LMFDB - Group of Dirichlet characters of modulus 840

sage:H = DirichletGroup(840)

pari:g = idealstar(,840,2)

Character group

sage:G.order() pari:g.no
Order	=	192
sage:H.invariants() pari:g.cyc
Structure	=	$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{12}$
sage:H.gens() pari:g.gen
Generators	=	$\chi_{840}(631,\cdot)$, $\chi_{840}(421,\cdot)$, $\chi_{840}(281,\cdot)$, $\chi_{840}(337,\cdot)$, $\chi_{840}(241,\cdot)$

First 32 of 192 characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

Character	Orbit	Order	Primitive	$-1$	$1$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$
$\chi_{840}(1,\cdot)$	840.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{840}(11,\cdot)$	840.cq	6	no	$1$	$1$	$e\left(\frac{1}{6}\right)$	$-1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{6}\right)$	$-1$	$1$
$\chi_{840}(13,\cdot)$	840.bj	4	no	$1$	$1$	$-1$	$i$	$i$	$-1$	$i$	$1$	$-1$	$i$	$-1$	$-i$
$\chi_{840}(17,\cdot)$	840.dn	12	no	$-1$	$1$	$e\left(\frac{1}{6}\right)$	$i$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{7}{12}\right)$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{7}{12}\right)$	$1$	$-i$
$\chi_{840}(19,\cdot)$	840.bz	6	no	$1$	$1$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$-1$
$\chi_{840}(23,\cdot)$	840.df	12	no	$-1$	$1$	$e\left(\frac{1}{3}\right)$	$i$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{11}{12}\right)$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{12}\right)$	$-1$	$-i$
$\chi_{840}(29,\cdot)$	840.be	2	no	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$
$\chi_{840}(31,\cdot)$	840.cr	6	no	$1$	$1$	$e\left(\frac{1}{6}\right)$	$-1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$-1$	$-1$
$\chi_{840}(37,\cdot)$	840.dm	12	no	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$i$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{12}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{12}\right)$	$1$	$i$
$\chi_{840}(41,\cdot)$	840.f	2	no	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$
$\chi_{840}(43,\cdot)$	840.br	4	no	$1$	$1$	$1$	$-i$	$-i$	$-1$	$-i$	$1$	$-1$	$i$	$1$	$i$
$\chi_{840}(47,\cdot)$	840.di	12	no	$1$	$1$	$e\left(\frac{1}{3}\right)$	$i$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{12}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{11}{12}\right)$	$1$	$i$
$\chi_{840}(53,\cdot)$	840.dc	12	yes	$1$	$1$	$e\left(\frac{2}{3}\right)$	$-i$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{12}\right)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{7}{12}\right)$	$-1$	$-i$
$\chi_{840}(59,\cdot)$	840.ct	6	yes	$-1$	$1$	$e\left(\frac{1}{6}\right)$	$-1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$-1$
$\chi_{840}(61,\cdot)$	840.cy	6	no	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{6}\right)$	$-1$	$-1$
$\chi_{840}(67,\cdot)$	840.dj	12	no	$1$	$1$	$e\left(\frac{2}{3}\right)$	$i$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{7}{12}\right)$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{12}\right)$	$1$	$-i$
$\chi_{840}(71,\cdot)$	840.ba	2	no	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$
$\chi_{840}(73,\cdot)$	840.dd	12	no	$1$	$1$	$e\left(\frac{2}{3}\right)$	$-i$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{7}{12}\right)$	$-1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{12}\right)$	$-1$	$i$
$\chi_{840}(79,\cdot)$	840.cs	6	no	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$-1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{6}\right)$	$1$	$1$
$\chi_{840}(83,\cdot)$	840.bm	4	yes	$1$	$1$	$-1$	$i$	$-i$	$1$	$i$	$-1$	$1$	$i$	$1$	$i$
$\chi_{840}(89,\cdot)$	840.da	6	no	$1$	$1$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{6}\right)$	$1$	$-1$
$\chi_{840}(97,\cdot)$	840.bt	4	no	$1$	$1$	$1$	$i$	$-i$	$1$	$-i$	$-1$	$-1$	$i$	$-1$	$-i$
$\chi_{840}(101,\cdot)$	840.cf	6	no	$1$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{6}\right)$	$1$	$-1$
$\chi_{840}(103,\cdot)$	840.do	12	no	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$-i$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{12}\right)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{12}\right)$	$-1$	$-i$
$\chi_{840}(107,\cdot)$	840.dp	12	yes	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$i$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{12}\right)$	$-1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{12}\right)$	$-1$	$-i$
$\chi_{840}(109,\cdot)$	840.db	6	no	$1$	$1$	$e\left(\frac{1}{6}\right)$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{6}\right)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$1$
$\chi_{840}(113,\cdot)$	840.bk	4	no	$1$	$1$	$-1$	$i$	$i$	$-1$	$-i$	$1$	$1$	$-i$	$-1$	$i$
$\chi_{840}(121,\cdot)$	840.bg	3	no	$1$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$1$
$\chi_{840}(127,\cdot)$	840.bl	4	no	$1$	$1$	$-1$	$-i$	$i$	$1$	$i$	$-1$	$-1$	$i$	$1$	$i$
$\chi_{840}(131,\cdot)$	840.cw	6	no	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$1$	$1$
$\chi_{840}(137,\cdot)$	840.dq	12	no	$1$	$1$	$e\left(\frac{1}{6}\right)$	$-i$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{7}{12}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{7}{12}\right)$	$-1$	$-i$
$\chi_{840}(139,\cdot)$	840.w	2	no	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$

