LMFDB - Group of Dirichlet characters of modulus 197

Show commands: PariGP / SageMath

sage: H = DirichletGroup(197)

pari: g = idealstar(,197,2)

Character group

sage: G.order() pari: g.no
Order	=	196
sage: H.invariants() pari: g.cyc
Structure	=	$C_{196}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{197}(2,\cdot)$

First 32 of 196 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	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$\chi_{197}(1,\cdot)$	197.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{197}(2,\cdot)$	197.i	196	yes	$-1$	$1$	$e\left(\frac{1}{196}\right)$	$e\left(\frac{181}{196}\right)$	$e\left(\frac{1}{98}\right)$	$e\left(\frac{89}{196}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{73}{98}\right)$	$e\left(\frac{3}{196}\right)$	$e\left(\frac{83}{98}\right)$	$e\left(\frac{45}{98}\right)$	$e\left(\frac{29}{196}\right)$
$\chi_{197}(3,\cdot)$	197.i	196	yes	$-1$	$1$	$e\left(\frac{181}{196}\right)$	$e\left(\frac{29}{196}\right)$	$e\left(\frac{83}{98}\right)$	$e\left(\frac{37}{196}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{81}{98}\right)$	$e\left(\frac{151}{196}\right)$	$e\left(\frac{29}{98}\right)$	$e\left(\frac{11}{98}\right)$	$e\left(\frac{153}{196}\right)$
$\chi_{197}(4,\cdot)$	197.h	98	yes	$1$	$1$	$e\left(\frac{1}{98}\right)$	$e\left(\frac{83}{98}\right)$	$e\left(\frac{1}{49}\right)$	$e\left(\frac{89}{98}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{24}{49}\right)$	$e\left(\frac{3}{98}\right)$	$e\left(\frac{34}{49}\right)$	$e\left(\frac{45}{49}\right)$	$e\left(\frac{29}{98}\right)$
$\chi_{197}(5,\cdot)$	197.i	196	yes	$-1$	$1$	$e\left(\frac{89}{196}\right)$	$e\left(\frac{37}{196}\right)$	$e\left(\frac{89}{98}\right)$	$e\left(\frac{81}{196}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{29}{98}\right)$	$e\left(\frac{71}{196}\right)$	$e\left(\frac{37}{98}\right)$	$e\left(\frac{85}{98}\right)$	$e\left(\frac{33}{196}\right)$
$\chi_{197}(6,\cdot)$	197.e	14	yes	$1$	$1$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{9}{14}\right)$	$1$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{13}{14}\right)$
$\chi_{197}(7,\cdot)$	197.h	98	yes	$1$	$1$	$e\left(\frac{73}{98}\right)$	$e\left(\frac{81}{98}\right)$	$e\left(\frac{24}{49}\right)$	$e\left(\frac{29}{98}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{37}{49}\right)$	$e\left(\frac{23}{98}\right)$	$e\left(\frac{32}{49}\right)$	$e\left(\frac{2}{49}\right)$	$e\left(\frac{59}{98}\right)$
$\chi_{197}(8,\cdot)$	197.i	196	yes	$-1$	$1$	$e\left(\frac{3}{196}\right)$	$e\left(\frac{151}{196}\right)$	$e\left(\frac{3}{98}\right)$	$e\left(\frac{71}{196}\right)$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{23}{98}\right)$	$e\left(\frac{9}{196}\right)$	$e\left(\frac{53}{98}\right)$	$e\left(\frac{37}{98}\right)$	$e\left(\frac{87}{196}\right)$
