LMFDB - Group of Dirichlet characters of modulus 345

Show commands: PariGP / SageMath

sage: H = DirichletGroup(345)

pari: g = idealstar(,345,2)

Character group

sage: G.order() pari: g.no
Order	=	176
sage: H.invariants() pari: g.cyc
Structure	=	$C_{2}\times C_{2}\times C_{44}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{345}(116,\cdot)$, $\chi_{345}(277,\cdot)$, $\chi_{345}(166,\cdot)$

First 32 of 176 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	$-1$	$1$	$2$	$4$	$7$	$8$	$11$	$13$	$14$	$16$	$17$	$19$
$\chi_{345}(1,\cdot)$	345.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{345}(2,\cdot)$	345.x	44	yes	$1$	$1$	$e\left(\frac{41}{44}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{43}{44}\right)$	$e\left(\frac{35}{44}\right)$	$e\left(\frac{7}{22}\right)$	$e\left(\frac{1}{44}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{17}{44}\right)$	$e\left(\frac{19}{22}\right)$
$\chi_{345}(4,\cdot)$	345.t	22	no	$1$	$1$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{17}{22}\right)$	$e\left(\frac{8}{11}\right)$
$\chi_{345}(7,\cdot)$	345.w	44	no	$1$	$1$	$e\left(\frac{43}{44}\right)$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{29}{44}\right)$	$e\left(\frac{41}{44}\right)$	$e\left(\frac{17}{22}\right)$	$e\left(\frac{37}{44}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{13}{44}\right)$	$e\left(\frac{5}{11}\right)$
$\chi_{345}(8,\cdot)$	345.x	44	yes	$1$	$1$	$e\left(\frac{35}{44}\right)$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{41}{44}\right)$	$e\left(\frac{17}{44}\right)$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{3}{44}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{7}{44}\right)$	$e\left(\frac{13}{22}\right)$
$\chi_{345}(11,\cdot)$	345.s	22	no	$1$	$1$	$e\left(\frac{7}{22}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{17}{22}\right)$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{3}{22}\right)$
$\chi_{345}(13,\cdot)$	345.v	44	no	$-1$	$1$	$e\left(\frac{1}{44}\right)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{37}{44}\right)$	$e\left(\frac{3}{44}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{7}{44}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{9}{44}\right)$	$e\left(\frac{1}{22}\right)$
$\chi_{345}(14,\cdot)$	345.n	22	yes	$1$	$1$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{15}{22}\right)$	$e\left(\frac{7}{22}\right)$
$\chi_{345}(16,\cdot)$	345.m	11	no	$1$	$1$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{5}{11}\right)$
$\chi_{345}(17,\cdot)$	345.u	44	yes	$-1$	$1$	$e\left(\frac{17}{44}\right)$	$e\left(\frac{17}{22}\right)$	$e\left(\frac{13}{44}\right)$	$e\left(\frac{7}{44}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{9}{44}\right)$	$e\left(\frac{15}{22}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{43}{44}\right)$	$e\left(\frac{3}{11}\right)$
$\chi_{345}(19,\cdot)$	345.r	22	no	$-1$	$1$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{3}{22}\right)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{7}{22}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{5}{22}\right)$
$\chi_{345}(22,\cdot)$	345.j	4	no	$1$	$1$	$i$	$-1$	$-i$	$-i$	$-1$	$-i$	$1$	$1$	$-i$	$1$
$\chi_{345}(26,\cdot)$	345.q	22	no	$-1$	$1$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{17}{22}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{10}{11}\right)$
$\chi_{345}(28,\cdot)$	345.w	44	no	$1$	$1$	$e\left(\frac{37}{44}\right)$	$e\left(\frac{15}{22}\right)$	$e\left(\frac{27}{44}\right)$	$e\left(\frac{23}{44}\right)$	$e\left(\frac{9}{22}\right)$	$e\left(\frac{39}{44}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{3}{44}\right)$	$e\left(\frac{2}{11}\right)$
$\chi_{345}(29,\cdot)$	345.p	22	yes	$-1$	$1$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{15}{22}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{3}{11}\right)$
$\chi_{345}(31,\cdot)$	345.m	11	no	$1$	$1$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{1}{11}\right)$
$\chi_{345}(32,\cdot)$	345.x	44	yes	$1$	$1$	$e\left(\frac{29}{44}\right)$	$e\left(\frac{7}{22}\right)$	$e\left(\frac{39}{44}\right)$	$e\left(\frac{43}{44}\right)$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{5}{44}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{41}{44}\right)$	$e\left(\frac{7}{22}\right)$
