LMFDB - Group of Dirichlet characters of modulus 31

sage:H = DirichletGroup(31)

pari:g = idealstar(,31,2)

Character group

sage:G.order() pari:g.no
Order	=	30
sage:H.invariants() pari:g.cyc
Structure	=	$C_{30}$
sage:H.gens() pari:g.gen
Generators	=	$\chi_{31}(3,\cdot)$

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	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$\chi_{31}(1,\cdot)$	31.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{31}(2,\cdot)$	31.d	5	yes	$1$	$1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{2}{5}\right)$	$1$	$1$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{2}{5}\right)$
$\chi_{31}(3,\cdot)$	31.h	30	yes	$-1$	$1$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{1}{30}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{23}{30}\right)$
$\chi_{31}(4,\cdot)$	31.d	5	yes	$1$	$1$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{4}{5}\right)$	$1$	$1$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{4}{5}\right)$
$\chi_{31}(5,\cdot)$	31.c	3	yes	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{31}(6,\cdot)$	31.e	6	yes	$-1$	$1$	$1$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{31}(7,\cdot)$	31.g	15	yes	$1$	$1$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{7}{15}\right)$
$\chi_{31}(8,\cdot)$	31.d	5	yes	$1$	$1$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{5}\right)$	$1$	$1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{1}{5}\right)$
$\chi_{31}(9,\cdot)$	31.g	15	yes	$1$	$1$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{8}{15}\right)$
$\chi_{31}(10,\cdot)$	31.g	15	yes	$1$	$1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{8}{15}\right)$	$e\left(\frac{11}{15}\right)$
$\chi_{31}(11,\cdot)$	31.h	30	yes	$-1$	$1$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{23}{30}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{8}{15}\right)$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{19}{30}\right)$
$\chi_{31}(12,\cdot)$	31.h	30	yes	$-1$	$1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{19}{30}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{17}{30}\right)$
$\chi_{31}(13,\cdot)$	31.h	30	yes	$-1$	$1$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{11}{30}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{13}{30}\right)$
$\chi_{31}(14,\cdot)$	31.g	15	yes	$1$	$1$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{8}{15}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{13}{15}\right)$
$\chi_{31}(15,\cdot)$	31.f	10	yes	$-1$	$1$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{3}{5}\right)$	$1$	$-1$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{1}{10}\right)$
$\chi_{31}(16,\cdot)$	31.d	5	yes	$1$	$1$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{3}{5}\right)$	$1$	$1$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{3}{5}\right)$
$\chi_{31}(17,\cdot)$	31.h	30	yes	$-1$	$1$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{7}{30}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{8}{15}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{11}{30}\right)$
$\chi_{31}(18,\cdot)$	31.g	15	yes	$1$	$1$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{14}{15}\right)$
$\chi_{31}(19,\cdot)$	31.g	15	yes	$1$	$1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{1}{15}\right)$
$\chi_{31}(20,\cdot)$	31.g	15	yes	$1$	$1$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{8}{15}\right)$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{2}{15}\right)$
$\chi_{31}(21,\cdot)$	31.h	30	yes	$-1$	$1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{29}{30}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{8}{15}\right)$	$e\left(\frac{7}{30}\right)$
$\chi_{31}(22,\cdot)$	31.h	30	yes	$-1$	$1$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{17}{30}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{1}{30}\right)$
$\chi_{31}(23,\cdot)$	31.f	10	yes	$-1$	$1$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{1}{5}\right)$	$1$	$-1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{7}{10}\right)$
$\chi_{31}(24,\cdot)$	31.h	30	yes	$-1$	$1$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{13}{30}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{29}{30}\right)$
$\chi_{31}(25,\cdot)$	31.c	3	yes	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{31}(26,\cdot)$	31.e	6	yes	$-1$	$1$	$1$	$e\left(\frac{1}{6}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{31}(27,\cdot)$	31.f	10	yes	$-1$	$1$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{4}{5}\right)$	$1$	$-1$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{3}{10}\right)$
$\chi_{31}(28,\cdot)$	31.g	15	yes	$1$	$1$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{8}{15}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{4}{15}\right)$
$\chi_{31}(29,\cdot)$	31.f	10	yes	$-1$	$1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{2}{5}\right)$	$1$	$-1$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{9}{10}\right)$
$\chi_{31}(30,\cdot)$	31.b	2	yes	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$

