LMFDB - Group of Dirichlet characters of modulus 578

Show commands: Magma / Pari/GP / SageMath

comment:Define the Dirichlet group

sage:G = DirichletGroup(578)

gp:g = idealstar(,578,2)

magma:G = FullDirichletGroup(578);

Character group

Order	=	272	comment:Order sage:G.order() gp:g.no magma:Order(G);
Structure	=	$C_{272}$	comment:Group structure sage:sorted(g.order() for g in G.gens()) gp:g.cyc magma:PrimaryInvariants(G);
Generators	=	$\chi_{578}(3,\cdot)$	comment:Generators sage:G.gens() gp:g.gen magma:Generators(G);

First 32 of 272 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	$-1$	$1$	$3$	$5$	$7$	$9$	$11$	$13$	$15$	$19$	$21$	$23$
$\chi_{578}(1,\cdot)$	578.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{578}(3,\cdot)$	578.j	272	no	$-1$	$1$	$e\left(\frac{1}{272}\right)$	$e\left(\frac{229}{272}\right)$	$e\left(\frac{155}{272}\right)$	$e\left(\frac{1}{136}\right)$	$e\left(\frac{23}{272}\right)$	$e\left(\frac{49}{68}\right)$	$e\left(\frac{115}{136}\right)$	$e\left(\frac{7}{136}\right)$	$e\left(\frac{39}{68}\right)$	$e\left(\frac{223}{272}\right)$
$\chi_{578}(5,\cdot)$	578.j	272	no	$-1$	$1$	$e\left(\frac{229}{272}\right)$	$e\left(\frac{217}{272}\right)$	$e\left(\frac{135}{272}\right)$	$e\left(\frac{93}{136}\right)$	$e\left(\frac{99}{272}\right)$	$e\left(\frac{1}{68}\right)$	$e\left(\frac{87}{136}\right)$	$e\left(\frac{107}{136}\right)$	$e\left(\frac{23}{68}\right)$	$e\left(\frac{203}{272}\right)$
$\chi_{578}(7,\cdot)$	578.j	272	no	$-1$	$1$	$e\left(\frac{155}{272}\right)$	$e\left(\frac{135}{272}\right)$	$e\left(\frac{89}{272}\right)$	$e\left(\frac{19}{136}\right)$	$e\left(\frac{29}{272}\right)$	$e\left(\frac{47}{68}\right)$	$e\left(\frac{9}{136}\right)$	$e\left(\frac{133}{136}\right)$	$e\left(\frac{61}{68}\right)$	$e\left(\frac{21}{272}\right)$
$\chi_{578}(9,\cdot)$	578.i	136	no	$1$	$1$	$e\left(\frac{1}{136}\right)$	$e\left(\frac{93}{136}\right)$	$e\left(\frac{19}{136}\right)$	$e\left(\frac{1}{68}\right)$	$e\left(\frac{23}{136}\right)$	$e\left(\frac{15}{34}\right)$	$e\left(\frac{47}{68}\right)$	$e\left(\frac{7}{68}\right)$	$e\left(\frac{5}{34}\right)$	$e\left(\frac{87}{136}\right)$
$\chi_{578}(11,\cdot)$	578.j	272	no	$-1$	$1$	$e\left(\frac{23}{272}\right)$	$e\left(\frac{99}{272}\right)$	$e\left(\frac{29}{272}\right)$	$e\left(\frac{23}{136}\right)$	$e\left(\frac{257}{272}\right)$	$e\left(\frac{39}{68}\right)$	$e\left(\frac{61}{136}\right)$	$e\left(\frac{25}{136}\right)$	$e\left(\frac{13}{68}\right)$	$e\left(\frac{233}{272}\right)$
$\chi_{578}(13,\cdot)$	578.h	68	no	$1$	$1$	$e\left(\frac{49}{68}\right)$	$e\left(\frac{1}{68}\right)$	$e\left(\frac{47}{68}\right)$	$e\left(\frac{15}{34}\right)$	$e\left(\frac{39}{68}\right)$	$e\left(\frac{4}{17}\right)$	$e\left(\frac{25}{34}\right)$	$e\left(\frac{3}{34}\right)$	$e\left(\frac{7}{17}\right)$	$e\left(\frac{47}{68}\right)$
