LMFDB - Group of Dirichlet characters of modulus 73

Show commands: PariGP / SageMath

sage: H = DirichletGroup(73)

pari: g = idealstar(,73,2)

Character group

sage: G.order() pari: g.no
Order	=	72
sage: H.invariants() pari: g.cyc
Structure	=	$C_{72}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{73}(5,\cdot)$

First 32 of 72 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	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$\chi_{73}(1,\cdot)$	73.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{73}(2,\cdot)$	73.g	9	yes	$1$	$1$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{1}{9}\right)$
$\chi_{73}(3,\cdot)$	73.h	12	yes	$1$	$1$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{6}\right)$	$-i$	$1$	$1$	$-i$	$e\left(\frac{7}{12}\right)$
$\chi_{73}(4,\cdot)$	73.g	9	yes	$1$	$1$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{2}{9}\right)$
$\chi_{73}(5,\cdot)$	73.l	72	yes	$-1$	$1$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{72}\right)$	$e\left(\frac{7}{36}\right)$	$e\left(\frac{11}{24}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{55}{72}\right)$
$\chi_{73}(6,\cdot)$	73.k	36	yes	$1$	$1$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{7}{36}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$-i$	$e\left(\frac{25}{36}\right)$
$\chi_{73}(7,\cdot)$	73.j	24	yes	$-1$	$1$	$e\left(\frac{2}{3}\right)$	$-i$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{11}{24}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{1}{8}\right)$	$1$	$-1$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{5}{24}\right)$
$\chi_{73}(8,\cdot)$	73.c	3	yes	$1$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$
$\chi_{73}(9,\cdot)$	73.e	6	yes	$1$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$-1$	$1$	$1$	$-1$	$e\left(\frac{1}{6}\right)$
$\chi_{73}(10,\cdot)$	73.f	8	yes	$-1$	$1$	$1$	$-i$	$1$	$e\left(\frac{1}{8}\right)$	$-i$	$e\left(\frac{1}{8}\right)$	$1$	$-1$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{7}{8}\right)$
$\chi_{73}(11,\cdot)$	73.l	72	yes	$-1$	$1$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{55}{72}\right)$	$e\left(\frac{25}{36}\right)$	$e\left(\frac{5}{24}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{72}\right)$
$\chi_{73}(12,\cdot)$	73.k	36	yes	$1$	$1$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{11}{36}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$-i$	$e\left(\frac{29}{36}\right)$
$\chi_{73}(13,\cdot)$	73.l	72	yes	$-1$	$1$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{59}{72}\right)$	$e\left(\frac{17}{36}\right)$	$e\left(\frac{1}{24}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{5}{72}\right)$
$\chi_{73}(14,\cdot)$	73.l	72	yes	$-1$	$1$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{41}{72}\right)$	$e\left(\frac{35}{36}\right)$	$e\left(\frac{19}{24}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{23}{72}\right)$
$\chi_{73}(15,\cdot)$	73.l	72	yes	$-1$	$1$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{7}{72}\right)$	$e\left(\frac{13}{36}\right)$	$e\left(\frac{5}{24}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{25}{72}\right)$
$\chi_{73}(16,\cdot)$	73.g	9	yes	$1$	$1$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{4}{9}\right)$
