LMFDB - Group of Dirichlet characters of modulus 209

Show commands: PariGP / SageMath

sage: H = DirichletGroup(209)

pari: g = idealstar(,209,2)

Character group

sage: G.order() pari: g.no
Order	=	180
sage: H.invariants() pari: g.cyc
Structure	=	$C_{2}\times C_{90}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{209}(134,\cdot)$, $\chi_{209}(78,\cdot)$

First 32 of 180 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	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$12$
$\chi_{209}(1,\cdot)$	209.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{209}(2,\cdot)$	209.w	90	yes	$1$	$1$	$e\left(\frac{7}{45}\right)$	$e\left(\frac{47}{90}\right)$	$e\left(\frac{14}{45}\right)$	$e\left(\frac{13}{45}\right)$	$e\left(\frac{61}{90}\right)$	$e\left(\frac{1}{30}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{2}{45}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{209}(3,\cdot)$	209.x	90	yes	$-1$	$1$	$e\left(\frac{47}{90}\right)$	$e\left(\frac{71}{90}\right)$	$e\left(\frac{2}{45}\right)$	$e\left(\frac{34}{45}\right)$	$e\left(\frac{14}{45}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{17}{30}\right)$	$e\left(\frac{26}{45}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{209}(4,\cdot)$	209.u	45	yes	$1$	$1$	$e\left(\frac{14}{45}\right)$	$e\left(\frac{2}{45}\right)$	$e\left(\frac{28}{45}\right)$	$e\left(\frac{26}{45}\right)$	$e\left(\frac{16}{45}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{4}{45}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{209}(5,\cdot)$	209.u	45	yes	$1$	$1$	$e\left(\frac{13}{45}\right)$	$e\left(\frac{34}{45}\right)$	$e\left(\frac{26}{45}\right)$	$e\left(\frac{37}{45}\right)$	$e\left(\frac{2}{45}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{23}{45}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{209}(6,\cdot)$	209.v	90	yes	$-1$	$1$	$e\left(\frac{61}{90}\right)$	$e\left(\frac{14}{45}\right)$	$e\left(\frac{16}{45}\right)$	$e\left(\frac{2}{45}\right)$	$e\left(\frac{89}{90}\right)$	$e\left(\frac{29}{30}\right)$	$e\left(\frac{1}{30}\right)$	$e\left(\frac{28}{45}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{209}(7,\cdot)$	209.s	30	yes	$-1$	$1$	$e\left(\frac{1}{30}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{29}{30}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{1}{6}\right)$	$1$
$\chi_{209}(8,\cdot)$	209.t	30	yes	$1$	$1$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{17}{30}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{1}{30}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{1}{3}\right)$	$-1$
$\chi_{209}(9,\cdot)$	209.u	45	yes	$1$	$1$	$e\left(\frac{2}{45}\right)$	$e\left(\frac{26}{45}\right)$	$e\left(\frac{4}{45}\right)$	$e\left(\frac{23}{45}\right)$	$e\left(\frac{28}{45}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{7}{45}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{209}(10,\cdot)$	209.p	18	yes	$1$	$1$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{209}(12,\cdot)$	209.i	6	no	$-1$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{6}\right)$	$-1$
$\chi_{209}(13,\cdot)$	209.w	90	yes	$1$	$1$	$e\left(\frac{17}{45}\right)$	$e\left(\frac{37}{90}\right)$	$e\left(\frac{34}{45}\right)$	$e\left(\frac{38}{45}\right)$	$e\left(\frac{71}{90}\right)$	$e\left(\frac{11}{30}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{37}{45}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{209}(14,\cdot)$	209.x	90	yes	$-1$	$1$	$e\left(\frac{17}{90}\right)$	$e\left(\frac{41}{90}\right)$	$e\left(\frac{17}{45}\right)$	$e\left(\frac{19}{45}\right)$	$e\left(\frac{29}{45}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{17}{30}\right)$	$e\left(\frac{41}{45}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{209}(15,\cdot)$	209.x	90	yes	$-1$	$1$	$e\left(\frac{73}{90}\right)$	$e\left(\frac{49}{90}\right)$	$e\left(\frac{28}{45}\right)$	$e\left(\frac{26}{45}\right)$	$e\left(\frac{16}{45}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{13}{30}\right)$	$e\left(\frac{4}{45}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{209}(16,\cdot)$	209.u	45	yes	$1$	$1$	$e\left(\frac{28}{45}\right)$	$e\left(\frac{4}{45}\right)$	$e\left(\frac{11}{45}\right)$	$e\left(\frac{7}{45}\right)$	$e\left(\frac{32}{45}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{8}{45}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{209}(17,\cdot)$	209.v	90	yes	$-1$	$1$	$e\left(\frac{41}{90}\right)$	$e\left(\frac{19}{45}\right)$	$e\left(\frac{41}{45}\right)$	$e\left(\frac{22}{45}\right)$	$e\left(\frac{79}{90}\right)$	$e\left(\frac{19}{30}\right)$	$e\left(\frac{11}{30}\right)$	$e\left(\frac{38}{45}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{209}(18,\cdot)$	209.k	10	yes	$1$	$1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{1}{5}\right)$	$1$	$-1$