$\chi_{578}(15,\cdot)$	578.i	136	no	$1$	$1$	$e\left(\frac{115}{136}\right)$	$e\left(\frac{87}{136}\right)$	$e\left(\frac{9}{136}\right)$	$e\left(\frac{47}{68}\right)$	$e\left(\frac{61}{136}\right)$	$e\left(\frac{25}{34}\right)$	$e\left(\frac{33}{68}\right)$	$e\left(\frac{57}{68}\right)$	$e\left(\frac{31}{34}\right)$	$e\left(\frac{77}{136}\right)$
$\chi_{578}(19,\cdot)$	578.i	136	no	$1$	$1$	$e\left(\frac{7}{136}\right)$	$e\left(\frac{107}{136}\right)$	$e\left(\frac{133}{136}\right)$	$e\left(\frac{7}{68}\right)$	$e\left(\frac{25}{136}\right)$	$e\left(\frac{3}{34}\right)$	$e\left(\frac{57}{68}\right)$	$e\left(\frac{49}{68}\right)$	$e\left(\frac{1}{34}\right)$	$e\left(\frac{65}{136}\right)$
$\chi_{578}(21,\cdot)$	578.h	68	no	$1$	$1$	$e\left(\frac{39}{68}\right)$	$e\left(\frac{23}{68}\right)$	$e\left(\frac{61}{68}\right)$	$e\left(\frac{5}{34}\right)$	$e\left(\frac{13}{68}\right)$	$e\left(\frac{7}{17}\right)$	$e\left(\frac{31}{34}\right)$	$e\left(\frac{1}{34}\right)$	$e\left(\frac{8}{17}\right)$	$e\left(\frac{61}{68}\right)$
$\chi_{578}(23,\cdot)$	578.j	272	no	$-1$	$1$	$e\left(\frac{223}{272}\right)$	$e\left(\frac{203}{272}\right)$	$e\left(\frac{21}{272}\right)$	$e\left(\frac{87}{136}\right)$	$e\left(\frac{233}{272}\right)$	$e\left(\frac{47}{68}\right)$	$e\left(\frac{77}{136}\right)$	$e\left(\frac{65}{136}\right)$	$e\left(\frac{61}{68}\right)$	$e\left(\frac{225}{272}\right)$
$\chi_{578}(25,\cdot)$	578.i	136	no	$1$	$1$	$e\left(\frac{93}{136}\right)$	$e\left(\frac{81}{136}\right)$	$e\left(\frac{135}{136}\right)$	$e\left(\frac{25}{68}\right)$	$e\left(\frac{99}{136}\right)$	$e\left(\frac{1}{34}\right)$	$e\left(\frac{19}{68}\right)$	$e\left(\frac{39}{68}\right)$	$e\left(\frac{23}{34}\right)$	$e\left(\frac{67}{136}\right)$
$\chi_{578}(27,\cdot)$	578.j	272	no	$-1$	$1$	$e\left(\frac{3}{272}\right)$	$e\left(\frac{143}{272}\right)$	$e\left(\frac{193}{272}\right)$	$e\left(\frac{3}{136}\right)$	$e\left(\frac{69}{272}\right)$	$e\left(\frac{11}{68}\right)$	$e\left(\frac{73}{136}\right)$	$e\left(\frac{21}{136}\right)$	$e\left(\frac{49}{68}\right)$	$e\left(\frac{125}{272}\right)$
$\chi_{578}(29,\cdot)$	578.j	272	no	$-1$	$1$	$e\left(\frac{125}{272}\right)$	$e\left(\frac{65}{272}\right)$	$e\left(\frac{63}{272}\right)$	$e\left(\frac{125}{136}\right)$	$e\left(\frac{155}{272}\right)$	$e\left(\frac{5}{68}\right)$	$e\left(\frac{95}{136}\right)$	$e\left(\frac{59}{136}\right)$	$e\left(\frac{47}{68}\right)$	$e\left(\frac{131}{272}\right)$
$\chi_{578}(31,\cdot)$	578.j	272	no	$-1$	$1$	$e\left(\frac{9}{272}\right)$	$e\left(\frac{157}{272}\right)$	$e\left(\frac{35}{272}\right)$	$e\left(\frac{9}{136}\right)$	$e\left(\frac{207}{272}\right)$	$e\left(\frac{33}{68}\right)$	$e\left(\frac{83}{136}\right)$	$e\left(\frac{63}{136}\right)$	$e\left(\frac{11}{68}\right)$	$e\left(\frac{103}{272}\right)$
$\chi_{578}(33,\cdot)$	578.g	34	no	$1$	$1$	$e\left(\frac{3}{34}\right)$	$e\left(\frac{7}{34}\right)$	$e\left(\frac{23}{34}\right)$	$e\left(\frac{3}{17}\right)$	$e\left(\frac{1}{34}\right)$	$e\left(\frac{5}{17}\right)$	$e\left(\frac{5}{17}\right)$	$e\left(\frac{4}{17}\right)$	$e\left(\frac{13}{17}\right)$	$e\left(\frac{23}{34}\right)$
