Properties

Modulus 77
Structure \(C_{30}\times C_{2}\)
Order 60

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Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(77)
 
pari: g = idealstar(,77,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 60
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{30}\times C_{2}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{77}(68,\cdot)$, $\chi_{77}(43,\cdot)$

First 32 of 60 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.

orbit label order primitive -1 1 2 3 4 5 6 8 9 10 12 13
\(\chi_{77}(1,\cdot)\) 77.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{77}(2,\cdot)\) 77.o 30 Yes \(-1\) \(1\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{10}\right)\)
\(\chi_{77}(3,\cdot)\) 77.p 30 Yes \(-1\) \(1\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{3}{10}\right)\)
\(\chi_{77}(4,\cdot)\) 77.m 15 Yes \(1\) \(1\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{5}\right)\)
\(\chi_{77}(5,\cdot)\) 77.p 30 Yes \(-1\) \(1\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{77}(6,\cdot)\) 77.l 10 Yes \(1\) \(1\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{2}{5}\right)\) \(1\) \(-1\) \(e\left(\frac{2}{5}\right)\)
\(\chi_{77}(8,\cdot)\) 77.k 10 No \(-1\) \(1\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{4}{5}\right)\) \(-1\) \(1\) \(e\left(\frac{3}{10}\right)\)
\(\chi_{77}(9,\cdot)\) 77.m 15 Yes \(1\) \(1\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{5}\right)\)
\(\chi_{77}(10,\cdot)\) 77.i 6 Yes \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\)
\(\chi_{77}(12,\cdot)\) 77.g 6 No \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\)
\(\chi_{77}(13,\cdot)\) 77.l 10 Yes \(1\) \(1\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{3}{5}\right)\) \(1\) \(-1\) \(e\left(\frac{3}{5}\right)\)
\(\chi_{77}(15,\cdot)\) 77.f 5 No \(1\) \(1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(1\) \(1\) \(e\left(\frac{1}{5}\right)\)
\(\chi_{77}(16,\cdot)\) 77.m 15 Yes \(1\) \(1\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{5}\right)\)
\(\chi_{77}(17,\cdot)\) 77.n 30 Yes \(1\) \(1\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{5}\right)\)
\(\chi_{77}(18,\cdot)\) 77.o 30 Yes \(-1\) \(1\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{77}(19,\cdot)\) 77.n 30 Yes \(1\) \(1\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{4}{5}\right)\)
\(\chi_{77}(20,\cdot)\) 77.j 10 Yes \(-1\) \(1\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(-1\) \(-1\) \(e\left(\frac{1}{10}\right)\)
\(\chi_{77}(23,\cdot)\) 77.e 3 No \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\)
\(\chi_{77}(24,\cdot)\) 77.n 30 Yes \(1\) \(1\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{3}{5}\right)\)
\(\chi_{77}(25,\cdot)\) 77.m 15 Yes \(1\) \(1\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{4}{5}\right)\)
\(\chi_{77}(26,\cdot)\) 77.p 30 Yes \(-1\) \(1\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{77}(27,\cdot)\) 77.j 10 Yes \(-1\) \(1\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(-1\) \(-1\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{77}(29,\cdot)\) 77.k 10 No \(-1\) \(1\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{1}{5}\right)\) \(-1\) \(1\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{77}(30,\cdot)\) 77.o 30 Yes \(-1\) \(1\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{10}\right)\)
\(\chi_{77}(31,\cdot)\) 77.p 30 Yes \(-1\) \(1\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{10}\right)\)
\(\chi_{77}(32,\cdot)\) 77.h 6 Yes \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\)
\(\chi_{77}(34,\cdot)\) 77.d 2 No \(-1\) \(1\) \(1\) \(-1\) \(1\) \(-1\) \(-1\) \(1\) \(1\) \(-1\) \(-1\) \(-1\)
\(\chi_{77}(36,\cdot)\) 77.f 5 No \(1\) \(1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(1\) \(1\) \(e\left(\frac{4}{5}\right)\)
\(\chi_{77}(37,\cdot)\) 77.m 15 Yes \(1\) \(1\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{5}\right)\)
\(\chi_{77}(38,\cdot)\) 77.p 30 Yes \(-1\) \(1\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{77}(39,\cdot)\) 77.o 30 Yes \(-1\) \(1\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{77}(40,\cdot)\) 77.n 30 Yes \(1\) \(1\) \(e\left(\frac{11}{30}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{5}\right)\)