LMFDB - Group of Dirichlet characters of modulus 99

Show commands: PariGP / SageMath

sage: H = DirichletGroup(99)

pari: g = idealstar(,99,2)

Character group

sage: G.order() pari: g.no
Order	=	60
sage: H.invariants() pari: g.cyc
Structure	=	$C_{2}\times C_{30}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{99}(56,\cdot)$, $\chi_{99}(46,\cdot)$

First 32 of 60 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	$-1$	$1$	$2$	$4$	$5$	$7$	$8$	$10$	$13$	$14$	$16$	$17$
$\chi_{99}(1,\cdot)$	99.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{99}(2,\cdot)$	99.p	30	yes	$1$	$1$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{8}{15}\right)$	$e\left(\frac{7}{30}\right)$	$e\left(\frac{11}{30}\right)$	$e\left(\frac{4}{5}\right)$	$-1$	$e\left(\frac{13}{30}\right)$	$e\left(\frac{19}{30}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{2}{5}\right)$
$\chi_{99}(4,\cdot)$	99.m	15	yes	$1$	$1$	$e\left(\frac{8}{15}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{3}{5}\right)$	$1$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{4}{5}\right)$
$\chi_{99}(5,\cdot)$	99.n	30	yes	$-1$	$1$	$e\left(\frac{7}{30}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{23}{30}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{7}{10}\right)$	$1$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{11}{30}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{1}{10}\right)$
$\chi_{99}(7,\cdot)$	99.o	30	yes	$-1$	$1$	$e\left(\frac{11}{30}\right)$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{17}{30}\right)$	$e\left(\frac{1}{10}\right)$	$-1$	$e\left(\frac{1}{30}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{3}{10}\right)$
$\chi_{99}(8,\cdot)$	99.j	10	no	$1$	$1$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{2}{5}\right)$	$-1$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{1}{5}\right)$
$\chi_{99}(10,\cdot)$	99.c	2	no	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$
$\chi_{99}(13,\cdot)$	99.o	30	yes	$-1$	$1$	$e\left(\frac{13}{30}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{1}{30}\right)$	$e\left(\frac{3}{10}\right)$	$-1$	$e\left(\frac{23}{30}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{9}{10}\right)$
$\chi_{99}(14,\cdot)$	99.n	30	yes	$-1$	$1$	$e\left(\frac{19}{30}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{11}{30}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{9}{10}\right)$	$1$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{17}{30}\right)$	$e\left(\frac{8}{15}\right)$	$e\left(\frac{7}{10}\right)$
$\chi_{99}(16,\cdot)$	99.m	15	yes	$1$	$1$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{1}{5}\right)$	$1$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{8}{15}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{3}{5}\right)$
$\chi_{99}(17,\cdot)$	99.j	10	no	$1$	$1$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{1}{5}\right)$	$-1$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{3}{5}\right)$
$\chi_{99}(19,\cdot)$	99.k	10	no	$-1$	$1$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{9}{10}\right)$	$-1$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{7}{10}\right)$
$\chi_{99}(20,\cdot)$	99.n	30	yes	$-1$	$1$	$e\left(\frac{23}{30}\right)$	$e\left(\frac{8}{15}\right)$	$e\left(\frac{7}{30}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{3}{10}\right)$	$1$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{19}{30}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{9}{10}\right)$
$\chi_{99}(23,\cdot)$	99.i	6	no	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$-1$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$-1$
$\chi_{99}(25,\cdot)$	99.m	15	yes	$1$	$1$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{8}{15}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{2}{5}\right)$	$1$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{1}{5}\right)$