$\chi_{578}(35,\cdot)$	578.f	17	no	$1$	$1$	$e\left(\frac{7}{17}\right)$	$e\left(\frac{5}{17}\right)$	$e\left(\frac{14}{17}\right)$	$e\left(\frac{14}{17}\right)$	$e\left(\frac{8}{17}\right)$	$e\left(\frac{12}{17}\right)$	$e\left(\frac{12}{17}\right)$	$e\left(\frac{13}{17}\right)$	$e\left(\frac{4}{17}\right)$	$e\left(\frac{14}{17}\right)$
$\chi_{578}(37,\cdot)$	578.j	272	no	$-1$	$1$	$e\left(\frac{129}{272}\right)$	$e\left(\frac{165}{272}\right)$	$e\left(\frac{139}{272}\right)$	$e\left(\frac{129}{136}\right)$	$e\left(\frac{247}{272}\right)$	$e\left(\frac{65}{68}\right)$	$e\left(\frac{11}{136}\right)$	$e\left(\frac{87}{136}\right)$	$e\left(\frac{67}{68}\right)$	$e\left(\frac{207}{272}\right)$
$\chi_{578}(39,\cdot)$	578.j	272	no	$-1$	$1$	$e\left(\frac{197}{272}\right)$	$e\left(\frac{233}{272}\right)$	$e\left(\frac{71}{272}\right)$	$e\left(\frac{61}{136}\right)$	$e\left(\frac{179}{272}\right)$	$e\left(\frac{65}{68}\right)$	$e\left(\frac{79}{136}\right)$	$e\left(\frac{19}{136}\right)$	$e\left(\frac{67}{68}\right)$	$e\left(\frac{139}{272}\right)$
$\chi_{578}(41,\cdot)$	578.j	272	no	$-1$	$1$	$e\left(\frac{171}{272}\right)$	$e\left(\frac{263}{272}\right)$	$e\left(\frac{121}{272}\right)$	$e\left(\frac{35}{136}\right)$	$e\left(\frac{125}{272}\right)$	$e\left(\frac{15}{68}\right)$	$e\left(\frac{81}{136}\right)$	$e\left(\frac{109}{136}\right)$	$e\left(\frac{5}{68}\right)$	$e\left(\frac{53}{272}\right)$
$\chi_{578}(43,\cdot)$	578.i	136	no	$1$	$1$	$e\left(\frac{113}{136}\right)$	$e\left(\frac{37}{136}\right)$	$e\left(\frac{107}{136}\right)$	$e\left(\frac{45}{68}\right)$	$e\left(\frac{15}{136}\right)$	$e\left(\frac{29}{34}\right)$	$e\left(\frac{7}{68}\right)$	$e\left(\frac{43}{68}\right)$	$e\left(\frac{21}{34}\right)$	$e\left(\frac{39}{136}\right)$
$\chi_{578}(45,\cdot)$	578.j	272	no	$-1$	$1$	$e\left(\frac{231}{272}\right)$	$e\left(\frac{131}{272}\right)$	$e\left(\frac{173}{272}\right)$	$e\left(\frac{95}{136}\right)$	$e\left(\frac{145}{272}\right)$	$e\left(\frac{31}{68}\right)$	$e\left(\frac{45}{136}\right)$	$e\left(\frac{121}{136}\right)$	$e\left(\frac{33}{68}\right)$	$e\left(\frac{105}{272}\right)$
$\chi_{578}(47,\cdot)$	578.h	68	no	$1$	$1$	$e\left(\frac{25}{68}\right)$	$e\left(\frac{13}{68}\right)$	$e\left(\frac{67}{68}\right)$	$e\left(\frac{25}{34}\right)$	$e\left(\frac{31}{68}\right)$	$e\left(\frac{1}{17}\right)$	$e\left(\frac{19}{34}\right)$	$e\left(\frac{5}{34}\right)$	$e\left(\frac{6}{17}\right)$	$e\left(\frac{67}{68}\right)$
$\chi_{578}(49,\cdot)$	578.i	136	no	$1$	$1$	$e\left(\frac{19}{136}\right)$	$e\left(\frac{135}{136}\right)$	$e\left(\frac{89}{136}\right)$	$e\left(\frac{19}{68}\right)$	$e\left(\frac{29}{136}\right)$	$e\left(\frac{13}{34}\right)$	$e\left(\frac{9}{68}\right)$	$e\left(\frac{65}{68}\right)$	$e\left(\frac{27}{34}\right)$	$e\left(\frac{21}{136}\right)$