$\chi_{209}(20,\cdot)$	209.f	5	no	$1$	$1$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{3}{5}\right)$	$1$	$1$
$\chi_{209}(21,\cdot)$	209.p	18	yes	$1$	$1$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{13}{18}\right)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{209}(23,\cdot)$	209.j	9	no	$1$	$1$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{209}(24,\cdot)$	209.v	90	yes	$-1$	$1$	$e\left(\frac{89}{90}\right)$	$e\left(\frac{16}{45}\right)$	$e\left(\frac{44}{45}\right)$	$e\left(\frac{28}{45}\right)$	$e\left(\frac{31}{90}\right)$	$e\left(\frac{1}{30}\right)$	$e\left(\frac{29}{30}\right)$	$e\left(\frac{32}{45}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{209}(25,\cdot)$	209.u	45	yes	$1$	$1$	$e\left(\frac{26}{45}\right)$	$e\left(\frac{23}{45}\right)$	$e\left(\frac{7}{45}\right)$	$e\left(\frac{29}{45}\right)$	$e\left(\frac{4}{45}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{1}{45}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{209}(26,\cdot)$	209.n	15	yes	$1$	$1$	$e\left(\frac{8}{15}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{7}{15}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{2}{3}\right)$	$1$
$\chi_{209}(27,\cdot)$	209.r	30	yes	$-1$	$1$	$e\left(\frac{17}{30}\right)$	$e\left(\frac{11}{30}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{5}{6}\right)$	$-1$
$\chi_{209}(28,\cdot)$	209.v	90	yes	$-1$	$1$	$e\left(\frac{31}{90}\right)$	$e\left(\frac{44}{45}\right)$	$e\left(\frac{31}{45}\right)$	$e\left(\frac{32}{45}\right)$	$e\left(\frac{29}{90}\right)$	$e\left(\frac{29}{30}\right)$	$e\left(\frac{1}{30}\right)$	$e\left(\frac{43}{45}\right)$	$e\left(\frac{1}{18}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{209}(29,\cdot)$	209.w	90	yes	$1$	$1$	$e\left(\frac{29}{45}\right)$	$e\left(\frac{79}{90}\right)$	$e\left(\frac{13}{45}\right)$	$e\left(\frac{41}{45}\right)$	$e\left(\frac{47}{90}\right)$	$e\left(\frac{17}{30}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{34}{45}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{209}(30,\cdot)$	209.s	30	yes	$-1$	$1$	$e\left(\frac{29}{30}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{1}{30}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{2}{15}\right)$	$e\left(\frac{5}{6}\right)$	$1$
$\chi_{209}(31,\cdot)$	209.r	30	yes	$-1$	$1$	$e\left(\frac{13}{30}\right)$	$e\left(\frac{19}{30}\right)$	$e\left(\frac{13}{15}\right)$	$e\left(\frac{11}{15}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{4}{15}\right)$	$e\left(\frac{1}{6}\right)$	$-1$
$\chi_{209}(32,\cdot)$	209.p	18	yes	$1$	$1$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{209}(34,\cdot)$	209.o	18	no	$-1$	$1$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{17}{18}\right)$	$e\left(\frac{2}{9}\right)$	$e\left(\frac{7}{9}\right)$	$e\left(\frac{5}{9}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{8}{9}\right)$	$e\left(\frac{7}{18}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{209}(35,\cdot)$	209.v	90	yes	$-1$	$1$	$e\left(\frac{29}{90}\right)$	$e\left(\frac{31}{45}\right)$	$e\left(\frac{29}{45}\right)$	$e\left(\frac{43}{45}\right)$	$e\left(\frac{1}{90}\right)$	$e\left(\frac{1}{30}\right)$	$e\left(\frac{29}{30}\right)$	$e\left(\frac{17}{45}\right)$	$e\left(\frac{5}{18}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{209}(36,\cdot)$	209.u	45	yes	$1$	$1$	$e\left(\frac{16}{45}\right)$	$e\left(\frac{28}{45}\right)$	$e\left(\frac{32}{45}\right)$	$e\left(\frac{4}{45}\right)$	$e\left(\frac{44}{45}\right)$	$e\left(\frac{14}{15}\right)$	$e\left(\frac{1}{15}\right)$	$e\left(\frac{11}{45}\right)$	$e\left(\frac{4}{9}\right)$	$e\left(\frac{1}{3}\right)$