$\chi_{197}(9,\cdot)$	197.h	98	yes	$1$	$1$	$e\left(\frac{83}{98}\right)$	$e\left(\frac{29}{98}\right)$	$e\left(\frac{34}{49}\right)$	$e\left(\frac{37}{98}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{32}{49}\right)$	$e\left(\frac{53}{98}\right)$	$e\left(\frac{29}{49}\right)$	$e\left(\frac{11}{49}\right)$	$e\left(\frac{55}{98}\right)$
$\chi_{197}(10,\cdot)$	197.h	98	yes	$1$	$1$	$e\left(\frac{45}{98}\right)$	$e\left(\frac{11}{98}\right)$	$e\left(\frac{45}{49}\right)$	$e\left(\frac{85}{98}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{2}{49}\right)$	$e\left(\frac{37}{98}\right)$	$e\left(\frac{11}{49}\right)$	$e\left(\frac{16}{49}\right)$	$e\left(\frac{31}{98}\right)$
$\chi_{197}(11,\cdot)$	197.i	196	yes	$-1$	$1$	$e\left(\frac{29}{196}\right)$	$e\left(\frac{153}{196}\right)$	$e\left(\frac{29}{98}\right)$	$e\left(\frac{33}{196}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{59}{98}\right)$	$e\left(\frac{87}{196}\right)$	$e\left(\frac{55}{98}\right)$	$e\left(\frac{31}{98}\right)$	$e\left(\frac{57}{196}\right)$
$\chi_{197}(12,\cdot)$	197.i	196	yes	$-1$	$1$	$e\left(\frac{183}{196}\right)$	$e\left(\frac{195}{196}\right)$	$e\left(\frac{85}{98}\right)$	$e\left(\frac{19}{196}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{31}{98}\right)$	$e\left(\frac{157}{196}\right)$	$e\left(\frac{97}{98}\right)$	$e\left(\frac{3}{98}\right)$	$e\left(\frac{15}{196}\right)$
$\chi_{197}(13,\cdot)$	197.i	196	yes	$-1$	$1$	$e\left(\frac{25}{196}\right)$	$e\left(\frac{17}{196}\right)$	$e\left(\frac{25}{98}\right)$	$e\left(\frac{69}{196}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{61}{98}\right)$	$e\left(\frac{75}{196}\right)$	$e\left(\frac{17}{98}\right)$	$e\left(\frac{47}{98}\right)$	$e\left(\frac{137}{196}\right)$
$\chi_{197}(14,\cdot)$	197.c	4	yes	$-1$	$1$	$-i$	$-i$	$-1$	$-i$	$-1$	$-1$	$i$	$-1$	$-1$	$-i$
$\chi_{197}(15,\cdot)$	197.h	98	yes	$1$	$1$	$e\left(\frac{37}{98}\right)$	$e\left(\frac{33}{98}\right)$	$e\left(\frac{37}{49}\right)$	$e\left(\frac{59}{98}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{6}{49}\right)$	$e\left(\frac{13}{98}\right)$	$e\left(\frac{33}{49}\right)$	$e\left(\frac{48}{49}\right)$	$e\left(\frac{93}{98}\right)$
$\chi_{197}(16,\cdot)$	197.g	49	yes	$1$	$1$	$e\left(\frac{1}{49}\right)$	$e\left(\frac{34}{49}\right)$	$e\left(\frac{2}{49}\right)$	$e\left(\frac{40}{49}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{48}{49}\right)$	$e\left(\frac{3}{49}\right)$	$e\left(\frac{19}{49}\right)$	$e\left(\frac{41}{49}\right)$	$e\left(\frac{29}{49}\right)$
$\chi_{197}(17,\cdot)$	197.i	196	yes	$-1$	$1$	$e\left(\frac{159}{196}\right)$	$e\left(\frac{163}{196}\right)$	$e\left(\frac{61}{98}\right)$	$e\left(\frac{39}{196}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{43}{98}\right)$	$e\left(\frac{85}{196}\right)$	$e\left(\frac{65}{98}\right)$	$e\left(\frac{1}{98}\right)$	$e\left(\frac{103}{196}\right)$