$\chi_{578}(53,\cdot)$	578.i	136	no	$1$	$1$	$e\left(\frac{103}{136}\right)$	$e\left(\frac{59}{136}\right)$	$e\left(\frac{53}{136}\right)$	$e\left(\frac{35}{68}\right)$	$e\left(\frac{57}{136}\right)$	$e\left(\frac{15}{34}\right)$	$e\left(\frac{13}{68}\right)$	$e\left(\frac{41}{68}\right)$	$e\left(\frac{5}{34}\right)$	$e\left(\frac{121}{136}\right)$
$\chi_{578}(55,\cdot)$	578.h	68	no	$1$	$1$	$e\left(\frac{63}{68}\right)$	$e\left(\frac{11}{68}\right)$	$e\left(\frac{41}{68}\right)$	$e\left(\frac{29}{34}\right)$	$e\left(\frac{21}{68}\right)$	$e\left(\frac{10}{17}\right)$	$e\left(\frac{3}{34}\right)$	$e\left(\frac{33}{34}\right)$	$e\left(\frac{9}{17}\right)$	$e\left(\frac{41}{68}\right)$
$\chi_{578}(57,\cdot)$	578.j	272	no	$-1$	$1$	$e\left(\frac{15}{272}\right)$	$e\left(\frac{171}{272}\right)$	$e\left(\frac{149}{272}\right)$	$e\left(\frac{15}{136}\right)$	$e\left(\frac{73}{272}\right)$	$e\left(\frac{55}{68}\right)$	$e\left(\frac{93}{136}\right)$	$e\left(\frac{105}{136}\right)$	$e\left(\frac{41}{68}\right)$	$e\left(\frac{81}{272}\right)$
$\chi_{578}(59,\cdot)$	578.i	136	no	$1$	$1$	$e\left(\frac{117}{136}\right)$	$e\left(\frac{1}{136}\right)$	$e\left(\frac{47}{136}\right)$	$e\left(\frac{49}{68}\right)$	$e\left(\frac{107}{136}\right)$	$e\left(\frac{21}{34}\right)$	$e\left(\frac{59}{68}\right)$	$e\left(\frac{3}{68}\right)$	$e\left(\frac{7}{34}\right)$	$e\left(\frac{115}{136}\right)$
$\chi_{578}(61,\cdot)$	578.j	272	no	$-1$	$1$	$e\left(\frac{259}{272}\right)$	$e\left(\frac{15}{272}\right)$	$e\left(\frac{161}{272}\right)$	$e\left(\frac{123}{136}\right)$	$e\left(\frac{245}{272}\right)$	$e\left(\frac{43}{68}\right)$	$e\left(\frac{1}{136}\right)$	$e\left(\frac{45}{136}\right)$	$e\left(\frac{37}{68}\right)$	$e\left(\frac{93}{272}\right)$
$\chi_{578}(63,\cdot)$	578.j	272	no	$-1$	$1$	$e\left(\frac{157}{272}\right)$	$e\left(\frac{49}{272}\right)$	$e\left(\frac{127}{272}\right)$	$e\left(\frac{21}{136}\right)$	$e\left(\frac{75}{272}\right)$	$e\left(\frac{9}{68}\right)$	$e\left(\frac{103}{136}\right)$	$e\left(\frac{11}{136}\right)$	$e\left(\frac{3}{68}\right)$	$e\left(\frac{195}{272}\right)$
$\chi_{578}(65,\cdot)$	578.e	16	no	$-1$	$1$	$e\left(\frac{9}{16}\right)$	$e\left(\frac{13}{16}\right)$	$e\left(\frac{3}{16}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{15}{16}\right)$	$i$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{7}{8}\right)$	$-i$	$e\left(\frac{7}{16}\right)$
$\chi_{578}(67,\cdot)$	578.g	34	no	$1$	$1$	$e\left(\frac{23}{34}\right)$	$e\left(\frac{31}{34}\right)$	$e\left(\frac{29}{34}\right)$	$e\left(\frac{6}{17}\right)$	$e\left(\frac{19}{34}\right)$	$e\left(\frac{10}{17}\right)$	$e\left(\frac{10}{17}\right)$	$e\left(\frac{8}{17}\right)$	$e\left(\frac{9}{17}\right)$	$e\left(\frac{29}{34}\right)$

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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	$578$
Structure	\(C_{272}\)
Order	$272$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(3\)	\(5\)	\(7\)	\(9\)	\(11\)	\(13\)	\(15\)	\(19\)	\(21\)	\(23\)