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	$31$
Structure	\(C_{30}\)
Order	$30$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(\chi_{31}(1,\cdot)\)	31.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{31}(2,\cdot)\)	31.d	5	yes	\(1\)	\(1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)	\(1\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)
\(\chi_{31}(3,\cdot)\)	31.h	30	yes	\(-1\)	\(1\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{30}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{23}{30}\right)\)
\(\chi_{31}(4,\cdot)\)	31.d	5	yes	\(1\)	\(1\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(1\)	\(1\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)
\(\chi_{31}(5,\cdot)\)	31.c	3	yes	\(1\)	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{31}(6,\cdot)\)	31.e	6	yes	\(-1\)	\(1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{31}(7,\cdot)\)	31.g	15	yes	\(1\)	\(1\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{7}{15}\right)\)
\(\chi_{31}(8,\cdot)\)	31.d	5	yes	\(1\)	\(1\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(1\)	\(1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)
\(\chi_{31}(9,\cdot)\)	31.g	15	yes	\(1\)	\(1\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{8}{15}\right)\)
\(\chi_{31}(10,\cdot)\)	31.g	15	yes	\(1\)	\(1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{8}{15}\right)\)	\(e\left(\frac{11}{15}\right)\)
\(\chi_{31}(11,\cdot)\)	31.h	30	yes	\(-1\)	\(1\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{23}{30}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{8}{15}\right)\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{19}{30}\right)\)
\(\chi_{31}(12,\cdot)\)	31.h	30	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{19}{30}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{17}{30}\right)\)
\(\chi_{31}(13,\cdot)\)	31.h	30	yes	\(-1\)	\(1\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{11}{30}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{13}{30}\right)\)
\(\chi_{31}(14,\cdot)\)	31.g	15	yes	\(1\)	\(1\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{8}{15}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{13}{15}\right)\)
\(\chi_{31}(15,\cdot)\)	31.f	10	yes	\(-1\)	\(1\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)
\(\chi_{31}(16,\cdot)\)	31.d	5	yes	\(1\)	\(1\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(1\)	\(1\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)
\(\chi_{31}(17,\cdot)\)	31.h	30	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{7}{30}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{8}{15}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{11}{30}\right)\)
\(\chi_{31}(18,\cdot)\)	31.g	15	yes	\(1\)	\(1\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{14}{15}\right)\)
\(\chi_{31}(19,\cdot)\)	31.g	15	yes	\(1\)	\(1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{1}{15}\right)\)
\(\chi_{31}(20,\cdot)\)	31.g	15	yes	\(1\)	\(1\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{8}{15}\right)\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{2}{15}\right)\)
\(\chi_{31}(21,\cdot)\)	31.h	30	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{29}{30}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{8}{15}\right)\)	\(e\left(\frac{7}{30}\right)\)
\(\chi_{31}(22,\cdot)\)	31.h	30	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{17}{30}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{1}{30}\right)\)
\(\chi_{31}(23,\cdot)\)	31.f	10	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)
\(\chi_{31}(24,\cdot)\)	31.h	30	yes	\(-1\)	\(1\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{13}{30}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{29}{30}\right)\)
\(\chi_{31}(25,\cdot)\)	31.c	3	yes	\(1\)	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{31}(26,\cdot)\)	31.e	6	yes	\(-1\)	\(1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{31}(27,\cdot)\)	31.f	10	yes	\(-1\)	\(1\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)
\(\chi_{31}(28,\cdot)\)	31.g	15	yes	\(1\)	\(1\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{8}{15}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{4}{15}\right)\)
\(\chi_{31}(29,\cdot)\)	31.f	10	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{9}{10}\right)\)
\(\chi_{31}(30,\cdot)\)	31.b	2	yes	\(-1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(-1\)

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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.