$\chi_{197}(18,\cdot)$	197.i	196	yes	$-1$	$1$	$e\left(\frac{167}{196}\right)$	$e\left(\frac{43}{196}\right)$	$e\left(\frac{69}{98}\right)$	$e\left(\frac{163}{196}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{39}{98}\right)$	$e\left(\frac{109}{196}\right)$	$e\left(\frac{43}{98}\right)$	$e\left(\frac{67}{98}\right)$	$e\left(\frac{139}{196}\right)$
$\chi_{197}(19,\cdot)$	197.e	14	yes	$1$	$1$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{4}{7}\right)$	$e\left(\frac{13}{14}\right)$	$1$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{5}{14}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{5}{7}\right)$	$e\left(\frac{11}{14}\right)$
$\chi_{197}(20,\cdot)$	197.f	28	yes	$-1$	$1$	$e\left(\frac{13}{28}\right)$	$e\left(\frac{1}{28}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{9}{28}\right)$	$-1$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{11}{28}\right)$	$e\left(\frac{1}{14}\right)$	$e\left(\frac{11}{14}\right)$	$e\left(\frac{13}{28}\right)$
$\chi_{197}(21,\cdot)$	197.i	196	yes	$-1$	$1$	$e\left(\frac{131}{196}\right)$	$e\left(\frac{191}{196}\right)$	$e\left(\frac{33}{98}\right)$	$e\left(\frac{95}{196}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{57}{98}\right)$	$e\left(\frac{1}{196}\right)$	$e\left(\frac{93}{98}\right)$	$e\left(\frac{15}{98}\right)$	$e\left(\frac{75}{196}\right)$
$\chi_{197}(22,\cdot)$	197.h	98	yes	$1$	$1$	$e\left(\frac{15}{98}\right)$	$e\left(\frac{69}{98}\right)$	$e\left(\frac{15}{49}\right)$	$e\left(\frac{61}{98}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{17}{49}\right)$	$e\left(\frac{45}{98}\right)$	$e\left(\frac{20}{49}\right)$	$e\left(\frac{38}{49}\right)$	$e\left(\frac{43}{98}\right)$
$\chi_{197}(23,\cdot)$	197.g	49	yes	$1$	$1$	$e\left(\frac{30}{49}\right)$	$e\left(\frac{40}{49}\right)$	$e\left(\frac{11}{49}\right)$	$e\left(\frac{24}{49}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{19}{49}\right)$	$e\left(\frac{41}{49}\right)$	$e\left(\frac{31}{49}\right)$	$e\left(\frac{5}{49}\right)$	$e\left(\frac{37}{49}\right)$
$\chi_{197}(24,\cdot)$	197.g	49	yes	$1$	$1$	$e\left(\frac{46}{49}\right)$	$e\left(\frac{45}{49}\right)$	$e\left(\frac{43}{49}\right)$	$e\left(\frac{27}{49}\right)$	$e\left(\frac{6}{7}\right)$	$e\left(\frac{3}{49}\right)$	$e\left(\frac{40}{49}\right)$	$e\left(\frac{41}{49}\right)$	$e\left(\frac{24}{49}\right)$	$e\left(\frac{11}{49}\right)$
$\chi_{197}(25,\cdot)$	197.h	98	yes	$1$	$1$	$e\left(\frac{89}{98}\right)$	$e\left(\frac{37}{98}\right)$	$e\left(\frac{40}{49}\right)$	$e\left(\frac{81}{98}\right)$	$e\left(\frac{2}{7}\right)$	$e\left(\frac{29}{49}\right)$	$e\left(\frac{71}{98}\right)$	$e\left(\frac{37}{49}\right)$	$e\left(\frac{36}{49}\right)$	$e\left(\frac{33}{98}\right)$