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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Modulus	$840$
Structure	\(C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{12}\)
Order	$192$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(11\)	\(13\)	\(17\)	\(19\)	\(23\)	\(29\)	\(31\)	\(37\)	\(41\)	\(43\)
\(\chi_{840}(1,\cdot)\)	840.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{840}(11,\cdot)\)	840.cq	6	no	\(1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(1\)
\(\chi_{840}(13,\cdot)\)	840.bj	4	no	\(1\)	\(1\)	\(-1\)	\(i\)	\(i\)	\(-1\)	\(i\)	\(1\)	\(-1\)	\(i\)	\(-1\)	\(-i\)
\(\chi_{840}(17,\cdot)\)	840.dn	12	no	\(-1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(i\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(1\)	\(-i\)
\(\chi_{840}(19,\cdot)\)	840.bz	6	no	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(-1\)
\(\chi_{840}(23,\cdot)\)	840.df	12	no	\(-1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(i\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(-1\)	\(-i\)
\(\chi_{840}(29,\cdot)\)	840.be	2	no	\(-1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(1\)
\(\chi_{840}(31,\cdot)\)	840.cr	6	no	\(1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(-1\)
\(\chi_{840}(37,\cdot)\)	840.dm	12	no	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(i\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(1\)	\(i\)
\(\chi_{840}(41,\cdot)\)	840.f	2	no	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(1\)
\(\chi_{840}(43,\cdot)\)	840.br	4	no	\(1\)	\(1\)	\(1\)	\(-i\)	\(-i\)	\(-1\)	\(-i\)	\(1\)	\(-1\)	\(i\)	\(1\)	\(i\)
\(\chi_{840}(47,\cdot)\)	840.di	12	no	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(i\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(1\)	\(i\)
\(\chi_{840}(53,\cdot)\)	840.dc	12	yes	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(-i\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(-1\)	\(-i\)
\(\chi_{840}(59,\cdot)\)	840.ct	6	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(-1\)
\(\chi_{840}(61,\cdot)\)	840.cy	6	no	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(-1\)
\(\chi_{840}(67,\cdot)\)	840.dj	12	no	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(i\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(1\)	\(-i\)
\(\chi_{840}(71,\cdot)\)	840.ba	2	no	\(1\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)
\(\chi_{840}(73,\cdot)\)	840.dd	12	no	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(-i\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(-1\)	\(i\)
\(\chi_{840}(79,\cdot)\)	840.cs	6	no	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)	\(1\)
\(\chi_{840}(83,\cdot)\)	840.bm	4	yes	\(1\)	\(1\)	\(-1\)	\(i\)	\(-i\)	\(1\)	\(i\)	\(-1\)	\(1\)	\(i\)	\(1\)	\(i\)
\(\chi_{840}(89,\cdot)\)	840.da	6	no	\(1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)	\(-1\)
\(\chi_{840}(97,\cdot)\)	840.bt	4	no	\(1\)	\(1\)	\(1\)	\(i\)	\(-i\)	\(1\)	\(-i\)	\(-1\)	\(-1\)	\(i\)	\(-1\)	\(-i\)
\(\chi_{840}(101,\cdot)\)	840.cf	6	no	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(-1\)
\(\chi_{840}(103,\cdot)\)	840.do	12	no	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(-i\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(-1\)	\(-i\)
\(\chi_{840}(107,\cdot)\)	840.dp	12	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(i\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(-1\)	\(-i\)
\(\chi_{840}(109,\cdot)\)	840.db	6	no	\(1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(1\)
\(\chi_{840}(113,\cdot)\)	840.bk	4	no	\(1\)	\(1\)	\(-1\)	\(i\)	\(i\)	\(-1\)	\(-i\)	\(1\)	\(1\)	\(-i\)	\(-1\)	\(i\)
\(\chi_{840}(121,\cdot)\)	840.bg	3	no	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(1\)
\(\chi_{840}(127,\cdot)\)	840.bl	4	no	\(1\)	\(1\)	\(-1\)	\(-i\)	\(i\)	\(1\)	\(i\)	\(-1\)	\(-1\)	\(i\)	\(1\)	\(i\)
\(\chi_{840}(131,\cdot)\)	840.cw	6	no	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)	\(1\)
\(\chi_{840}(137,\cdot)\)	840.dq	12	no	\(1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(-i\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(-1\)	\(-i\)
\(\chi_{840}(139,\cdot)\)	840.w	2	no	\(1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)

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First 32 of 192 characters

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.