# Properties

 Label 560.2.bw.a Level $560$ Weight $2$ Character orbit 560.bw Analytic conductor $4.472$ Analytic rank $1$ Dimension $4$ 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.bw (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$1$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} ) q^{3} + ( -1 + \beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{2} + \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( -1 - \beta_{2} ) q^{3} + ( -1 + \beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{2} + \beta_{3} ) q^{7} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{11} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{13} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{15} + ( -2 + \beta_{1} - \beta_{2} ) q^{17} + ( 1 - 2 \beta_{1} + \beta_{3} ) q^{19} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{21} + ( -4 + \beta_{2} - 2 \beta_{3} ) q^{23} + ( -4 - \beta_{1} + \beta_{3} ) q^{25} + ( -3 + 6 \beta_{2} ) q^{27} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{29} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{31} + ( 3 + 3 \beta_{3} ) q^{33} + ( 6 \beta_{2} + \beta_{3} ) q^{35} + ( -9 + 4 \beta_{2} - \beta_{3} ) q^{37} + ( 2 + 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{39} + ( -8 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{41} + ( 2 + \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{43} + ( -5 + 2 \beta_{2} - \beta_{3} ) q^{47} + ( 4 - 3 \beta_{1} - 4 \beta_{2} ) q^{49} + ( 1 - 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{51} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{53} + ( -3 - 2 \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{55} + ( -1 + 3 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{57} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{59} + ( -2 + 4 \beta_{1} + 7 \beta_{2} - 2 \beta_{3} ) q^{61} + ( 10 - 2 \beta_{2} - 2 \beta_{3} ) q^{65} + ( -5 + 4 \beta_{1} - \beta_{2} ) q^{67} + ( 5 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{69} + ( -4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{71} + ( 2 + \beta_{1} + 3 \beta_{2} ) q^{73} + ( 4 + \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{75} + ( -4 + 3 \beta_{1} + 11 \beta_{2} ) q^{77} + ( 1 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{79} + ( 9 - 9 \beta_{2} ) q^{81} + ( 2 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{83} + ( -3 - 2 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} ) q^{85} + ( -1 + 3 \beta_{1} + 2 \beta_{2} ) q^{87} -7 \beta_{2} q^{89} + ( 2 + 2 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} ) q^{91} + ( 1 - 2 \beta_{2} - 3 \beta_{3} ) q^{93} + ( 9 + \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{95} + ( 4 - 8 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{3} - 2 q^{5} - 3 q^{7} + O(q^{10})$$ $$4 q - 6 q^{3} - 2 q^{5} - 3 q^{7} - 3 q^{11} + 3 q^{15} - 9 q^{17} + q^{19} - 12 q^{23} - 18 q^{25} - 2 q^{29} - q^{31} + 9 q^{33} + 11 q^{35} - 27 q^{37} + 6 q^{39} - 30 q^{41} - 15 q^{47} + 5 q^{49} + 9 q^{51} + 3 q^{53} - 27 q^{55} - q^{59} + 12 q^{61} + 38 q^{65} - 18 q^{67} + 24 q^{69} - 12 q^{71} + 15 q^{73} + 27 q^{75} + 9 q^{77} - 7 q^{79} + 18 q^{81} - 5 q^{85} + 3 q^{87} - 14 q^{89} - 10 q^{91} + 3 q^{93} + 28 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} - 4 \nu - 5$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu + 5$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{3} + 4 \beta_{1} + 5$$