$\chi_{345}(34,\cdot)$	345.r	22	no	$-1$	$1$	$e\left(\frac{7}{22}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{15}{22}\right)$	$e\left(\frac{5}{22}\right)$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{3}{22}\right)$
$\chi_{345}(37,\cdot)$	345.w	44	no	$1$	$1$	$e\left(\frac{7}{44}\right)$	$e\left(\frac{7}{22}\right)$	$e\left(\frac{17}{44}\right)$	$e\left(\frac{21}{44}\right)$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{5}{44}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{41}{44}\right)$	$e\left(\frac{9}{11}\right)$
$\chi_{345}(38,\cdot)$	345.u	44	yes	$-1$	$1$	$e\left(\frac{35}{44}\right)$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{19}{44}\right)$	$e\left(\frac{17}{44}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{3}{44}\right)$	$e\left(\frac{5}{22}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{29}{44}\right)$	$e\left(\frac{1}{11}\right)$
$\chi_{345}(41,\cdot)$	345.q	22	no	$-1$	$1$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{17}{22}\right)$	$e\left(\frac{9}{22}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{7}{22}\right)$	$e\left(\frac{2}{11}\right)$
$\chi_{345}(43,\cdot)$	345.w	44	no	$1$	$1$	$e\left(\frac{9}{44}\right)$	$e\left(\frac{9}{22}\right)$	$e\left(\frac{3}{44}\right)$	$e\left(\frac{27}{44}\right)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{19}{44}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{15}{44}\right)$	$e\left(\frac{10}{11}\right)$
$\chi_{345}(44,\cdot)$	345.n	22	yes	$1$	$1$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{17}{22}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{3}{22}\right)$	$e\left(\frac{19}{22}\right)$
$\chi_{345}(47,\cdot)$	345.i	4	no	$1$	$1$	$-i$	$-1$	$i$	$i$	$-1$	$-i$	$1$	$1$	$-i$	$-1$
$\chi_{345}(49,\cdot)$	345.t	22	no	$1$	$1$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{7}{22}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{15}{22}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{10}{11}\right)$
$\chi_{345}(52,\cdot)$	345.v	44	no	$-1$	$1$	$e\left(\frac{39}{44}\right)$	$e\left(\frac{17}{22}\right)$	$e\left(\frac{35}{44}\right)$	$e\left(\frac{29}{44}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{9}{44}\right)$	$e\left(\frac{15}{22}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{43}{44}\right)$	$e\left(\frac{17}{22}\right)$
$\chi_{345}(53,\cdot)$	345.u	44	yes	$-1$	$1$	$e\left(\frac{43}{44}\right)$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{7}{44}\right)$	$e\left(\frac{41}{44}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{15}{44}\right)$	$e\left(\frac{3}{22}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{13}{44}\right)$	$e\left(\frac{5}{11}\right)$
$\chi_{345}(56,\cdot)$	345.s	22	no	$1$	$1$	$e\left(\frac{17}{22}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{7}{22}\right)$	$e\left(\frac{8}{11}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{4}{11}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{1}{22}\right)$
$\chi_{345}(58,\cdot)$	345.v	44	no	$-1$	$1$	$e\left(\frac{25}{44}\right)$	$e\left(\frac{3}{22}\right)$	$e\left(\frac{1}{44}\right)$	$e\left(\frac{31}{44}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{43}{44}\right)$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{5}{44}\right)$	$e\left(\frac{3}{22}\right)$
$\chi_{345}(59,\cdot)$	345.p	22	yes	$-1$	$1$	$e\left(\frac{3}{11}\right)$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{13}{22}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{5}{22}\right)$	$e\left(\frac{9}{22}\right)$	$e\left(\frac{19}{22}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{5}{11}\right)$	$e\left(\frac{6}{11}\right)$
$\chi_{345}(61,\cdot)$	345.o	22	no	$-1$	$1$	$e\left(\frac{6}{11}\right)$	$e\left(\frac{1}{11}\right)$	$e\left(\frac{15}{22}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{9}{11}\right)$	$e\left(\frac{5}{22}\right)$	$e\left(\frac{2}{11}\right)$	$e\left(\frac{9}{22}\right)$	$e\left(\frac{13}{22}\right)$
$\chi_{345}(62,\cdot)$	345.x	44	yes	$1$	$1$	$e\left(\frac{21}{44}\right)$	$e\left(\frac{21}{22}\right)$	$e\left(\frac{7}{44}\right)$	$e\left(\frac{19}{44}\right)$	$e\left(\frac{17}{22}\right)$	$e\left(\frac{37}{44}\right)$	$e\left(\frac{7}{11}\right)$	$e\left(\frac{10}{11}\right)$	$e\left(\frac{13}{44}\right)$	$e\left(\frac{21}{22}\right)$