$\chi_{197}(26,\cdot)$	197.h	98	yes	$1$	$1$	$e\left(\frac{13}{98}\right)$	$e\left(\frac{1}{98}\right)$	$e\left(\frac{13}{49}\right)$	$e\left(\frac{79}{98}\right)$	$e\left(\frac{1}{7}\right)$	$e\left(\frac{18}{49}\right)$	$e\left(\frac{39}{98}\right)$	$e\left(\frac{1}{49}\right)$	$e\left(\frac{46}{49}\right)$	$e\left(\frac{83}{98}\right)$
$\chi_{197}(27,\cdot)$	197.i	196	yes	$-1$	$1$	$e\left(\frac{151}{196}\right)$	$e\left(\frac{87}{196}\right)$	$e\left(\frac{53}{98}\right)$	$e\left(\frac{111}{196}\right)$	$e\left(\frac{3}{14}\right)$	$e\left(\frac{47}{98}\right)$	$e\left(\frac{61}{196}\right)$	$e\left(\frac{87}{98}\right)$	$e\left(\frac{33}{98}\right)$	$e\left(\frac{67}{196}\right)$
$\chi_{197}(28,\cdot)$	197.g	49	yes	$1$	$1$	$e\left(\frac{37}{49}\right)$	$e\left(\frac{33}{49}\right)$	$e\left(\frac{25}{49}\right)$	$e\left(\frac{10}{49}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{12}{49}\right)$	$e\left(\frac{13}{49}\right)$	$e\left(\frac{17}{49}\right)$	$e\left(\frac{47}{49}\right)$	$e\left(\frac{44}{49}\right)$
$\chi_{197}(29,\cdot)$	197.g	49	yes	$1$	$1$	$e\left(\frac{9}{49}\right)$	$e\left(\frac{12}{49}\right)$	$e\left(\frac{18}{49}\right)$	$e\left(\frac{17}{49}\right)$	$e\left(\frac{3}{7}\right)$	$e\left(\frac{40}{49}\right)$	$e\left(\frac{27}{49}\right)$	$e\left(\frac{24}{49}\right)$	$e\left(\frac{26}{49}\right)$	$e\left(\frac{16}{49}\right)$
$\chi_{197}(30,\cdot)$	197.i	196	yes	$-1$	$1$	$e\left(\frac{75}{196}\right)$	$e\left(\frac{51}{196}\right)$	$e\left(\frac{75}{98}\right)$	$e\left(\frac{11}{196}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{85}{98}\right)$	$e\left(\frac{29}{196}\right)$	$e\left(\frac{51}{98}\right)$	$e\left(\frac{43}{98}\right)$	$e\left(\frac{19}{196}\right)$
$\chi_{197}(31,\cdot)$	197.i	196	yes	$-1$	$1$	$e\left(\frac{141}{196}\right)$	$e\left(\frac{41}{196}\right)$	$e\left(\frac{43}{98}\right)$	$e\left(\frac{5}{196}\right)$	$e\left(\frac{13}{14}\right)$	$e\left(\frac{3}{98}\right)$	$e\left(\frac{31}{196}\right)$	$e\left(\frac{41}{98}\right)$	$e\left(\frac{73}{98}\right)$	$e\left(\frac{169}{196}\right)$
$\chi_{197}(32,\cdot)$	197.i	196	yes	$-1$	$1$	$e\left(\frac{5}{196}\right)$	$e\left(\frac{121}{196}\right)$	$e\left(\frac{5}{98}\right)$	$e\left(\frac{53}{196}\right)$	$e\left(\frac{9}{14}\right)$	$e\left(\frac{71}{98}\right)$	$e\left(\frac{15}{196}\right)$	$e\left(\frac{23}{98}\right)$	$e\left(\frac{29}{98}\right)$	$e\left(\frac{145}{196}\right)$

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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	$197$
Structure	\(C_{196}\)