\(\chi_{578}(1,\cdot)\)	578.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{578}(3,\cdot)\)	578.j	272	no	\(-1\)	\(1\)	\(e\left(\frac{1}{272}\right)\)	\(e\left(\frac{229}{272}\right)\)	\(e\left(\frac{155}{272}\right)\)	\(e\left(\frac{1}{136}\right)\)	\(e\left(\frac{23}{272}\right)\)	\(e\left(\frac{49}{68}\right)\)	\(e\left(\frac{115}{136}\right)\)	\(e\left(\frac{7}{136}\right)\)	\(e\left(\frac{39}{68}\right)\)	\(e\left(\frac{223}{272}\right)\)
\(\chi_{578}(5,\cdot)\)	578.j	272	no	\(-1\)	\(1\)	\(e\left(\frac{229}{272}\right)\)	\(e\left(\frac{217}{272}\right)\)	\(e\left(\frac{135}{272}\right)\)	\(e\left(\frac{93}{136}\right)\)	\(e\left(\frac{99}{272}\right)\)	\(e\left(\frac{1}{68}\right)\)	\(e\left(\frac{87}{136}\right)\)	\(e\left(\frac{107}{136}\right)\)	\(e\left(\frac{23}{68}\right)\)	\(e\left(\frac{203}{272}\right)\)
\(\chi_{578}(7,\cdot)\)	578.j	272	no	\(-1\)	\(1\)	\(e\left(\frac{155}{272}\right)\)	\(e\left(\frac{135}{272}\right)\)	\(e\left(\frac{89}{272}\right)\)	\(e\left(\frac{19}{136}\right)\)	\(e\left(\frac{29}{272}\right)\)	\(e\left(\frac{47}{68}\right)\)	\(e\left(\frac{9}{136}\right)\)	\(e\left(\frac{133}{136}\right)\)	\(e\left(\frac{61}{68}\right)\)	\(e\left(\frac{21}{272}\right)\)
\(\chi_{578}(9,\cdot)\)	578.i	136	no	\(1\)	\(1\)	\(e\left(\frac{1}{136}\right)\)	\(e\left(\frac{93}{136}\right)\)	\(e\left(\frac{19}{136}\right)\)	\(e\left(\frac{1}{68}\right)\)	\(e\left(\frac{23}{136}\right)\)	\(e\left(\frac{15}{34}\right)\)	\(e\left(\frac{47}{68}\right)\)	\(e\left(\frac{7}{68}\right)\)	\(e\left(\frac{5}{34}\right)\)	\(e\left(\frac{87}{136}\right)\)
\(\chi_{578}(11,\cdot)\)	578.j	272	no	\(-1\)	\(1\)	\(e\left(\frac{23}{272}\right)\)	\(e\left(\frac{99}{272}\right)\)	\(e\left(\frac{29}{272}\right)\)	\(e\left(\frac{23}{136}\right)\)	\(e\left(\frac{257}{272}\right)\)	\(e\left(\frac{39}{68}\right)\)	\(e\left(\frac{61}{136}\right)\)	\(e\left(\frac{25}{136}\right)\)	\(e\left(\frac{13}{68}\right)\)	\(e\left(\frac{233}{272}\right)\)
\(\chi_{578}(13,\cdot)\)	578.h	68	no	\(1\)	\(1\)	\(e\left(\frac{49}{68}\right)\)	\(e\left(\frac{1}{68}\right)\)	\(e\left(\frac{47}{68}\right)\)	\(e\left(\frac{15}{34}\right)\)	\(e\left(\frac{39}{68}\right)\)	\(e\left(\frac{4}{17}\right)\)	\(e\left(\frac{25}{34}\right)\)	\(e\left(\frac{3}{34}\right)\)	\(e\left(\frac{7}{17}\right)\)	\(e\left(\frac{47}{68}\right)\)
\(\chi_{578}(15,\cdot)\)	578.i	136	no	\(1\)	\(1\)	\(e\left(\frac{115}{136}\right)\)	\(e\left(\frac{87}{136}\right)\)	\(e\left(\frac{9}{136}\right)\)	\(e\left(\frac{47}{68}\right)\)	\(e\left(\frac{61}{136}\right)\)	\(e\left(\frac{25}{34}\right)\)	\(e\left(\frac{33}{68}\right)\)	\(e\left(\frac{57}{68}\right)\)	\(e\left(\frac{31}{34}\right)\)	\(e\left(\frac{77}{136}\right)\)
\(\chi_{578}(19,\cdot)\)	578.i	136	no	\(1\)	\(1\)	\(e\left(\frac{7}{136}\right)\)	\(e\left(\frac{107}{136}\right)\)	\(e\left(\frac{133}{136}\right)\)	\(e\left(\frac{7}{68}\right)\)	\(e\left(\frac{25}{136}\right)\)	\(e\left(\frac{3}{34}\right)\)	\(e\left(\frac{57}{68}\right)\)	\(e\left(\frac{49}{68}\right)\)	\(e\left(\frac{1}{34}\right)\)	\(e\left(\frac{65}{136}\right)\)
