Properties

Modulus $47$
Structure \(C_{46}\)
Order $46$

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Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(47)
 
pari: g = idealstar(,47,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 46
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{46}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{47}(5,\cdot)$

First 32 of 46 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 \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{47}(1,\cdot)\) 47.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{47}(2,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{17}{23}\right)\)
\(\chi_{47}(3,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{1}{23}\right)\)
\(\chi_{47}(4,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{11}{23}\right)\)
\(\chi_{47}(5,\cdot)\) 47.d 46 yes \(-1\) \(1\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{1}{46}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{19}{46}\right)\) \(e\left(\frac{7}{46}\right)\)
\(\chi_{47}(6,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{18}{23}\right)\)
\(\chi_{47}(7,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{20}{23}\right)\)
\(\chi_{47}(8,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{5}{23}\right)\)
\(\chi_{47}(9,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{2}{23}\right)\)
\(\chi_{47}(10,\cdot)\) 47.d 46 yes \(-1\) \(1\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{19}{46}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{39}{46}\right)\) \(e\left(\frac{41}{46}\right)\)
\(\chi_{47}(11,\cdot)\) 47.d 46 yes \(-1\) \(1\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{7}{46}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{41}{46}\right)\) \(e\left(\frac{3}{46}\right)\)
\(\chi_{47}(12,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{12}{23}\right)\)
\(\chi_{47}(13,\cdot)\) 47.d 46 yes \(-1\) \(1\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{11}{46}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{25}{46}\right)\) \(e\left(\frac{31}{46}\right)\)
\(\chi_{47}(14,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{14}{23}\right)\)
\(\chi_{47}(15,\cdot)\) 47.d 46 yes \(-1\) \(1\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{21}{46}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{31}{46}\right)\) \(e\left(\frac{9}{46}\right)\)
\(\chi_{47}(16,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{22}{23}\right)\)
\(\chi_{47}(17,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{10}{23}\right)\)
\(\chi_{47}(18,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{19}{23}\right)\)
\(\chi_{47}(19,\cdot)\) 47.d 46 yes \(-1\) \(1\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{45}{46}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{27}{46}\right)\) \(e\left(\frac{39}{46}\right)\)
\(\chi_{47}(20,\cdot)\) 47.d 46 yes \(-1\) \(1\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{37}{46}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{13}{46}\right)\) \(e\left(\frac{29}{46}\right)\)
\(\chi_{47}(21,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{21}{23}\right)\)
\(\chi_{47}(22,\cdot)\) 47.d 46 yes \(-1\) \(1\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{25}{46}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{15}{46}\right)\) \(e\left(\frac{37}{46}\right)\)
\(\chi_{47}(23,\cdot)\) 47.d 46 yes \(-1\) \(1\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{5}{46}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{3}{46}\right)\) \(e\left(\frac{35}{46}\right)\)
\(\chi_{47}(24,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{6}{23}\right)\)
\(\chi_{47}(25,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{20}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{7}{23}\right)\)
\(\chi_{47}(26,\cdot)\) 47.d 46 yes \(-1\) \(1\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{29}{46}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{1}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{45}{46}\right)\) \(e\left(\frac{19}{46}\right)\)
\(\chi_{47}(27,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{17}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{3}{23}\right)\)
\(\chi_{47}(28,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{13}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{19}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{8}{23}\right)\)
\(\chi_{47}(29,\cdot)\) 47.d 46 yes \(-1\) \(1\) \(e\left(\frac{16}{23}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{9}{23}\right)\) \(e\left(\frac{35}{46}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{21}{46}\right)\) \(e\left(\frac{15}{46}\right)\)
\(\chi_{47}(30,\cdot)\) 47.d 46 yes \(-1\) \(1\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{39}{46}\right)\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{18}{23}\right)\) \(e\left(\frac{21}{23}\right)\) \(e\left(\frac{5}{46}\right)\) \(e\left(\frac{43}{46}\right)\)
\(\chi_{47}(31,\cdot)\) 47.d 46 yes \(-1\) \(1\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{7}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{3}{46}\right)\) \(e\left(\frac{11}{23}\right)\) \(e\left(\frac{2}{23}\right)\) \(e\left(\frac{12}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{11}{46}\right)\) \(e\left(\frac{21}{46}\right)\)
\(\chi_{47}(32,\cdot)\) 47.c 23 yes \(1\) \(1\) \(e\left(\frac{5}{23}\right)\) \(e\left(\frac{3}{23}\right)\) \(e\left(\frac{10}{23}\right)\) \(e\left(\frac{22}{23}\right)\) \(e\left(\frac{8}{23}\right)\) \(e\left(\frac{14}{23}\right)\) \(e\left(\frac{15}{23}\right)\) \(e\left(\frac{6}{23}\right)\) \(e\left(\frac{4}{23}\right)\) \(e\left(\frac{16}{23}\right)\)