Order	$196$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(\chi_{197}(1,\cdot)\)	197.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{197}(2,\cdot)\)	197.i	196	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{196}\right)\)	\(e\left(\frac{181}{196}\right)\)	\(e\left(\frac{1}{98}\right)\)	\(e\left(\frac{89}{196}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{73}{98}\right)\)	\(e\left(\frac{3}{196}\right)\)	\(e\left(\frac{83}{98}\right)\)	\(e\left(\frac{45}{98}\right)\)	\(e\left(\frac{29}{196}\right)\)
\(\chi_{197}(3,\cdot)\)	197.i	196	yes	\(-1\)	\(1\)	\(e\left(\frac{181}{196}\right)\)	\(e\left(\frac{29}{196}\right)\)	\(e\left(\frac{83}{98}\right)\)	\(e\left(\frac{37}{196}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{81}{98}\right)\)	\(e\left(\frac{151}{196}\right)\)	\(e\left(\frac{29}{98}\right)\)	\(e\left(\frac{11}{98}\right)\)	\(e\left(\frac{153}{196}\right)\)
\(\chi_{197}(4,\cdot)\)	197.h	98	yes	\(1\)	\(1\)	\(e\left(\frac{1}{98}\right)\)	\(e\left(\frac{83}{98}\right)\)	\(e\left(\frac{1}{49}\right)\)	\(e\left(\frac{89}{98}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{24}{49}\right)\)	\(e\left(\frac{3}{98}\right)\)	\(e\left(\frac{34}{49}\right)\)	\(e\left(\frac{45}{49}\right)\)	\(e\left(\frac{29}{98}\right)\)
\(\chi_{197}(5,\cdot)\)	197.i	196	yes	\(-1\)	\(1\)	\(e\left(\frac{89}{196}\right)\)	\(e\left(\frac{37}{196}\right)\)	\(e\left(\frac{89}{98}\right)\)	\(e\left(\frac{81}{196}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{29}{98}\right)\)	\(e\left(\frac{71}{196}\right)\)	\(e\left(\frac{37}{98}\right)\)	\(e\left(\frac{85}{98}\right)\)	\(e\left(\frac{33}{196}\right)\)
\(\chi_{197}(6,\cdot)\)	197.e	14	yes	\(1\)	\(1\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(1\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{13}{14}\right)\)
\(\chi_{197}(7,\cdot)\)	197.h	98	yes	\(1\)	\(1\)	\(e\left(\frac{73}{98}\right)\)	\(e\left(\frac{81}{98}\right)\)	\(e\left(\frac{24}{49}\right)\)	\(e\left(\frac{29}{98}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{37}{49}\right)\)	\(e\left(\frac{23}{98}\right)\)	\(e\left(\frac{32}{49}\right)\)	\(e\left(\frac{2}{49}\right)\)	\(e\left(\frac{59}{98}\right)\)
\(\chi_{197}(8,\cdot)\)	197.i	196	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{196}\right)\)	\(e\left(\frac{151}{196}\right)\)	\(e\left(\frac{3}{98}\right)\)	\(e\left(\frac{71}{196}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{23}{98}\right)\)	\(e\left(\frac{9}{196}\right)\)	\(e\left(\frac{53}{98}\right)\)	\(e\left(\frac{37}{98}\right)\)	\(e\left(\frac{87}{196}\right)\)
\(\chi_{197}(9,\cdot)\)	197.h	98	yes	\(1\)	\(1\)	\(e\left(\frac{83}{98}\right)\)	\(e\left(\frac{29}{98}\right)\)	\(e\left(\frac{34}{49}\right)\)	\(e\left(\frac{37}{98}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{32}{49}\right)\)	\(e\left(\frac{53}{98}\right)\)	\(e\left(\frac{29}{49}\right)\)	\(e\left(\frac{11}{49}\right)\)	\(e\left(\frac{55}{98}\right)\)
