $$1 + 2 T + 2 T^{2} + 86 T^{3} + 1849 T^{4}$$
$47$ $$1 + 6 T^{2} + 2209 T^{4}$$
$53$ $$1 - 6 T + 18 T^{2} - 318 T^{3} + 2809 T^{4}$$
$59$ $$1 + 2 T + 2 T^{2} + 118 T^{3} + 3481 T^{4}$$
$61$ $$1 + 14 T + 98 T^{2} + 854 T^{3} + 3721 T^{4}$$
$67$ $$1 + 6 T + 18 T^{2} + 402 T^{3} + 4489 T^{4}$$
$71$ $$1 - 106 T^{2} + 5041 T^{4}$$
$73$ $$( 1 + 10 T + 73 T^{2} )^{2}$$
$79$ $$( 1 + 8 T + 79 T^{2} )^{2}$$
$83$ $$1 + 10 T + 50 T^{2} + 830 T^{3} + 6889 T^{4}$$
$89$ $$( 1 - 89 T^{2} )^{2}$$
$97$ $$( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} )$$