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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	$345$
Structure	\(C_{2}\times C_{2}\times C_{44}\)
Order	$176$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(4\)	\(7\)	\(8\)	\(11\)	\(13\)	\(14\)	\(16\)	\(17\)	\(19\)
\(\chi_{345}(1,\cdot)\)	345.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{345}(2,\cdot)\)	345.x	44	yes	\(1\)	\(1\)	\(e\left(\frac{41}{44}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{43}{44}\right)\)	\(e\left(\frac{35}{44}\right)\)	\(e\left(\frac{7}{22}\right)\)	\(e\left(\frac{1}{44}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{17}{44}\right)\)	\(e\left(\frac{19}{22}\right)\)
\(\chi_{345}(4,\cdot)\)	345.t	22	no	\(1\)	\(1\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{8}{11}\right)\)
\(\chi_{345}(7,\cdot)\)	345.w	44	no	\(1\)	\(1\)	\(e\left(\frac{43}{44}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{29}{44}\right)\)	\(e\left(\frac{41}{44}\right)\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{37}{44}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{13}{44}\right)\)	\(e\left(\frac{5}{11}\right)\)
\(\chi_{345}(8,\cdot)\)	345.x	44	yes	\(1\)	\(1\)	\(e\left(\frac{35}{44}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{41}{44}\right)\)	\(e\left(\frac{17}{44}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{3}{44}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{7}{44}\right)\)	\(e\left(\frac{13}{22}\right)\)
\(\chi_{345}(11,\cdot)\)	345.s	22	no	\(1\)	\(1\)	\(e\left(\frac{7}{22}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{3}{22}\right)\)
\(\chi_{345}(13,\cdot)\)	345.v	44	no	\(-1\)	\(1\)	\(e\left(\frac{1}{44}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{37}{44}\right)\)	\(e\left(\frac{3}{44}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{7}{44}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{9}{44}\right)\)	\(e\left(\frac{1}{22}\right)\)
\(\chi_{345}(14,\cdot)\)	345.n	22	yes	\(1\)	\(1\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{15}{22}\right)\)	\(e\left(\frac{7}{22}\right)\)
\(\chi_{345}(16,\cdot)\)	345.m	11	no	\(1\)	\(1\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{5}{11}\right)\)
\(\chi_{345}(17,\cdot)\)	345.u	44	yes	\(-1\)	\(1\)	\(e\left(\frac{17}{44}\right)\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{13}{44}\right)\)	\(e\left(\frac{7}{44}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{9}{44}\right)\)	\(e\left(\frac{15}{22}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{43}{44}\right)\)	\(e\left(\frac{3}{11}\right)\)
\(\chi_{345}(19,\cdot)\)	345.r	22	no	\(-1\)	\(1\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{3}{22}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{7}{22}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{5}{22}\right)\)
\(\chi_{345}(22,\cdot)\)	345.j	4	no	\(1\)	\(1\)	\(i\)	\(-1\)	\(-i\)	\(-i\)	\(-1\)	\(-i\)	\(1\)	\(1\)	\(-i\)	\(1\)
\(\chi_{345}(26,\cdot)\)	345.q	22	no	\(-1\)	\(1\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{10}{11}\right)\)
\(\chi_{345}(28,\cdot)\)	345.w	44	no	\(1\)	\(1\)	\(e\left(\frac{37}{44}\right)\)	\(e\left(\frac{15}{22}\right)\)	\(e\left(\frac{27}{44}\right)\)	\(e\left(\frac{23}{44}\right)\)	\(e\left(\frac{9}{22}\right)\)	\(e\left(\frac{39}{44}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{3}{44}\right)\)	\(e\left(\frac{2}{11}\right)\)
\(\chi_{345}(29,\cdot)\)	345.p	22	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{15}{22}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{3}{11}\right)\)
\(\chi_{345}(31,\cdot)\)	345.m	11	no	\(1\)	\(1\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{1}{11}\right)\)
\(\chi_{345}(32,\cdot)\)	345.x	44	yes	\(1\)	\(1\)	\(e\left(\frac{29}{44}\right)\)	\(e\left(\frac{7}{22}\right)\)	\(e\left(\frac{39}{44}\right)\)	\(e\left(\frac{43}{44}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{5}{44}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{41}{44}\right)\)	\(e\left(\frac{7}{22}\right)\)