\(\chi_{197}(10,\cdot)\)	197.h	98	yes	\(1\)	\(1\)	\(e\left(\frac{45}{98}\right)\)	\(e\left(\frac{11}{98}\right)\)	\(e\left(\frac{45}{49}\right)\)	\(e\left(\frac{85}{98}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{2}{49}\right)\)	\(e\left(\frac{37}{98}\right)\)	\(e\left(\frac{11}{49}\right)\)	\(e\left(\frac{16}{49}\right)\)	\(e\left(\frac{31}{98}\right)\)
\(\chi_{197}(11,\cdot)\)	197.i	196	yes	\(-1\)	\(1\)	\(e\left(\frac{29}{196}\right)\)	\(e\left(\frac{153}{196}\right)\)	\(e\left(\frac{29}{98}\right)\)	\(e\left(\frac{33}{196}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{59}{98}\right)\)	\(e\left(\frac{87}{196}\right)\)	\(e\left(\frac{55}{98}\right)\)	\(e\left(\frac{31}{98}\right)\)	\(e\left(\frac{57}{196}\right)\)
\(\chi_{197}(12,\cdot)\)	197.i	196	yes	\(-1\)	\(1\)	\(e\left(\frac{183}{196}\right)\)	\(e\left(\frac{195}{196}\right)\)	\(e\left(\frac{85}{98}\right)\)	\(e\left(\frac{19}{196}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{31}{98}\right)\)	\(e\left(\frac{157}{196}\right)\)	\(e\left(\frac{97}{98}\right)\)	\(e\left(\frac{3}{98}\right)\)	\(e\left(\frac{15}{196}\right)\)
\(\chi_{197}(13,\cdot)\)	197.i	196	yes	\(-1\)	\(1\)	\(e\left(\frac{25}{196}\right)\)	\(e\left(\frac{17}{196}\right)\)	\(e\left(\frac{25}{98}\right)\)	\(e\left(\frac{69}{196}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{61}{98}\right)\)	\(e\left(\frac{75}{196}\right)\)	\(e\left(\frac{17}{98}\right)\)	\(e\left(\frac{47}{98}\right)\)	\(e\left(\frac{137}{196}\right)\)
\(\chi_{197}(14,\cdot)\)	197.c	4	yes	\(-1\)	\(1\)	\(-i\)	\(-i\)	\(-1\)	\(-i\)	\(-1\)	\(-1\)	\(i\)	\(-1\)	\(-1\)	\(-i\)
\(\chi_{197}(15,\cdot)\)	197.h	98	yes	\(1\)	\(1\)	\(e\left(\frac{37}{98}\right)\)	\(e\left(\frac{33}{98}\right)\)	\(e\left(\frac{37}{49}\right)\)	\(e\left(\frac{59}{98}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{6}{49}\right)\)	\(e\left(\frac{13}{98}\right)\)	\(e\left(\frac{33}{49}\right)\)	\(e\left(\frac{48}{49}\right)\)	\(e\left(\frac{93}{98}\right)\)
\(\chi_{197}(16,\cdot)\)	197.g	49	yes	\(1\)	\(1\)	\(e\left(\frac{1}{49}\right)\)	\(e\left(\frac{34}{49}\right)\)	\(e\left(\frac{2}{49}\right)\)	\(e\left(\frac{40}{49}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{48}{49}\right)\)	\(e\left(\frac{3}{49}\right)\)	\(e\left(\frac{19}{49}\right)\)	\(e\left(\frac{41}{49}\right)\)	\(e\left(\frac{29}{49}\right)\)
\(\chi_{197}(17,\cdot)\)	197.i	196	yes	\(-1\)	\(1\)	\(e\left(\frac{159}{196}\right)\)	\(e\left(\frac{163}{196}\right)\)	\(e\left(\frac{61}{98}\right)\)	\(e\left(\frac{39}{196}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{43}{98}\right)\)	\(e\left(\frac{85}{196}\right)\)	\(e\left(\frac{65}{98}\right)\)	\(e\left(\frac{1}{98}\right)\)	\(e\left(\frac{103}{196}\right)\)