$\chi_{99}(26,\cdot)$	99.l	10	no	$-1$	$1$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{10}\right)$	$1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{3}{10}\right)$
$\chi_{99}(28,\cdot)$	99.k	10	no	$-1$	$1$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{7}{10}\right)$	$-1$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{1}{10}\right)$
$\chi_{99}(29,\cdot)$	99.p	30	yes	$1$	$1$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{19}{30}\right)$	$e\left(\frac{17}{30}\right)$	$e\left(\frac{3}{5}\right)$	$-1$	$e\left(\frac{1}{30}\right)$	$e\left(\frac{13}{30}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{4}{5}\right)$
$\chi_{99}(31,\cdot)$	99.m	15	yes	$1$	$1$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{8}{15}\right)$	$e\left(\frac{4}{5}\right)$	$1$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{2}{5}\right)$
$\chi_{99}(32,\cdot)$	99.g	6	yes	$1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{6}\right)$	$1$	$-1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$1$
$\chi_{99}(34,\cdot)$	99.e	3	no	$1$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$
$\chi_{99}(35,\cdot)$	99.j	10	no	$1$	$1$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{4}{5}\right)$	$-1$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{2}{5}\right)$
$\chi_{99}(37,\cdot)$	99.f	5	no	$1$	$1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{3}{5}\right)$	$1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{4}{5}\right)$
$\chi_{99}(38,\cdot)$	99.n	30	yes	$-1$	$1$	$e\left(\frac{17}{30}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{13}{30}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{7}{10}\right)$	$1$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{1}{30}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{1}{10}\right)$
$\chi_{99}(40,\cdot)$	99.o	30	yes	$-1$	$1$	$e\left(\frac{1}{30}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{7}{30}\right)$	$e\left(\frac{1}{10}\right)$	$-1$	$e\left(\frac{11}{30}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{3}{10}\right)$
$\chi_{99}(41,\cdot)$	99.p	30	yes	$1$	$1$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{11}{30}\right)$	$e\left(\frac{13}{30}\right)$	$e\left(\frac{2}{5}\right)$	$-1$	$e\left(\frac{29}{30}\right)$	$e\left(\frac{17}{30}\right)$	$e\left(\frac{8}{15}\right)$	$e\left(\frac{1}{5}\right)$
$\chi_{99}(43,\cdot)$	99.h	6	yes	$-1$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$-1$	$-1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$-1$
$\chi_{99}(46,\cdot)$	99.k	10	no	$-1$	$1$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{3}{10}\right)$	$-1$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{9}{10}\right)$
$\chi_{99}(47,\cdot)$	99.n	30	yes	$-1$	$1$	$e\left(\frac{29}{30}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{1}{30}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{9}{10}\right)$	$1$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{7}{30}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{7}{10}\right)$
$\chi_{99}(49,\cdot)$	99.m	15	yes	$1$	$1$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{1}{5}\right)$	$1$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{3}{5}\right)$
$\chi_{99}(50,\cdot)$	99.p	30	yes	$1$	$1$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{23}{30}\right)$	$e\left(\frac{19}{30}\right)$	$e\left(\frac{1}{5}\right)$	$-1$	$e\left(\frac{17}{30}\right)$	$e\left(\frac{11}{30}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{3}{5}\right)$
$\chi_{99}(52,\cdot)$	99.o	30	yes	$-1$	$1$	$e\left(\frac{29}{30}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{8}{15}\right)$	$e\left(\frac{23}{30}\right)$	$e\left(\frac{9}{10}\right)$	$-1$	$e\left(\frac{19}{30}\right)$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{7}{10}\right)$