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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	$209$
Structure	\(C_{2}\times C_{90}\)
Order	$180$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(12\)
\(\chi_{209}(1,\cdot)\)	209.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{209}(2,\cdot)\)	209.w	90	yes	\(1\)	\(1\)	\(e\left(\frac{7}{45}\right)\)	\(e\left(\frac{47}{90}\right)\)	\(e\left(\frac{14}{45}\right)\)	\(e\left(\frac{13}{45}\right)\)	\(e\left(\frac{61}{90}\right)\)	\(e\left(\frac{1}{30}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{2}{45}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{209}(3,\cdot)\)	209.x	90	yes	\(-1\)	\(1\)	\(e\left(\frac{47}{90}\right)\)	\(e\left(\frac{71}{90}\right)\)	\(e\left(\frac{2}{45}\right)\)	\(e\left(\frac{34}{45}\right)\)	\(e\left(\frac{14}{45}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{17}{30}\right)\)	\(e\left(\frac{26}{45}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{209}(4,\cdot)\)	209.u	45	yes	\(1\)	\(1\)	\(e\left(\frac{14}{45}\right)\)	\(e\left(\frac{2}{45}\right)\)	\(e\left(\frac{28}{45}\right)\)	\(e\left(\frac{26}{45}\right)\)	\(e\left(\frac{16}{45}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{4}{45}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{209}(5,\cdot)\)	209.u	45	yes	\(1\)	\(1\)	\(e\left(\frac{13}{45}\right)\)	\(e\left(\frac{34}{45}\right)\)	\(e\left(\frac{26}{45}\right)\)	\(e\left(\frac{37}{45}\right)\)	\(e\left(\frac{2}{45}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{23}{45}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{209}(6,\cdot)\)	209.v	90	yes	\(-1\)	\(1\)	\(e\left(\frac{61}{90}\right)\)	\(e\left(\frac{14}{45}\right)\)	\(e\left(\frac{16}{45}\right)\)	\(e\left(\frac{2}{45}\right)\)	\(e\left(\frac{89}{90}\right)\)	\(e\left(\frac{29}{30}\right)\)	\(e\left(\frac{1}{30}\right)\)	\(e\left(\frac{28}{45}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{209}(7,\cdot)\)	209.s	30	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{30}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{29}{30}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)
\(\chi_{209}(8,\cdot)\)	209.t	30	yes	\(1\)	\(1\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{17}{30}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{1}{30}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)
\(\chi_{209}(9,\cdot)\)	209.u	45	yes	\(1\)	\(1\)	\(e\left(\frac{2}{45}\right)\)	\(e\left(\frac{26}{45}\right)\)	\(e\left(\frac{4}{45}\right)\)	\(e\left(\frac{23}{45}\right)\)	\(e\left(\frac{28}{45}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{7}{45}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{209}(10,\cdot)\)	209.p	18	yes	\(1\)	\(1\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{209}(12,\cdot)\)	209.i	6	no	\(-1\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)
\(\chi_{209}(13,\cdot)\)	209.w	90	yes	\(1\)	\(1\)	\(e\left(\frac{17}{45}\right)\)	\(e\left(\frac{37}{90}\right)\)	\(e\left(\frac{34}{45}\right)\)	\(e\left(\frac{38}{45}\right)\)	\(e\left(\frac{71}{90}\right)\)	\(e\left(\frac{11}{30}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{37}{45}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{209}(14,\cdot)\)	209.x	90	yes	\(-1\)	\(1\)	\(e\left(\frac{17}{90}\right)\)	\(e\left(\frac{41}{90}\right)\)	\(e\left(\frac{17}{45}\right)\)	\(e\left(\frac{19}{45}\right)\)	\(e\left(\frac{29}{45}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{17}{30}\right)\)	\(e\left(\frac{41}{45}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{209}(15,\cdot)\)	209.x	90	yes	\(-1\)	\(1\)	\(e\left(\frac{73}{90}\right)\)	\(e\left(\frac{49}{90}\right)\)	\(e\left(\frac{28}{45}\right)\)	\(e\left(\frac{26}{45}\right)\)	\(e\left(\frac{16}{45}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{13}{30}\right)\)	\(e\left(\frac{4}{45}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{209}(16,\cdot)\)	209.u	45	yes	\(1\)	\(1\)	\(e\left(\frac{28}{45}\right)\)	\(e\left(\frac{4}{45}\right)\)	\(e\left(\frac{11}{45}\right)\)	\(e\left(\frac{7}{45}\right)\)	\(e\left(\frac{32}{45}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{8}{45}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{209}(17,\cdot)\)	209.v	90	yes	\(-1\)	\(1\)	\(e\left(\frac{41}{90}\right)\)	\(e\left(\frac{19}{45}\right)\)	\(e\left(\frac{41}{45}\right)\)	\(e\left(\frac{22}{45}\right)\)	\(e\left(\frac{79}{90}\right)\)	\(e\left(\frac{19}{30}\right)\)	\(e\left(\frac{11}{30}\right)\)	\(e\left(\frac{38}{45}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{209}(18,\cdot)\)	209.k	10	yes	\(1\)	\(1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(1\)	\(-1\)
