# Properties

 Label 560.2.q.k Level $560$ Weight $2$ Character orbit 560.q Analytic conductor $4.472$ Analytic rank $0$ 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.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + \beta_{2} q^{5} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{7} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + \beta_{2} q^{5} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{7} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{9} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{11} + ( -2 - 2 \beta_{3} ) q^{13} + ( -1 + \beta_{3} ) q^{15} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{17} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{19} + ( 4 + \beta_{2} - 2 \beta_{3} ) q^{21} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{23} + ( -1 - \beta_{2} ) q^{25} + ( -1 - \beta_{3} ) q^{27} - q^{29} + ( -6 - 6 \beta_{2} ) q^{31} + ( 4 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} ) q^{33} + ( 1 + 2 \beta_{1} + \beta_{3} ) q^{35} + ( 2 + 2 \beta_{2} ) q^{39} + ( -5 - 2 \beta_{3} ) q^{41} + ( -5 + \beta_{3} ) q^{43} -2 \beta_{1} q^{45} -2 \beta_{2} q^{47} + ( -2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{49} + 2 \beta_{2} q^{51} + ( 4 + 2 \beta_{1} + 4 \beta_{2} ) q^{53} + ( -2 + 2 \beta_{3} ) q^{55} + ( -4 + 2 \beta_{3} ) q^{57} + ( -4 - 6 \beta_{1} - 4 \beta_{2} ) q^{59} + ( -6 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} ) q^{61} + ( 4 + 8 \beta_{2} - 2 \beta_{3} ) q^{63} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{65} + ( 11 + \beta_{1} + 11 \beta_{2} ) q^{67} + ( -3 + 2 \beta_{3} ) q^{69} + ( 4 - 6 \beta_{3} ) q^{71} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{73} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{75} + ( 8 + 2 \beta_{2} - 4 \beta_{3} ) q^{77} + ( 2 \beta_{1} - 12 \beta_{2} + 2 \beta_{3} ) q^{79} + ( 1 + 6 \beta_{1} + \beta_{2} ) q^{81} + ( -1 - 9 \beta_{3} ) q^{83} + ( 2 + 2 \beta_{3} ) q^{85} + ( -1 - \beta_{1} - \beta_{2} ) q^{87} + ( 4 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{89} + ( 6 - 2 \beta_{2} + 4 \beta_{3} ) q^{91} + ( -6 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} ) q^{93} -2 \beta_{1} q^{95} + ( 6 + 4 \beta_{3} ) q^{97} + ( -8 + 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} - 2q^{5} - 2q^{7} + O(q^{10})$$ $$4q + 2q^{3} - 2q^{5} - 2q^{7} + 4q^{11} - 8q^{13} - 4q^{15} - 4q^{17} + 14q^{21} - 2q^{23} - 2q^{25} - 4q^{27} - 4q^{29} - 12q^{31} - 12q^{33} + 4q^{35} + 4q^{39} - 20q^{41} - 20q^{43} + 4q^{47} + 10q^{49} - 4q^{51} + 8q^{53} - 8q^{55} - 16q^{57} - 8q^{59} + 6q^{61} + 4q^{65} + 22q^{67} - 12q^{69} + 16q^{71} - 4q^{73} + 2q^{75} + 28q^{77} + 24q^{79} + 2q^{81} - 4q^{83} + 8q^{85} - 2q^{87} + 6q^{89} + 28q^{91} + 12q^{93} + 24q^{97} - 32q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
0 −0.207107 + 0.358719i 0 −0.500000 0.866025i 0 −2.62132 0.358719i 0 1.41421 + 2.44949i 0
81.2 0 1.20711 2.09077i 0 −0.500000 0.866025i 0 1.62132 + 2.09077i 0 −1.41421 2.44949i 0
401.1 0 −0.207107 0.358719i 0 −0.500000 + 0.866025i 0 −2.62132 + 0.358719i 0 1.41421 2.44949i 0
401.2 0 1.20711 + 2.09077i 0 −0.500000 + 0.866025i 0 1.62132 2.09077i 0 −1.41421 + 2.44949i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.q.k 4
4.b odd 2 1 35.2.e.a 4
7.c even 3 1 inner 560.2.q.k 4
7.c even 3 1 3920.2.a.bq 2
7.d odd 6 1 3920.2.a.bv 2
12.b even 2 1 315.2.j.e 4
20.d odd 2 1 175.2.e.c 4
20.e even 4 2 175.2.k.a 8
28.d even 2 1 245.2.e.e 4
28.f even 6 1 245.2.a.g 2
28.f even 6 1 245.2.e.e 4
28.g odd 6 1 35.2.e.a 4
28.g odd 6 1 245.2.a.h 2
84.j odd 6 1 2205.2.a.q 2
84.n even 6 1 315.2.j.e 4
84.n even 6 1 2205.2.a.n 2
140.p odd 6 1 175.2.e.c 4
140.p odd 6 1 1225.2.a.k 2
140.s even 6 1 1225.2.a.m 2
140.w even 12 2 175.2.k.a 8
140.w even 12 2 1225.2.b.g 4
140.x odd 12 2 1225.2.b.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 4.b odd 2 1
35.2.e.a 4 28.g odd 6 1
175.2.e.c 4 20.d odd 2 1
175.2.e.c 4 140.p odd 6 1
175.2.k.a 8 20.e even 4 2
175.2.k.a 8 140.w even 12 2
245.2.a.g 2 28.f even 6 1
245.2.a.h 2 28.g odd 6 1
245.2.e.e 4 28.d even 2 1
245.2.e.e 4 28.f even 6 1
315.2.j.e 4 12.b even 2 1
315.2.j.e 4 84.n even 6 1
560.2.q.k 4 1.a even 1 1 trivial
560.2.q.k 4 7.c even 3 1 inner
1225.2.a.k 2 140.p odd 6 1
1225.2.a.m 2 140.s even 6 1
1225.2.b.g 4 140.w even 12 2
1225.2.b.h 4 140.x odd 12 2
2205.2.a.n 2 84.n even 6 1
2205.2.a.q 2 84.j odd 6 1
3920.2.a.bq 2 7.c even 3 1
3920.2.a.bv 2 7.d odd 6 1