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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 $\F_{q}$

Modulus	$99$
Structure	\(C_{2}\times C_{30}\)
Order	$60$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(4\)	\(5\)	\(7\)	\(8\)	\(10\)	\(13\)	\(14\)	\(16\)	\(17\)
\(\chi_{99}(1,\cdot)\)	99.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{99}(2,\cdot)\)	99.p	30	yes	\(1\)	\(1\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{8}{15}\right)\)	\(e\left(\frac{7}{30}\right)\)	\(e\left(\frac{11}{30}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(-1\)	\(e\left(\frac{13}{30}\right)\)	\(e\left(\frac{19}{30}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{2}{5}\right)\)
\(\chi_{99}(4,\cdot)\)	99.m	15	yes	\(1\)	\(1\)	\(e\left(\frac{8}{15}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(1\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{4}{5}\right)\)
\(\chi_{99}(5,\cdot)\)	99.n	30	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{30}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{23}{30}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(1\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{11}{30}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{1}{10}\right)\)
\(\chi_{99}(7,\cdot)\)	99.o	30	yes	\(-1\)	\(1\)	\(e\left(\frac{11}{30}\right)\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{17}{30}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(-1\)	\(e\left(\frac{1}{30}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{3}{10}\right)\)
\(\chi_{99}(8,\cdot)\)	99.j	10	no	\(1\)	\(1\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(-1\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)
\(\chi_{99}(10,\cdot)\)	99.c	2	no	\(-1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(-1\)
\(\chi_{99}(13,\cdot)\)	99.o	30	yes	\(-1\)	\(1\)	\(e\left(\frac{13}{30}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{1}{30}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(-1\)	\(e\left(\frac{23}{30}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{9}{10}\right)\)
\(\chi_{99}(14,\cdot)\)	99.n	30	yes	\(-1\)	\(1\)	\(e\left(\frac{19}{30}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{11}{30}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(1\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{17}{30}\right)\)	\(e\left(\frac{8}{15}\right)\)	\(e\left(\frac{7}{10}\right)\)
\(\chi_{99}(16,\cdot)\)	99.m	15	yes	\(1\)	\(1\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(1\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{8}{15}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{3}{5}\right)\)
\(\chi_{99}(17,\cdot)\)	99.j	10	no	\(1\)	\(1\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(-1\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)
\(\chi_{99}(19,\cdot)\)	99.k	10	no	\(-1\)	\(1\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(-1\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)
\(\chi_{99}(20,\cdot)\)	99.n	30	yes	\(-1\)	\(1\)	\(e\left(\frac{23}{30}\right)\)	\(e\left(\frac{8}{15}\right)\)	\(e\left(\frac{7}{30}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(1\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{19}{30}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{9}{10}\right)\)
\(\chi_{99}(23,\cdot)\)	99.i	6	no	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)
\(\chi_{99}(25,\cdot)\)	99.m	15	yes	\(1\)	\(1\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{8}{15}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{1}{5}\right)\)
\(\chi_{99}(26,\cdot)\)	99.l	10	no	\(-1\)	\(1\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)
\(\chi_{99}(28,\cdot)\)	99.k	10	no	\(-1\)	\(1\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(-1\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)
\(\chi_{99}(29,\cdot)\)	99.p	30	yes	\(1\)	\(1\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{19}{30}\right)\)	\(e\left(\frac{17}{30}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(-1\)	\(e\left(\frac{1}{30}\right)\)	\(e\left(\frac{13}{30}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{4}{5}\right)\)
\(\chi_{99}(31,\cdot)\)	99.m	15	yes	\(1\)	\(1\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{8}{15}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(1\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{2}{5}\right)\)
\(\chi_{99}(32,\cdot)\)	99.g	6	yes	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)
\(\chi_{99}(34,\cdot)\)	99.e	3	no	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)
\(\chi_{99}(35,\cdot)\)	99.j	10	no	\(1\)	\(1\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(-1\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)
\(\chi_{99}(37,\cdot)\)	99.f	5	no	\(1\)	\(1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)
\(\chi_{99}(38,\cdot)\)	99.n	30	yes	\(-1\)	\(1\)	\(e\left(\frac{17}{30}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{13}{30}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(1\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{1}{30}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{1}{10}\right)\)
\(\chi_{99}(40,\cdot)\)	99.o	30	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{30}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{7}{30}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(-1\)	\(e\left(\frac{11}{30}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{3}{10}\right)\)
\(\chi_{99}(41,\cdot)\)	99.p	30	yes	\(1\)	\(1\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{11}{30}\right)\)	\(e\left(\frac{13}{30}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(-1\)	\(e\left(\frac{29}{30}\right)\)	\(e\left(\frac{17}{30}\right)\)	\(e\left(\frac{8}{15}\right)\)	\(e\left(\frac{1}{5}\right)\)
\(\chi_{99}(43,\cdot)\)	99.h	6	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)
\(\chi_{99}(46,\cdot)\)	99.k	10	no	\(-1\)	\(1\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(-1\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{9}{10}\right)\)
\(\chi_{99}(47,\cdot)\)	99.n	30	yes	\(-1\)	\(1\)	\(e\left(\frac{29}{30}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{1}{30}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(1\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{7}{30}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{7}{10}\right)\)
\(\chi_{99}(49,\cdot)\)	99.m	15	yes	\(1\)	\(1\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(1\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{3}{5}\right)\)
\(\chi_{99}(50,\cdot)\)	99.p	30	yes	\(1\)	\(1\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{23}{30}\right)\)	\(e\left(\frac{19}{30}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(-1\)	\(e\left(\frac{17}{30}\right)\)	\(e\left(\frac{11}{30}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{3}{5}\right)\)
\(\chi_{99}(52,\cdot)\)	99.o	30	yes	\(-1\)	\(1\)	\(e\left(\frac{29}{30}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{8}{15}\right)\)	\(e\left(\frac{23}{30}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(-1\)	\(e\left(\frac{19}{30}\right)\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{7}{10}\right)\)

Introduction

L-functions

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Character group

First 32 of 60 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.