\(\chi_{578}(21,\cdot)\)	578.h	68	no	\(1\)	\(1\)	\(e\left(\frac{39}{68}\right)\)	\(e\left(\frac{23}{68}\right)\)	\(e\left(\frac{61}{68}\right)\)	\(e\left(\frac{5}{34}\right)\)	\(e\left(\frac{13}{68}\right)\)	\(e\left(\frac{7}{17}\right)\)	\(e\left(\frac{31}{34}\right)\)	\(e\left(\frac{1}{34}\right)\)	\(e\left(\frac{8}{17}\right)\)	\(e\left(\frac{61}{68}\right)\)
\(\chi_{578}(23,\cdot)\)	578.j	272	no	\(-1\)	\(1\)	\(e\left(\frac{223}{272}\right)\)	\(e\left(\frac{203}{272}\right)\)	\(e\left(\frac{21}{272}\right)\)	\(e\left(\frac{87}{136}\right)\)	\(e\left(\frac{233}{272}\right)\)	\(e\left(\frac{47}{68}\right)\)	\(e\left(\frac{77}{136}\right)\)	\(e\left(\frac{65}{136}\right)\)	\(e\left(\frac{61}{68}\right)\)	\(e\left(\frac{225}{272}\right)\)
\(\chi_{578}(25,\cdot)\)	578.i	136	no	\(1\)	\(1\)	\(e\left(\frac{93}{136}\right)\)	\(e\left(\frac{81}{136}\right)\)	\(e\left(\frac{135}{136}\right)\)	\(e\left(\frac{25}{68}\right)\)	\(e\left(\frac{99}{136}\right)\)	\(e\left(\frac{1}{34}\right)\)	\(e\left(\frac{19}{68}\right)\)	\(e\left(\frac{39}{68}\right)\)	\(e\left(\frac{23}{34}\right)\)	\(e\left(\frac{67}{136}\right)\)
\(\chi_{578}(27,\cdot)\)	578.j	272	no	\(-1\)	\(1\)	\(e\left(\frac{3}{272}\right)\)	\(e\left(\frac{143}{272}\right)\)	\(e\left(\frac{193}{272}\right)\)	\(e\left(\frac{3}{136}\right)\)	\(e\left(\frac{69}{272}\right)\)	\(e\left(\frac{11}{68}\right)\)	\(e\left(\frac{73}{136}\right)\)	\(e\left(\frac{21}{136}\right)\)	\(e\left(\frac{49}{68}\right)\)	\(e\left(\frac{125}{272}\right)\)
\(\chi_{578}(29,\cdot)\)	578.j	272	no	\(-1\)	\(1\)	\(e\left(\frac{125}{272}\right)\)	\(e\left(\frac{65}{272}\right)\)	\(e\left(\frac{63}{272}\right)\)	\(e\left(\frac{125}{136}\right)\)	\(e\left(\frac{155}{272}\right)\)	\(e\left(\frac{5}{68}\right)\)	\(e\left(\frac{95}{136}\right)\)	\(e\left(\frac{59}{136}\right)\)	\(e\left(\frac{47}{68}\right)\)	\(e\left(\frac{131}{272}\right)\)
\(\chi_{578}(31,\cdot)\)	578.j	272	no	\(-1\)	\(1\)	\(e\left(\frac{9}{272}\right)\)	\(e\left(\frac{157}{272}\right)\)	\(e\left(\frac{35}{272}\right)\)	\(e\left(\frac{9}{136}\right)\)	\(e\left(\frac{207}{272}\right)\)	\(e\left(\frac{33}{68}\right)\)	\(e\left(\frac{83}{136}\right)\)	\(e\left(\frac{63}{136}\right)\)	\(e\left(\frac{11}{68}\right)\)	\(e\left(\frac{103}{272}\right)\)
\(\chi_{578}(33,\cdot)\)	578.g	34	no	\(1\)	\(1\)	\(e\left(\frac{3}{34}\right)\)	\(e\left(\frac{7}{34}\right)\)	\(e\left(\frac{23}{34}\right)\)	\(e\left(\frac{3}{17}\right)\)	\(e\left(\frac{1}{34}\right)\)	\(e\left(\frac{5}{17}\right)\)	\(e\left(\frac{5}{17}\right)\)	\(e\left(\frac{4}{17}\right)\)	\(e\left(\frac{13}{17}\right)\)	\(e\left(\frac{23}{34}\right)\)