\(\chi_{209}(20,\cdot)\)	209.f	5	no	\(1\)	\(1\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(1\)	\(1\)
\(\chi_{209}(21,\cdot)\)	209.p	18	yes	\(1\)	\(1\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{5}{6}\right)\)
\(\chi_{209}(23,\cdot)\)	209.j	9	no	\(1\)	\(1\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{209}(24,\cdot)\)	209.v	90	yes	\(-1\)	\(1\)	\(e\left(\frac{89}{90}\right)\)	\(e\left(\frac{16}{45}\right)\)	\(e\left(\frac{44}{45}\right)\)	\(e\left(\frac{28}{45}\right)\)	\(e\left(\frac{31}{90}\right)\)	\(e\left(\frac{1}{30}\right)\)	\(e\left(\frac{29}{30}\right)\)	\(e\left(\frac{32}{45}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{209}(25,\cdot)\)	209.u	45	yes	\(1\)	\(1\)	\(e\left(\frac{26}{45}\right)\)	\(e\left(\frac{23}{45}\right)\)	\(e\left(\frac{7}{45}\right)\)	\(e\left(\frac{29}{45}\right)\)	\(e\left(\frac{4}{45}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{1}{45}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{209}(26,\cdot)\)	209.n	15	yes	\(1\)	\(1\)	\(e\left(\frac{8}{15}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{7}{15}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)
\(\chi_{209}(27,\cdot)\)	209.r	30	yes	\(-1\)	\(1\)	\(e\left(\frac{17}{30}\right)\)	\(e\left(\frac{11}{30}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)
\(\chi_{209}(28,\cdot)\)	209.v	90	yes	\(-1\)	\(1\)	\(e\left(\frac{31}{90}\right)\)	\(e\left(\frac{44}{45}\right)\)	\(e\left(\frac{31}{45}\right)\)	\(e\left(\frac{32}{45}\right)\)	\(e\left(\frac{29}{90}\right)\)	\(e\left(\frac{29}{30}\right)\)	\(e\left(\frac{1}{30}\right)\)	\(e\left(\frac{43}{45}\right)\)	\(e\left(\frac{1}{18}\right)\)	\(e\left(\frac{2}{3}\right)\)
\(\chi_{209}(29,\cdot)\)	209.w	90	yes	\(1\)	\(1\)	\(e\left(\frac{29}{45}\right)\)	\(e\left(\frac{79}{90}\right)\)	\(e\left(\frac{13}{45}\right)\)	\(e\left(\frac{41}{45}\right)\)	\(e\left(\frac{47}{90}\right)\)	\(e\left(\frac{17}{30}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{34}{45}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{209}(30,\cdot)\)	209.s	30	yes	\(-1\)	\(1\)	\(e\left(\frac{29}{30}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{1}{30}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{2}{15}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)
\(\chi_{209}(31,\cdot)\)	209.r	30	yes	\(-1\)	\(1\)	\(e\left(\frac{13}{30}\right)\)	\(e\left(\frac{19}{30}\right)\)	\(e\left(\frac{13}{15}\right)\)	\(e\left(\frac{11}{15}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{4}{15}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)
\(\chi_{209}(32,\cdot)\)	209.p	18	yes	\(1\)	\(1\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{209}(34,\cdot)\)	209.o	18	no	\(-1\)	\(1\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{17}{18}\right)\)	\(e\left(\frac{2}{9}\right)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{1}{6}\right)\)
\(\chi_{209}(35,\cdot)\)	209.v	90	yes	\(-1\)	\(1\)	\(e\left(\frac{29}{90}\right)\)	\(e\left(\frac{31}{45}\right)\)	\(e\left(\frac{29}{45}\right)\)	\(e\left(\frac{43}{45}\right)\)	\(e\left(\frac{1}{90}\right)\)	\(e\left(\frac{1}{30}\right)\)	\(e\left(\frac{29}{30}\right)\)	\(e\left(\frac{17}{45}\right)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{1}{3}\right)\)
\(\chi_{209}(36,\cdot)\)	209.u	45	yes	\(1\)	\(1\)	\(e\left(\frac{16}{45}\right)\)	\(e\left(\frac{28}{45}\right)\)	\(e\left(\frac{32}{45}\right)\)	\(e\left(\frac{4}{45}\right)\)	\(e\left(\frac{44}{45}\right)\)	\(e\left(\frac{14}{15}\right)\)	\(e\left(\frac{1}{15}\right)\)	\(e\left(\frac{11}{45}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)

Introduction

L-functions

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First 32 of 180 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.