# Properties

 Label 560.2.g.f Level $560$ Weight $2$ Character orbit 560.g Analytic conductor $4.472$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.5161984.1 Defining polynomial: $$x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{3} + ( \beta_{2} + \beta_{5} ) q^{5} + \beta_{4} q^{7} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{3} + ( \beta_{2} + \beta_{5} ) q^{5} + \beta_{4} q^{7} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{13} + ( 3 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{15} + ( -\beta_{1} + \beta_{2} - \beta_{5} ) q^{17} + ( 2 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{19} + \beta_{3} q^{21} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{23} + ( -2 - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{25} + ( \beta_{1} - \beta_{2} - 6 \beta_{4} + \beta_{5} ) q^{27} + ( 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{29} + ( -4 - 2 \beta_{3} ) q^{31} + ( \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{33} + ( -\beta_{1} - \beta_{3} ) q^{35} -6 \beta_{4} q^{37} + ( \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{39} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -4 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{45} + ( -3 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} ) q^{47} - q^{49} + ( -6 - \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{51} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} ) q^{53} + ( 2 - \beta_{1} - 3 \beta_{2} + 4 \beta_{4} - \beta_{5} ) q^{55} + ( 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{4} - 2 \beta_{5} ) q^{57} + ( 2 - 3 \beta_{1} - 3 \beta_{2} ) q^{59} + ( 8 - \beta_{1} - \beta_{2} ) q^{61} + ( \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} ) q^{63} + ( -5 + \beta_{1} - \beta_{2} - 4 \beta_{3} + 5 \beta_{4} - \beta_{5} ) q^{65} + ( -2 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} ) q^{67} + ( -4 + 6 \beta_{3} ) q^{69} + ( -2 - 2 \beta_{3} ) q^{71} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{73} + ( 6 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{75} + ( -\beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{77} + ( 4 - \beta_{1} - \beta_{2} - 5 \beta_{3} ) q^{79} + ( 3 - 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{81} + ( -\beta_{1} + \beta_{2} + 6 \beta_{4} + 6 \beta_{5} ) q^{83} + ( 2 + 3 \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{85} + ( 3 \beta_{1} - 3 \beta_{2} - 6 \beta_{4} - 3 \beta_{5} ) q^{87} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{89} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{91} + ( -2 \beta_{1} + 2 \beta_{2} + 8 \beta_{4} + 2 \beta_{5} ) q^{93} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 7 \beta_{4} + 3 \beta_{5} ) q^{95} + ( -\beta_{1} + \beta_{2} + 4 \beta_{4} + 3 \beta_{5} ) q^{97} + ( 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 8q^{9} + O(q^{10})$$ $$6q - 8q^{9} - 14q^{11} + 18q^{15} + 8q^{19} - 2q^{21} - 10q^{25} - 6q^{29} - 20q^{31} + 2q^{35} - 10q^{39} + 36q^{41} - 28q^{45} - 6q^{49} - 42q^{51} + 12q^{55} + 12q^{59} + 48q^{61} - 22q^{65} - 36q^{69} - 8q^{71} + 40q^{75} + 34q^{79} + 30q^{81} + 14q^{85} - 10q^{91} + 4q^{95} + 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{5} - 2 \nu^{4} - 25 \nu^{3} + 10 \nu^{2} - 121 \nu + 100$$$$)/121$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{5} + 27 \nu^{4} + 35 \nu^{3} - 14 \nu^{2} + 223$$$$)/121$$ $$\beta_{4}$$ $$=$$ $$($$$$-25 \nu^{5} - 10 \nu^{4} - 4 \nu^{3} + 50 \nu^{2} - 605 \nu + 258$$$$)/242$$ $$\beta_{5}$$ $$=$$ $$($$$$-65 \nu^{5} - 26 \nu^{4} + 38 \nu^{3} + 372 \nu^{2} - 1331 \nu + 574$$$$)/242$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - 3 \beta_{4} + \beta_{2} - \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{4} - 5 \beta_{2} + 2$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{3} + 7 \beta_{2} + 7 \beta_{1} - 15$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{5} - 16 \beta_{4} - 2 \beta_{3} - 29 \beta_{1} + 16$$