## Hecke kernels

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

 $$T_{3}^{4} - 2 T_{3}^{3} + 5 T_{3}^{2} + 2 T_{3} + 1$$ $$T_{11}^{4} - 4 T_{11}^{3} + 20 T_{11}^{2} + 16 T_{11} + 16$$ $$T_{13}^{2} + 4 T_{13} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$49 + 14 T - 3 T^{2} + 2 T^{3} + T^{4}$$
$11$ $$16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4}$$
$13$ $$( -4 + 4 T + T^{2} )^{2}$$
$17$ $$16 - 16 T + 20 T^{2} + 4 T^{3} + T^{4}$$
$19$ $$64 + 8 T^{2} + T^{4}$$
$23$ $$1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4}$$
$29$ $$( 1 + T )^{4}$$
$31$ $$( 36 + 6 T + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$( 17 + 10 T + T^{2} )^{2}$$
$43$ $$( 23 + 10 T + T^{2} )^{2}$$
$47$ $$( 4 - 2 T + T^{2} )^{2}$$
$53$ $$64 - 64 T + 56 T^{2} - 8 T^{3} + T^{4}$$
$59$ $$3136 - 448 T + 120 T^{2} + 8 T^{3} + T^{4}$$
$61$ $$3969 + 378 T + 99 T^{2} - 6 T^{3} + T^{4}$$
$67$ $$14161 - 2618 T + 365 T^{2} - 22 T^{3} + T^{4}$$
$71$ $$( -56 - 8 T + T^{2} )^{2}$$
$73$ $$16 - 16 T + 20 T^{2} + 4 T^{3} + T^{4}$$
$79$ $$18496 - 3264 T + 440 T^{2} - 24 T^{3} + T^{4}$$
$83$ $$( -161 + 2 T + T^{2} )^{2}$$
$89$ $$529 + 138 T + 59 T^{2} - 6 T^{3} + T^{4}$$
$97$ $$( 4 - 12 T + T^{2} )^{2}$$