LMFDB - Group of Dirichlet characters of modulus 58

Show commands: PariGP / SageMath

sage: H = DirichletGroup(58)

pari: g = idealstar(,58,2)

Character group

sage: G.order() pari: g.no
Order	=	28
sage: H.invariants() pari: g.cyc
Structure	=	$C_{28}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{58}(31,\cdot)$

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	$-1$	$1$	$3$	$5$	$7$	$9$	$11$	$13$	$15$	$17$	$19$	$21$
$\chi_{58}(1,\cdot)$	58.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{58}(3,\cdot)$	58.f	28	no	$-1$	$1$	$e\left(\frac{25}{28}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{13}{28}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{23}{28}\right)$	$-i$	$e\left(\frac{17}{28}\right)$	$e\left(\frac{1}{28}\right)$
$\chi_{58}(5,\cdot)$	58.e	14	no	$1$	$1$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{3}{14}\right)$	$-1$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{5}{14}\right)$
$\chi_{58}(7,\cdot)$	58.d	7	no	$1$	$1$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{4}{7}\right)$	$1$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{2}{7}\right)$
$\chi_{58}(9,\cdot)$	58.e	14	no	$1$	$1$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{9}{14}\right)$	$-1$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{1}{14}\right)$
$\chi_{58}(11,\cdot)$	58.f	28	no	$-1$	$1$	$e\left(\frac{13}{28}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{9}{28}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{3}{28}\right)$	$-i$	$e\left(\frac{1}{28}\right)$	$e\left(\frac{5}{28}\right)$
$\chi_{58}(13,\cdot)$	58.e	14	no	$1$	$1$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{5}{14}\right)$	$-1$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{13}{14}\right)$
$\chi_{58}(15,\cdot)$	58.f	28	no	$-1$	$1$	$e\left(\frac{23}{28}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{3}{28}\right)$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{1}{28}\right)$	$i$	$e\left(\frac{19}{28}\right)$	$e\left(\frac{11}{28}\right)$
$\chi_{58}(17,\cdot)$	58.c	4	no	$-1$	$1$	$-i$	$-1$	$1$	$-1$	$-i$	$-1$	$i$	$-i$	$-i$	$-i$
$\chi_{58}(19,\cdot)$	58.f	28	no	$-1$	$1$	$e\left(\frac{17}{28}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{1}{28}\right)$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{19}{28}\right)$	$-i$	$e\left(\frac{25}{28}\right)$	$e\left(\frac{13}{28}\right)$
$\chi_{58}(21,\cdot)$	58.f	28	no	$-1$	$1$	$e\left(\frac{1}{28}\right)$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{5}{28}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{11}{28}\right)$	$-i$	$e\left(\frac{13}{28}\right)$	$e\left(\frac{9}{28}\right)$
$\chi_{58}(23,\cdot)$	58.d	7	no	$1$	$1$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{2}{7}\right)$	$1$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{1}{7}\right)$
$\chi_{58}(25,\cdot)$	58.d	7	no	$1$	$1$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{3}{7}\right)$	$1$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{5}{7}\right)$
$\chi_{58}(27,\cdot)$	58.f	28	no	$-1$	$1$	$e\left(\frac{19}{28}\right)$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{11}{28}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{13}{28}\right)$	$i$	$e\left(\frac{23}{28}\right)$	$e\left(\frac{3}{28}\right)$
$\chi_{58}(31,\cdot)$	58.f	28	no	$-1$	$1$	$e\left(\frac{5}{28}\right)$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{25}{28}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{27}{28}\right)$	$-i$	$e\left(\frac{9}{28}\right)$	$e\left(\frac{17}{28}\right)$
$\chi_{58}(33,\cdot)$	58.e	14	no	$1$	$1$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{13}{14}\right)$	$-1$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{3}{14}\right)$
$\chi_{58}(35,\cdot)$	58.e	14	no	$1$	$1$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{11}{14}\right)$	$-1$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{9}{14}\right)$
$\chi_{58}(37,\cdot)$	58.f	28	no	$-1$	$1$	$e\left(\frac{15}{28}\right)$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{19}{28}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{25}{28}\right)$	$i$	$e\left(\frac{27}{28}\right)$	$e\left(\frac{23}{28}\right)$
$\chi_{58}(39,\cdot)$	58.f	28	no	$-1$	$1$	$e\left(\frac{3}{28}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{15}{28}\right)$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{5}{28}\right)$	$i$	$e\left(\frac{11}{28}\right)$	$e\left(\frac{27}{28}\right)$
$\chi_{58}(41,\cdot)$	58.c	4	no	$-1$	$1$	$i$	$-1$	$1$	$-1$	$i$	$-1$	$-i$	$i$	$i$	$i$
$\chi_{58}(43,\cdot)$	58.f	28	no	$-1$	$1$	$e\left(\frac{9}{28}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{17}{28}\right)$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{15}{28}\right)$	$-i$	$e\left(\frac{5}{28}\right)$	$e\left(\frac{25}{28}\right)$
$\chi_{58}(45,\cdot)$	58.d	7	no	$1$	$1$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{6}{7}\right)$	$1$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{3}{7}\right)$
$\chi_{58}(47,\cdot)$	58.f	28	no	$-1$	$1$	$e\left(\frac{27}{28}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{23}{28}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{17}{28}\right)$	$i$	$e\left(\frac{15}{28}\right)$	$e\left(\frac{19}{28}\right)$
$\chi_{58}(49,\cdot)$	58.d	7	no	$1$	$1$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{1}{7}\right)$	$1$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{4}{7}\right)$
$\chi_{58}(51,\cdot)$	58.e	14	no	$1$	$1$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{1}{14}\right)$	$-1$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{11}{14}\right)$
$\chi_{58}(53,\cdot)$	58.d	7	no	$1$	$1$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{5}{7}\right)$	$1$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{6}{7}\right)$
$\chi_{58}(55,\cdot)$	58.f	28	no	$-1$	$1$	$e\left(\frac{11}{28}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{27}{28}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{9}{28}\right)$	$i$	$e\left(\frac{3}{28}\right)$	$e\left(\frac{15}{28}\right)$
$\chi_{58}(57,\cdot)$	58.b	2	no	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$-1$