\(\chi_{578}(35,\cdot)\)	578.f	17	no	\(1\)	\(1\)	\(e\left(\frac{7}{17}\right)\)	\(e\left(\frac{5}{17}\right)\)	\(e\left(\frac{14}{17}\right)\)	\(e\left(\frac{14}{17}\right)\)	\(e\left(\frac{8}{17}\right)\)	\(e\left(\frac{12}{17}\right)\)	\(e\left(\frac{12}{17}\right)\)	\(e\left(\frac{13}{17}\right)\)	\(e\left(\frac{4}{17}\right)\)	\(e\left(\frac{14}{17}\right)\)
\(\chi_{578}(37,\cdot)\)	578.j	272	no	\(-1\)	\(1\)	\(e\left(\frac{129}{272}\right)\)	\(e\left(\frac{165}{272}\right)\)	\(e\left(\frac{139}{272}\right)\)	\(e\left(\frac{129}{136}\right)\)	\(e\left(\frac{247}{272}\right)\)	\(e\left(\frac{65}{68}\right)\)	\(e\left(\frac{11}{136}\right)\)	\(e\left(\frac{87}{136}\right)\)	\(e\left(\frac{67}{68}\right)\)	\(e\left(\frac{207}{272}\right)\)
\(\chi_{578}(39,\cdot)\)	578.j	272	no	\(-1\)	\(1\)	\(e\left(\frac{197}{272}\right)\)	\(e\left(\frac{233}{272}\right)\)	\(e\left(\frac{71}{272}\right)\)	\(e\left(\frac{61}{136}\right)\)	\(e\left(\frac{179}{272}\right)\)	\(e\left(\frac{65}{68}\right)\)	\(e\left(\frac{79}{136}\right)\)	\(e\left(\frac{19}{136}\right)\)	\(e\left(\frac{67}{68}\right)\)	\(e\left(\frac{139}{272}\right)\)
\(\chi_{578}(41,\cdot)\)	578.j	272	no	\(-1\)	\(1\)	\(e\left(\frac{171}{272}\right)\)	\(e\left(\frac{263}{272}\right)\)	\(e\left(\frac{121}{272}\right)\)	\(e\left(\frac{35}{136}\right)\)	\(e\left(\frac{125}{272}\right)\)	\(e\left(\frac{15}{68}\right)\)	\(e\left(\frac{81}{136}\right)\)	\(e\left(\frac{109}{136}\right)\)	\(e\left(\frac{5}{68}\right)\)	\(e\left(\frac{53}{272}\right)\)
\(\chi_{578}(43,\cdot)\)	578.i	136	no	\(1\)	\(1\)	\(e\left(\frac{113}{136}\right)\)	\(e\left(\frac{37}{136}\right)\)	\(e\left(\frac{107}{136}\right)\)	\(e\left(\frac{45}{68}\right)\)	\(e\left(\frac{15}{136}\right)\)	\(e\left(\frac{29}{34}\right)\)	\(e\left(\frac{7}{68}\right)\)	\(e\left(\frac{43}{68}\right)\)	\(e\left(\frac{21}{34}\right)\)	\(e\left(\frac{39}{136}\right)\)
\(\chi_{578}(45,\cdot)\)	578.j	272	no	\(-1\)	\(1\)	\(e\left(\frac{231}{272}\right)\)	\(e\left(\frac{131}{272}\right)\)	\(e\left(\frac{173}{272}\right)\)	\(e\left(\frac{95}{136}\right)\)	\(e\left(\frac{145}{272}\right)\)	\(e\left(\frac{31}{68}\right)\)	\(e\left(\frac{45}{136}\right)\)	\(e\left(\frac{121}{136}\right)\)	\(e\left(\frac{33}{68}\right)\)	\(e\left(\frac{105}{272}\right)\)
\(\chi_{578}(47,\cdot)\)	578.h	68	no	\(1\)	\(1\)	\(e\left(\frac{25}{68}\right)\)	\(e\left(\frac{13}{68}\right)\)	\(e\left(\frac{67}{68}\right)\)	\(e\left(\frac{25}{34}\right)\)	\(e\left(\frac{31}{68}\right)\)	\(e\left(\frac{1}{17}\right)\)	\(e\left(\frac{19}{34}\right)\)	\(e\left(\frac{5}{34}\right)\)	\(e\left(\frac{6}{17}\right)\)	\(e\left(\frac{67}{68}\right)\)
\(\chi_{578}(49,\cdot)\)	578.i	136	no	\(1\)	\(1\)	\(e\left(\frac{19}{136}\right)\)	\(e\left(\frac{135}{136}\right)\)	\(e\left(\frac{89}{136}\right)\)	\(e\left(\frac{19}{68}\right)\)	\(e\left(\frac{29}{136}\right)\)	\(e\left(\frac{13}{34}\right)\)	\(e\left(\frac{9}{68}\right)\)	\(e\left(\frac{65}{68}\right)\)	\(e\left(\frac{27}{34}\right)\)	\(e\left(\frac{21}{136}\right)\)