## 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
 1.32001 + 1.32001i 0.432320 − 0.432320i −1.75233 + 1.75233i −1.75233 − 1.75233i 0.432320 + 0.432320i 1.32001 − 1.32001i
0 3.12489i 0 1.32001 + 1.80487i 0 1.00000i 0 −6.76491 0
449.2 0 1.76156i 0 0.432320 + 2.19388i 0 1.00000i 0 −0.103084 0
449.3 0 0.363328i 0 −1.75233 1.38900i 0 1.00000i 0 2.86799 0
449.4 0 0.363328i 0 −1.75233 + 1.38900i 0 1.00000i 0 2.86799 0
449.5 0 1.76156i 0 0.432320 2.19388i 0 1.00000i 0 −0.103084 0
449.6 0 3.12489i 0 1.32001 1.80487i 0 1.00000i 0 −6.76491 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.6 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.f 6
3.b odd 2 1 5040.2.t.y 6
4.b odd 2 1 280.2.g.b 6
5.b even 2 1 inner 560.2.g.f 6
5.c odd 4 1 2800.2.a.bq 3
5.c odd 4 1 2800.2.a.br 3
8.b even 2 1 2240.2.g.m 6
8.d odd 2 1 2240.2.g.l 6
12.b even 2 1 2520.2.t.g 6
15.d odd 2 1 5040.2.t.y 6
20.d odd 2 1 280.2.g.b 6
20.e even 4 1 1400.2.a.s 3
20.e even 4 1 1400.2.a.t 3
28.d even 2 1 1960.2.g.c 6
40.e odd 2 1 2240.2.g.l 6
40.f even 2 1 2240.2.g.m 6
60.h even 2 1 2520.2.t.g 6
140.c even 2 1 1960.2.g.c 6
140.j odd 4 1 9800.2.a.cd 3
140.j odd 4 1 9800.2.a.cg 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.g.b 6 4.b odd 2 1
280.2.g.b 6 20.d odd 2 1
560.2.g.f 6 1.a even 1 1 trivial
560.2.g.f 6 5.b even 2 1 inner
1400.2.a.s 3 20.e even 4 1
1400.2.a.t 3 20.e even 4 1
1960.2.g.c 6 28.d even 2 1
1960.2.g.c 6 140.c even 2 1
2240.2.g.l 6 8.d odd 2 1
2240.2.g.l 6 40.e odd 2 1
2240.2.g.m 6 8.b even 2 1
2240.2.g.m 6 40.f even 2 1
2520.2.t.g 6 12.b even 2 1
2520.2.t.g 6 60.h even 2 1
2800.2.a.bq 3 5.c odd 4 1
2800.2.a.br 3 5.c odd 4 1
5040.2.t.y 6 3.b odd 2 1
5040.2.t.y 6 15.d odd 2 1
9800.2.a.cd 3 140.j odd 4 1
9800.2.a.cg 3 140.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 $$S_{2}^{\mathrm{new}}(560, [\chi])$$:

 $$T_{3}^{6} + 13 T_{3}^{4} + 32 T_{3}^{2} + 4$$ $$T_{11}^{3} + 7 T_{11}^{2} + 8 T_{11} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$4 + 32 T^{2} + 13 T^{4} + T^{6}$$
$5$ $$125 + 25 T^{2} + 8 T^{3} + 5 T^{4} + T^{6}$$
$7$ $$( 1 + T^{2} )^{3}$$
$11$ $$( -8 + 8 T + 7 T^{2} + T^{3} )^{2}$$
$13$ $$11236 + 1544 T^{2} + 69 T^{4} + T^{6}$$
$17$ $$400 + 536 T^{2} + 49 T^{4} + T^{6}$$
$19$ $$( -8 - 14 T - 4 T^{2} + T^{3} )^{2}$$
$23$ $$18496 + 2416 T^{2} + 92 T^{4} + T^{6}$$
$29$ $$( -108 - 72 T + 3 T^{2} + T^{3} )^{2}$$
$31$ $$( -80 + 8 T + 10 T^{2} + T^{3} )^{2}$$
$37$ $$( 36 + T^{2} )^{3}$$
$41$ $$( 88 + 68 T - 18 T^{2} + T^{3} )^{2}$$
$43$ $$4096 + 1600 T^{2} + 80 T^{4} + T^{6}$$
$47$ $$53824 + 6480 T^{2} + 177 T^{4} + T^{6}$$
$53$ $$222784 + 11376 T^{2} + 188 T^{4} + T^{6}$$
$59$ $$( -44 - 78 T - 6 T^{2} + T^{3} )^{2}$$
$61$ $$( -440 + 182 T - 24 T^{2} + T^{3} )^{2}$$
$67$ $$262144 + 14336 T^{2} + 228 T^{4} + T^{6}$$
$71$ $$( -64 - 20 T + 4 T^{2} + T^{3} )^{2}$$
$73$ $$18496 + 2416 T^{2} + 92 T^{4} + T^{6}$$
$79$ $$( 548 - 32 T - 17 T^{2} + T^{3} )^{2}$$
$83$ $$678976 + 39940 T^{2} + 428 T^{4} + T^{6}$$
$89$ $$( 464 - 172 T + T^{3} )^{2}$$
$97$ $$1936 + 1048 T^{2} + 113 T^{4} + T^{6}$$