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 $\F_{q}$

Modulus	$58$
Structure	\(C_{28}\)
Order	$28$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(3\)	\(5\)	\(7\)	\(9\)	\(11\)	\(13\)	\(15\)	\(17\)	\(19\)	\(21\)
\(\chi_{58}(1,\cdot)\)	58.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{58}(3,\cdot)\)	58.f	28	no	\(-1\)	\(1\)	\(e\left(\frac{25}{28}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{13}{28}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{23}{28}\right)\)	\(-i\)	\(e\left(\frac{17}{28}\right)\)	\(e\left(\frac{1}{28}\right)\)
\(\chi_{58}(5,\cdot)\)	58.e	14	no	\(1\)	\(1\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(-1\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{5}{14}\right)\)
\(\chi_{58}(7,\cdot)\)	58.d	7	no	\(1\)	\(1\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(1\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)
\(\chi_{58}(9,\cdot)\)	58.e	14	no	\(1\)	\(1\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(-1\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{1}{14}\right)\)
\(\chi_{58}(11,\cdot)\)	58.f	28	no	\(-1\)	\(1\)	\(e\left(\frac{13}{28}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{9}{28}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{3}{28}\right)\)	\(-i\)	\(e\left(\frac{1}{28}\right)\)	\(e\left(\frac{5}{28}\right)\)
\(\chi_{58}(13,\cdot)\)	58.e	14	no	\(1\)	\(1\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{5}{14}\right)\)	\(-1\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{13}{14}\right)\)
\(\chi_{58}(15,\cdot)\)	58.f	28	no	\(-1\)	\(1\)	\(e\left(\frac{23}{28}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{3}{28}\right)\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{1}{28}\right)\)	\(i\)	\(e\left(\frac{19}{28}\right)\)	\(e\left(\frac{11}{28}\right)\)
\(\chi_{58}(17,\cdot)\)	58.c	4	no	\(-1\)	\(1\)	\(-i\)	\(-1\)	\(1\)	\(-1\)	\(-i\)	\(-1\)	\(i\)	\(-i\)	\(-i\)	\(-i\)
\(\chi_{58}(19,\cdot)\)	58.f	28	no	\(-1\)	\(1\)	\(e\left(\frac{17}{28}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{1}{28}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{19}{28}\right)\)	\(-i\)	\(e\left(\frac{25}{28}\right)\)	\(e\left(\frac{13}{28}\right)\)
\(\chi_{58}(21,\cdot)\)	58.f	28	no	\(-1\)	\(1\)	\(e\left(\frac{1}{28}\right)\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{5}{28}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{11}{28}\right)\)	\(-i\)	\(e\left(\frac{13}{28}\right)\)	\(e\left(\frac{9}{28}\right)\)
\(\chi_{58}(23,\cdot)\)	58.d	7	no	\(1\)	\(1\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(1\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)
\(\chi_{58}(25,\cdot)\)	58.d	7	no	\(1\)	\(1\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(1\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)
\(\chi_{58}(27,\cdot)\)	58.f	28	no	\(-1\)	\(1\)	\(e\left(\frac{19}{28}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{11}{28}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{13}{28}\right)\)	\(i\)	\(e\left(\frac{23}{28}\right)\)	\(e\left(\frac{3}{28}\right)\)