\(\chi_{578}(53,\cdot)\)	578.i	136	no	\(1\)	\(1\)	\(e\left(\frac{103}{136}\right)\)	\(e\left(\frac{59}{136}\right)\)	\(e\left(\frac{53}{136}\right)\)	\(e\left(\frac{35}{68}\right)\)	\(e\left(\frac{57}{136}\right)\)	\(e\left(\frac{15}{34}\right)\)	\(e\left(\frac{13}{68}\right)\)	\(e\left(\frac{41}{68}\right)\)	\(e\left(\frac{5}{34}\right)\)	\(e\left(\frac{121}{136}\right)\)
\(\chi_{578}(55,\cdot)\)	578.h	68	no	\(1\)	\(1\)	\(e\left(\frac{63}{68}\right)\)	\(e\left(\frac{11}{68}\right)\)	\(e\left(\frac{41}{68}\right)\)	\(e\left(\frac{29}{34}\right)\)	\(e\left(\frac{21}{68}\right)\)	\(e\left(\frac{10}{17}\right)\)	\(e\left(\frac{3}{34}\right)\)	\(e\left(\frac{33}{34}\right)\)	\(e\left(\frac{9}{17}\right)\)	\(e\left(\frac{41}{68}\right)\)
\(\chi_{578}(57,\cdot)\)	578.j	272	no	\(-1\)	\(1\)	\(e\left(\frac{15}{272}\right)\)	\(e\left(\frac{171}{272}\right)\)	\(e\left(\frac{149}{272}\right)\)	\(e\left(\frac{15}{136}\right)\)	\(e\left(\frac{73}{272}\right)\)	\(e\left(\frac{55}{68}\right)\)	\(e\left(\frac{93}{136}\right)\)	\(e\left(\frac{105}{136}\right)\)	\(e\left(\frac{41}{68}\right)\)	\(e\left(\frac{81}{272}\right)\)
\(\chi_{578}(59,\cdot)\)	578.i	136	no	\(1\)	\(1\)	\(e\left(\frac{117}{136}\right)\)	\(e\left(\frac{1}{136}\right)\)	\(e\left(\frac{47}{136}\right)\)	\(e\left(\frac{49}{68}\right)\)	\(e\left(\frac{107}{136}\right)\)	\(e\left(\frac{21}{34}\right)\)	\(e\left(\frac{59}{68}\right)\)	\(e\left(\frac{3}{68}\right)\)	\(e\left(\frac{7}{34}\right)\)	\(e\left(\frac{115}{136}\right)\)
\(\chi_{578}(61,\cdot)\)	578.j	272	no	\(-1\)	\(1\)	\(e\left(\frac{259}{272}\right)\)	\(e\left(\frac{15}{272}\right)\)	\(e\left(\frac{161}{272}\right)\)	\(e\left(\frac{123}{136}\right)\)	\(e\left(\frac{245}{272}\right)\)	\(e\left(\frac{43}{68}\right)\)	\(e\left(\frac{1}{136}\right)\)	\(e\left(\frac{45}{136}\right)\)	\(e\left(\frac{37}{68}\right)\)	\(e\left(\frac{93}{272}\right)\)
\(\chi_{578}(63,\cdot)\)	578.j	272	no	\(-1\)	\(1\)	\(e\left(\frac{157}{272}\right)\)	\(e\left(\frac{49}{272}\right)\)	\(e\left(\frac{127}{272}\right)\)	\(e\left(\frac{21}{136}\right)\)	\(e\left(\frac{75}{272}\right)\)	\(e\left(\frac{9}{68}\right)\)	\(e\left(\frac{103}{136}\right)\)	\(e\left(\frac{11}{136}\right)\)	\(e\left(\frac{3}{68}\right)\)	\(e\left(\frac{195}{272}\right)\)
\(\chi_{578}(65,\cdot)\)	578.e	16	no	\(-1\)	\(1\)	\(e\left(\frac{9}{16}\right)\)	\(e\left(\frac{13}{16}\right)\)	\(e\left(\frac{3}{16}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{15}{16}\right)\)	\(i\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(-i\)	\(e\left(\frac{7}{16}\right)\)
\(\chi_{578}(67,\cdot)\)	578.g	34	no	\(1\)	\(1\)	\(e\left(\frac{23}{34}\right)\)	\(e\left(\frac{31}{34}\right)\)	\(e\left(\frac{29}{34}\right)\)	\(e\left(\frac{6}{17}\right)\)	\(e\left(\frac{19}{34}\right)\)	\(e\left(\frac{10}{17}\right)\)	\(e\left(\frac{10}{17}\right)\)	\(e\left(\frac{8}{17}\right)\)	\(e\left(\frac{9}{17}\right)\)	\(e\left(\frac{29}{34}\right)\)

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