$\chi_{73}(17,\cdot)$	73.j	24	yes	$-1$	$1$	$e\left(\frac{1}{3}\right)$	$-i$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{7}{24}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{5}{8}\right)$	$1$	$-1$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{1}{24}\right)$
$\chi_{73}(18,\cdot)$	73.i	18	yes	$1$	$1$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{5}{18}\right)$
$\chi_{73}(19,\cdot)$	73.k	36	yes	$1$	$1$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{31}{36}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$-i$	$e\left(\frac{13}{36}\right)$
$\chi_{73}(20,\cdot)$	73.l	72	yes	$-1$	$1$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{17}{72}\right)$	$e\left(\frac{11}{36}\right)$	$e\left(\frac{19}{24}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{71}{72}\right)$
$\chi_{73}(21,\cdot)$	73.j	24	yes	$-1$	$1$	$e\left(\frac{1}{3}\right)$	$i$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{13}{24}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{7}{8}\right)$	$1$	$-1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{19}{24}\right)$
$\chi_{73}(22,\cdot)$	73.f	8	yes	$-1$	$1$	$1$	$i$	$1$	$e\left(\frac{7}{8}\right)$	$i$	$e\left(\frac{7}{8}\right)$	$1$	$-1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{8}\right)$
$\chi_{73}(23,\cdot)$	73.k	36	yes	$1$	$1$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{23}{36}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$-i$	$e\left(\frac{5}{36}\right)$
$\chi_{73}(24,\cdot)$	73.h	12	yes	$1$	$1$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{5}{6}\right)$	$-i$	$1$	$1$	$-i$	$e\left(\frac{11}{12}\right)$
$\chi_{73}(25,\cdot)$	73.k	36	yes	$1$	$1$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{36}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$i$	$e\left(\frac{19}{36}\right)$
$\chi_{73}(26,\cdot)$	73.l	72	yes	$-1$	$1$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{67}{72}\right)$	$e\left(\frac{1}{36}\right)$	$e\left(\frac{17}{24}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{13}{72}\right)$
$\chi_{73}(27,\cdot)$	73.d	4	yes	$1$	$1$	$1$	$-1$	$1$	$i$	$-1$	$i$	$1$	$1$	$i$	$-i$
$\chi_{73}(28,\cdot)$	73.l	72	yes	$-1$	$1$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{49}{72}\right)$	$e\left(\frac{19}{36}\right)$	$e\left(\frac{11}{24}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{31}{72}\right)$
$\chi_{73}(29,\cdot)$	73.l	72	yes	$-1$	$1$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{35}{72}\right)$	$e\left(\frac{29}{36}\right)$	$e\left(\frac{1}{24}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{53}{72}\right)$
$\chi_{73}(30,\cdot)$	73.j	24	yes	$-1$	$1$	$e\left(\frac{2}{3}\right)$	$i$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{24}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{7}{8}\right)$	$1$	$-1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{11}{24}\right)$
$\chi_{73}(31,\cdot)$	73.l	72	yes	$-1$	$1$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{11}{72}\right)$	$e\left(\frac{5}{36}\right)$	$e\left(\frac{1}{24}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{29}{72}\right)$
$\chi_{73}(32,\cdot)$	73.g	9	yes	$1$	$1$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{5}{9}\right)$