## Character values

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

 $$n$$ $$241$$ $$337$$ $$351$$ $$421$$ $$\chi(n)$$ $$-\beta_{2}$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 2.13746 − 0.656712i −1.63746 + 1.52274i −1.63746 − 1.52274i 2.13746 + 0.656712i
0 −1.50000 + 0.866025i 0 −0.500000 2.17945i 0 1.13746 + 2.38876i 0 0 0
289.2 0 −1.50000 + 0.866025i 0 −0.500000 + 2.17945i 0 −2.63746 + 0.209313i 0 0 0
529.1 0 −1.50000 0.866025i 0 −0.500000 2.17945i 0 −2.63746 0.209313i 0 0 0
529.2 0 −1.50000 0.866025i 0 −0.500000 + 2.17945i 0 1.13746 2.38876i 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.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.bw.a 4
4.b odd 2 1 140.2.q.b yes 4
5.b even 2 1 560.2.bw.e 4
7.c even 3 1 560.2.bw.e 4
12.b even 2 1 1260.2.bm.b 4
20.d odd 2 1 140.2.q.a 4
20.e even 4 2 700.2.i.f 8
28.d even 2 1 980.2.q.b 4
28.f even 6 1 980.2.e.c 4
28.f even 6 1 980.2.q.g 4
28.g odd 6 1 140.2.q.a 4
28.g odd 6 1 980.2.e.f 4
35.j even 6 1 inner 560.2.bw.a 4
60.h even 2 1 1260.2.bm.a 4
84.n even 6 1 1260.2.bm.a 4
140.c even 2 1 980.2.q.g 4
140.p odd 6 1 140.2.q.b yes 4
140.p odd 6 1 980.2.e.f 4
140.s even 6 1 980.2.e.c 4
140.s even 6 1 980.2.q.b 4
140.w even 12 2 700.2.i.f 8
140.w even 12 2 4900.2.a.be 4
140.x odd 12 2 4900.2.a.bf 4
420.ba even 6 1 1260.2.bm.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.q.a 4 20.d odd 2 1
140.2.q.a 4 28.g odd 6 1
140.2.q.b yes 4 4.b odd 2 1
140.2.q.b yes 4 140.p odd 6 1
560.2.bw.a 4 1.a even 1 1 trivial
560.2.bw.a 4 35.j even 6 1 inner
560.2.bw.e 4 5.b even 2 1
560.2.bw.e 4 7.c even 3 1
700.2.i.f 8 20.e even 4 2
700.2.i.f 8 140.w even 12 2
980.2.e.c 4 28.f even 6 1
980.2.e.c 4 140.s even 6 1
980.2.e.f 4 28.g odd 6 1
980.2.e.f 4 140.p odd 6 1
980.2.q.b 4 28.d even 2 1
980.2.q.b 4 140.s even 6 1
980.2.q.g 4 28.f even 6 1
980.2.q.g 4 140.c even 2 1
1260.2.bm.a 4 60.h even 2 1
1260.2.bm.a 4 84.n even 6 1
1260.2.bm.b 4 12.b even 2 1
1260.2.bm.b 4 420.ba even 6 1
4900.2.a.be 4 140.w even 12 2
4900.2.a.bf 4 140.x odd 12 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + 3 T + T^{2} )^{2}$$
$5$ $$( 5 + T + T^{2} )^{2}$$
$7$ $$49 + 21 T + 2 T^{2} + 3 T^{3} + T^{4}$$
$11$ $$144 - 36 T + 21 T^{2} + 3 T^{3} + T^{4}$$
$13$ $$256 + 44 T^{2} + T^{4}$$
$17$ $$4 + 18 T + 29 T^{2} + 9 T^{3} + T^{4}$$
$19$ $$196 + 14 T + 15 T^{2} - T^{3} + T^{4}$$
$23$ $$49 - 84 T + 41 T^{2} + 12 T^{3} + T^{4}$$
$29$ $$( -14 + T + T^{2} )^{2}$$
$31$ $$196 - 14 T + 15 T^{2} + T^{3} + T^{4}$$
$37$ $$3136 + 1512 T + 299 T^{2} + 27 T^{3} + T^{4}$$
$41$ $$( 42 + 15 T + T^{2} )^{2}$$
$43$ $$196 + 47 T^{2} + T^{4}$$
$47$ $$196 + 210 T + 89 T^{2} + 15 T^{3} + T^{4}$$
$53$ $$1764 + 126 T - 39 T^{2} - 3 T^{3} + T^{4}$$
$59$ $$196 - 14 T + 15 T^{2} + T^{3} + T^{4}$$
$61$ $$441 + 252 T + 165 T^{2} - 12 T^{3} + T^{4}$$
$67$ $$2401 - 882 T + 59 T^{2} + 18 T^{3} + T^{4}$$
$71$ $$( -48 + 6 T + T^{2} )^{2}$$
$73$ $$196 - 210 T + 89 T^{2} - 15 T^{3} + T^{4}$$
$79$ $$4 - 14 T + 51 T^{2} + 7 T^{3} + T^{4}$$
$83$ $$1764 + 87 T^{2} + T^{4}$$
$89$ $$( 49 + 7 T + T^{2} )^{2}$$
$97$ $$( 48 + T^{2} )^{2}$$