\(\chi_{197}(18,\cdot)\)	197.i	196	yes	\(-1\)	\(1\)	\(e\left(\frac{167}{196}\right)\)	\(e\left(\frac{43}{196}\right)\)	\(e\left(\frac{69}{98}\right)\)	\(e\left(\frac{163}{196}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{39}{98}\right)\)	\(e\left(\frac{109}{196}\right)\)	\(e\left(\frac{43}{98}\right)\)	\(e\left(\frac{67}{98}\right)\)	\(e\left(\frac{139}{196}\right)\)
\(\chi_{197}(19,\cdot)\)	197.e	14	yes	\(1\)	\(1\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(1\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{5}{14}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{5}{7}\right)\)	\(e\left(\frac{11}{14}\right)\)
\(\chi_{197}(20,\cdot)\)	197.f	28	yes	\(-1\)	\(1\)	\(e\left(\frac{13}{28}\right)\)	\(e\left(\frac{1}{28}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{9}{28}\right)\)	\(-1\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{11}{28}\right)\)	\(e\left(\frac{1}{14}\right)\)	\(e\left(\frac{11}{14}\right)\)	\(e\left(\frac{13}{28}\right)\)
\(\chi_{197}(21,\cdot)\)	197.i	196	yes	\(-1\)	\(1\)	\(e\left(\frac{131}{196}\right)\)	\(e\left(\frac{191}{196}\right)\)	\(e\left(\frac{33}{98}\right)\)	\(e\left(\frac{95}{196}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{57}{98}\right)\)	\(e\left(\frac{1}{196}\right)\)	\(e\left(\frac{93}{98}\right)\)	\(e\left(\frac{15}{98}\right)\)	\(e\left(\frac{75}{196}\right)\)
\(\chi_{197}(22,\cdot)\)	197.h	98	yes	\(1\)	\(1\)	\(e\left(\frac{15}{98}\right)\)	\(e\left(\frac{69}{98}\right)\)	\(e\left(\frac{15}{49}\right)\)	\(e\left(\frac{61}{98}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{17}{49}\right)\)	\(e\left(\frac{45}{98}\right)\)	\(e\left(\frac{20}{49}\right)\)	\(e\left(\frac{38}{49}\right)\)	\(e\left(\frac{43}{98}\right)\)
\(\chi_{197}(23,\cdot)\)	197.g	49	yes	\(1\)	\(1\)	\(e\left(\frac{30}{49}\right)\)	\(e\left(\frac{40}{49}\right)\)	\(e\left(\frac{11}{49}\right)\)	\(e\left(\frac{24}{49}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{19}{49}\right)\)	\(e\left(\frac{41}{49}\right)\)	\(e\left(\frac{31}{49}\right)\)	\(e\left(\frac{5}{49}\right)\)	\(e\left(\frac{37}{49}\right)\)
\(\chi_{197}(24,\cdot)\)	197.g	49	yes	\(1\)	\(1\)	\(e\left(\frac{46}{49}\right)\)	\(e\left(\frac{45}{49}\right)\)	\(e\left(\frac{43}{49}\right)\)	\(e\left(\frac{27}{49}\right)\)	\(e\left(\frac{6}{7}\right)\)	\(e\left(\frac{3}{49}\right)\)	\(e\left(\frac{40}{49}\right)\)	\(e\left(\frac{41}{49}\right)\)	\(e\left(\frac{24}{49}\right)\)	\(e\left(\frac{11}{49}\right)\)
\(\chi_{197}(25,\cdot)\)	197.h	98	yes	\(1\)	\(1\)	\(e\left(\frac{89}{98}\right)\)	\(e\left(\frac{37}{98}\right)\)	\(e\left(\frac{40}{49}\right)\)	\(e\left(\frac{81}{98}\right)\)	\(e\left(\frac{2}{7}\right)\)	\(e\left(\frac{29}{49}\right)\)	\(e\left(\frac{71}{98}\right)\)	\(e\left(\frac{37}{49}\right)\)	\(e\left(\frac{36}{49}\right)\)	\(e\left(\frac{33}{98}\right)\)