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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	$73$
Structure	\(C_{72}\)
Order	$72$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(\chi_{73}(1,\cdot)\)	73.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{73}(2,\cdot)\)	73.g	9	yes	\(1\)	\(1\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{1}{9}\right)\)
\(\chi_{73}(3,\cdot)\)	73.h	12	yes	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-i\)	\(1\)	\(1\)	\(-i\)	\(e\left(\frac{7}{12}\right)\)
\(\chi_{73}(4,\cdot)\)	73.g	9	yes	\(1\)	\(1\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{2}{9}\right)\)
\(\chi_{73}(5,\cdot)\)	73.l	72	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{72}\right)\)	\(e\left(\frac{7}{36}\right)\)	\(e\left(\frac{11}{24}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{55}{72}\right)\)
\(\chi_{73}(6,\cdot)\)	73.k	36	yes	\(1\)	\(1\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{7}{36}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-i\)	\(e\left(\frac{25}{36}\right)\)
\(\chi_{73}(7,\cdot)\)	73.j	24	yes	\(-1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(-i\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{11}{24}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{5}{24}\right)\)
\(\chi_{73}(8,\cdot)\)	73.c	3	yes	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(1\)	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{73}(9,\cdot)\)	73.e	6	yes	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{73}(10,\cdot)\)	73.f	8	yes	\(-1\)	\(1\)	\(1\)	\(-i\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(-i\)	\(e\left(\frac{1}{8}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)
\(\chi_{73}(11,\cdot)\)	73.l	72	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{55}{72}\right)\)	\(e\left(\frac{25}{36}\right)\)	\(e\left(\frac{5}{24}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{72}\right)\)
\(\chi_{73}(12,\cdot)\)	73.k	36	yes	\(1\)	\(1\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{11}{36}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-i\)	\(e\left(\frac{29}{36}\right)\)
\(\chi_{73}(13,\cdot)\)	73.l	72	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{59}{72}\right)\)	\(e\left(\frac{17}{36}\right)\)	\(e\left(\frac{1}{24}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{5}{72}\right)\)
\(\chi_{73}(14,\cdot)\)	73.l	72	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{41}{72}\right)\)	\(e\left(\frac{35}{36}\right)\)	\(e\left(\frac{19}{24}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{23}{72}\right)\)
\(\chi_{73}(15,\cdot)\)	73.l	72	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{7}{72}\right)\)	\(e\left(\frac{13}{36}\right)\)	\(e\left(\frac{5}{24}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{25}{72}\right)\)
\(\chi_{73}(16,\cdot)\)	73.g	9	yes	\(1\)	\(1\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{4}{9}\right)\)
\(\chi_{73}(17,\cdot)\)	73.j	24	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(-i\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{7}{24}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{1}{24}\right)\)
\(\chi_{73}(18,\cdot)\)	73.i	18	yes	\(1\)	\(1\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(e\left(\frac{5}{18}\right)\)
\(\chi_{73}(19,\cdot)\)	73.k	36	yes	\(1\)	\(1\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{31}{36}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-i\)	\(e\left(\frac{13}{36}\right)\)
\(\chi_{73}(20,\cdot)\)	73.l	72	yes	\(-1\)	\(1\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{17}{72}\right)\)	\(e\left(\frac{11}{36}\right)\)	\(e\left(\frac{19}{24}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{71}{72}\right)\)
\(\chi_{73}(21,\cdot)\)	73.j	24	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(i\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{13}{24}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{19}{24}\right)\)
\(\chi_{73}(22,\cdot)\)	73.f	8	yes	\(-1\)	\(1\)	\(1\)	\(i\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(i\)	\(e\left(\frac{7}{8}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)
\(\chi_{73}(23,\cdot)\)	73.k	36	yes	\(1\)	\(1\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{23}{36}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-i\)	\(e\left(\frac{5}{36}\right)\)
\(\chi_{73}(24,\cdot)\)	73.h	12	yes	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(-i\)	\(1\)	\(1\)	\(-i\)	\(e\left(\frac{11}{12}\right)\)
\(\chi_{73}(25,\cdot)\)	73.k	36	yes	\(1\)	\(1\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{36}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(i\)	\(e\left(\frac{19}{36}\right)\)
\(\chi_{73}(26,\cdot)\)	73.l	72	yes	\(-1\)	\(1\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{67}{72}\right)\)	\(e\left(\frac{1}{36}\right)\)	\(e\left(\frac{17}{24}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{13}{72}\right)\)
\(\chi_{73}(27,\cdot)\)	73.d	4	yes	\(1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(i\)	\(-1\)	\(i\)	\(1\)	\(1\)	\(i\)	\(-i\)
\(\chi_{73}(28,\cdot)\)	73.l	72	yes	\(-1\)	\(1\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{49}{72}\right)\)	\(e\left(\frac{19}{36}\right)\)	\(e\left(\frac{11}{24}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{31}{72}\right)\)
\(\chi_{73}(29,\cdot)\)	73.l	72	yes	\(-1\)	\(1\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{35}{72}\right)\)	\(e\left(\frac{29}{36}\right)\)	\(e\left(\frac{1}{24}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{53}{72}\right)\)
\(\chi_{73}(30,\cdot)\)	73.j	24	yes	\(-1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(i\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{24}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{11}{24}\right)\)
\(\chi_{73}(31,\cdot)\)	73.l	72	yes	\(-1\)	\(1\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{11}{72}\right)\)	\(e\left(\frac{5}{36}\right)\)	\(e\left(\frac{1}{24}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{29}{72}\right)\)
\(\chi_{73}(32,\cdot)\)	73.g	9	yes	\(1\)	\(1\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{5}{9}\right)\)

Introduction

L-functions

Modular forms

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Character group

First 32 of 72 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.