\(\chi_{58}(31,\cdot)\)	58.f	28	no	\(-1\)	\(1\)	\(e\left(\frac{5}{28}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{25}{28}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{27}{28}\right)\)	\(-i\)	\(e\left(\frac{9}{28}\right)\)	\(e\left(\frac{17}{28}\right)\)
\(\chi_{58}(33,\cdot)\)	58.e	14	no	\(1\)	\(1\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(-1\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{3}{14}\right)\)
\(\chi_{58}(35,\cdot)\)	58.e	14	no	\(1\)	\(1\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(-1\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{9}{14}\right)\)
\(\chi_{58}(37,\cdot)\)	58.f	28	no	\(-1\)	\(1\)	\(e\left(\frac{15}{28}\right)\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{19}{28}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{25}{28}\right)\)	\(i\)	\(e\left(\frac{27}{28}\right)\)	\(e\left(\frac{23}{28}\right)\)
\(\chi_{58}(39,\cdot)\)	58.f	28	no	\(-1\)	\(1\)	\(e\left(\frac{3}{28}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{15}{28}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{5}{28}\right)\)	\(i\)	\(e\left(\frac{11}{28}\right)\)	\(e\left(\frac{27}{28}\right)\)
\(\chi_{58}(41,\cdot)\)	58.c	4	no	\(-1\)	\(1\)	\(i\)	\(-1\)	\(1\)	\(-1\)	\(i\)	\(-1\)	\(-i\)	\(i\)	\(i\)	\(i\)
\(\chi_{58}(43,\cdot)\)	58.f	28	no	\(-1\)	\(1\)	\(e\left(\frac{9}{28}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{17}{28}\right)\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{15}{28}\right)\)	\(-i\)	\(e\left(\frac{5}{28}\right)\)	\(e\left(\frac{25}{28}\right)\)
\(\chi_{58}(45,\cdot)\)	58.d	7	no	\(1\)	\(1\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(1\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)
\(\chi_{58}(47,\cdot)\)	58.f	28	no	\(-1\)	\(1\)	\(e\left(\frac{27}{28}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{23}{28}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{17}{28}\right)\)	\(i\)	\(e\left(\frac{15}{28}\right)\)	\(e\left(\frac{19}{28}\right)\)
\(\chi_{58}(49,\cdot)\)	58.d	7	no	\(1\)	\(1\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(1\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)
\(\chi_{58}(51,\cdot)\)	58.e	14	no	\(1\)	\(1\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(-1\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{11}{14}\right)\)
\(\chi_{58}(53,\cdot)\)	58.d	7	no	\(1\)	\(1\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(1\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{6}{7}\right)\)
\(\chi_{58}(55,\cdot)\)	58.f	28	no	\(-1\)	\(1\)	\(e\left(\frac{11}{28}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{27}{28}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{9}{28}\right)\)	\(i\)	\(e\left(\frac{3}{28}\right)\)	\(e\left(\frac{15}{28}\right)\)
\(\chi_{58}(57,\cdot)\)	58.b	2	no	\(1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)

Introduction

L-functions

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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.