\(\chi_{197}(26,\cdot)\)	197.h	98	yes	\(1\)	\(1\)	\(e\left(\frac{13}{98}\right)\)	\(e\left(\frac{1}{98}\right)\)	\(e\left(\frac{13}{49}\right)\)	\(e\left(\frac{79}{98}\right)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{18}{49}\right)\)	\(e\left(\frac{39}{98}\right)\)	\(e\left(\frac{1}{49}\right)\)	\(e\left(\frac{46}{49}\right)\)	\(e\left(\frac{83}{98}\right)\)
\(\chi_{197}(27,\cdot)\)	197.i	196	yes	\(-1\)	\(1\)	\(e\left(\frac{151}{196}\right)\)	\(e\left(\frac{87}{196}\right)\)	\(e\left(\frac{53}{98}\right)\)	\(e\left(\frac{111}{196}\right)\)	\(e\left(\frac{3}{14}\right)\)	\(e\left(\frac{47}{98}\right)\)	\(e\left(\frac{61}{196}\right)\)	\(e\left(\frac{87}{98}\right)\)	\(e\left(\frac{33}{98}\right)\)	\(e\left(\frac{67}{196}\right)\)
\(\chi_{197}(28,\cdot)\)	197.g	49	yes	\(1\)	\(1\)	\(e\left(\frac{37}{49}\right)\)	\(e\left(\frac{33}{49}\right)\)	\(e\left(\frac{25}{49}\right)\)	\(e\left(\frac{10}{49}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{12}{49}\right)\)	\(e\left(\frac{13}{49}\right)\)	\(e\left(\frac{17}{49}\right)\)	\(e\left(\frac{47}{49}\right)\)	\(e\left(\frac{44}{49}\right)\)
\(\chi_{197}(29,\cdot)\)	197.g	49	yes	\(1\)	\(1\)	\(e\left(\frac{9}{49}\right)\)	\(e\left(\frac{12}{49}\right)\)	\(e\left(\frac{18}{49}\right)\)	\(e\left(\frac{17}{49}\right)\)	\(e\left(\frac{3}{7}\right)\)	\(e\left(\frac{40}{49}\right)\)	\(e\left(\frac{27}{49}\right)\)	\(e\left(\frac{24}{49}\right)\)	\(e\left(\frac{26}{49}\right)\)	\(e\left(\frac{16}{49}\right)\)
\(\chi_{197}(30,\cdot)\)	197.i	196	yes	\(-1\)	\(1\)	\(e\left(\frac{75}{196}\right)\)	\(e\left(\frac{51}{196}\right)\)	\(e\left(\frac{75}{98}\right)\)	\(e\left(\frac{11}{196}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{85}{98}\right)\)	\(e\left(\frac{29}{196}\right)\)	\(e\left(\frac{51}{98}\right)\)	\(e\left(\frac{43}{98}\right)\)	\(e\left(\frac{19}{196}\right)\)
\(\chi_{197}(31,\cdot)\)	197.i	196	yes	\(-1\)	\(1\)	\(e\left(\frac{141}{196}\right)\)	\(e\left(\frac{41}{196}\right)\)	\(e\left(\frac{43}{98}\right)\)	\(e\left(\frac{5}{196}\right)\)	\(e\left(\frac{13}{14}\right)\)	\(e\left(\frac{3}{98}\right)\)	\(e\left(\frac{31}{196}\right)\)	\(e\left(\frac{41}{98}\right)\)	\(e\left(\frac{73}{98}\right)\)	\(e\left(\frac{169}{196}\right)\)
\(\chi_{197}(32,\cdot)\)	197.i	196	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{196}\right)\)	\(e\left(\frac{121}{196}\right)\)	\(e\left(\frac{5}{98}\right)\)	\(e\left(\frac{53}{196}\right)\)	\(e\left(\frac{9}{14}\right)\)	\(e\left(\frac{71}{98}\right)\)	\(e\left(\frac{15}{196}\right)\)	\(e\left(\frac{23}{98}\right)\)	\(e\left(\frac{29}{98}\right)\)	\(e\left(\frac{145}{196}\right)\)

Introduction

L-functions

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Character group

First 32 of 196 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.