\(\chi_{345}(34,\cdot)\)	345.r	22	no	\(-1\)	\(1\)	\(e\left(\frac{7}{22}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{15}{22}\right)\)	\(e\left(\frac{5}{22}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{3}{22}\right)\)
\(\chi_{345}(37,\cdot)\)	345.w	44	no	\(1\)	\(1\)	\(e\left(\frac{7}{44}\right)\)	\(e\left(\frac{7}{22}\right)\)	\(e\left(\frac{17}{44}\right)\)	\(e\left(\frac{21}{44}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{5}{44}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{41}{44}\right)\)	\(e\left(\frac{9}{11}\right)\)
\(\chi_{345}(38,\cdot)\)	345.u	44	yes	\(-1\)	\(1\)	\(e\left(\frac{35}{44}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{19}{44}\right)\)	\(e\left(\frac{17}{44}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{3}{44}\right)\)	\(e\left(\frac{5}{22}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{29}{44}\right)\)	\(e\left(\frac{1}{11}\right)\)
\(\chi_{345}(41,\cdot)\)	345.q	22	no	\(-1\)	\(1\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{9}{22}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{7}{22}\right)\)	\(e\left(\frac{2}{11}\right)\)
\(\chi_{345}(43,\cdot)\)	345.w	44	no	\(1\)	\(1\)	\(e\left(\frac{9}{44}\right)\)	\(e\left(\frac{9}{22}\right)\)	\(e\left(\frac{3}{44}\right)\)	\(e\left(\frac{27}{44}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{19}{44}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{15}{44}\right)\)	\(e\left(\frac{10}{11}\right)\)
\(\chi_{345}(44,\cdot)\)	345.n	22	yes	\(1\)	\(1\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{3}{22}\right)\)	\(e\left(\frac{19}{22}\right)\)
\(\chi_{345}(47,\cdot)\)	345.i	4	no	\(1\)	\(1\)	\(-i\)	\(-1\)	\(i\)	\(i\)	\(-1\)	\(-i\)	\(1\)	\(1\)	\(-i\)	\(-1\)
\(\chi_{345}(49,\cdot)\)	345.t	22	no	\(1\)	\(1\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{7}{22}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{15}{22}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{10}{11}\right)\)
\(\chi_{345}(52,\cdot)\)	345.v	44	no	\(-1\)	\(1\)	\(e\left(\frac{39}{44}\right)\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{35}{44}\right)\)	\(e\left(\frac{29}{44}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{9}{44}\right)\)	\(e\left(\frac{15}{22}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{43}{44}\right)\)	\(e\left(\frac{17}{22}\right)\)
\(\chi_{345}(53,\cdot)\)	345.u	44	yes	\(-1\)	\(1\)	\(e\left(\frac{43}{44}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{7}{44}\right)\)	\(e\left(\frac{41}{44}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{15}{44}\right)\)	\(e\left(\frac{3}{22}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{13}{44}\right)\)	\(e\left(\frac{5}{11}\right)\)
\(\chi_{345}(56,\cdot)\)	345.s	22	no	\(1\)	\(1\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{7}{22}\right)\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{1}{22}\right)\)
\(\chi_{345}(58,\cdot)\)	345.v	44	no	\(-1\)	\(1\)	\(e\left(\frac{25}{44}\right)\)	\(e\left(\frac{3}{22}\right)\)	\(e\left(\frac{1}{44}\right)\)	\(e\left(\frac{31}{44}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{43}{44}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{5}{44}\right)\)	\(e\left(\frac{3}{22}\right)\)
\(\chi_{345}(59,\cdot)\)	345.p	22	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{5}{22}\right)\)	\(e\left(\frac{9}{22}\right)\)	\(e\left(\frac{19}{22}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{6}{11}\right)\)
\(\chi_{345}(61,\cdot)\)	345.o	22	no	\(-1\)	\(1\)	\(e\left(\frac{6}{11}\right)\)	\(e\left(\frac{1}{11}\right)\)	\(e\left(\frac{15}{22}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{9}{11}\right)\)	\(e\left(\frac{5}{22}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{9}{22}\right)\)	\(e\left(\frac{13}{22}\right)\)
\(\chi_{345}(62,\cdot)\)	345.x	44	yes	\(1\)	\(1\)	\(e\left(\frac{21}{44}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{7}{44}\right)\)	\(e\left(\frac{19}{44}\right)\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{37}{44}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{13}{44}\right)\)	\(e\left(\frac{21}{22}\right)\)

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.