Properties

Modulus 78
Structure \(C_{12}\times C_{2}\)
Order 24

Learn more about

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(78)
pari: g = idealstar(,78,2)

Character group

sage: G.order()
pari: g.no
Order = 24
sage: H.invariants()
pari: g.cyc
Structure = \(C_{12}\times C_{2}\)
sage: H.gens()
pari: g.gen
Generators = $\chi_{78}(67,\cdot)$, $\chi_{78}(53,\cdot)$

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 5 7 11 17 19 23 25 29 31 35
\(\chi_{78}(1,\cdot)\) 78.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{78}(5,\cdot)\) 78.g 4 No \(1\) \(1\) \(i\) \(i\) \(-i\) \(1\) \(-i\) \(1\) \(-1\) \(-1\) \(-i\) \(-1\)
\(\chi_{78}(7,\cdot)\) 78.l 12 No \(-1\) \(1\) \(i\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(i\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{78}(11,\cdot)\) 78.k 12 No \(1\) \(1\) \(-i\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(i\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{78}(17,\cdot)\) 78.j 6 No \(-1\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{78}(19,\cdot)\) 78.l 12 No \(-1\) \(1\) \(-i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(-i\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{78}(23,\cdot)\) 78.j 6 No \(-1\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{78}(25,\cdot)\) 78.b 2 No \(1\) \(1\) \(-1\) \(-1\) \(-1\) \(1\) \(-1\) \(1\) \(1\) \(1\) \(-1\) \(1\)
\(\chi_{78}(29,\cdot)\) 78.h 6 No \(-1\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{78}(31,\cdot)\) 78.f 4 No \(-1\) \(1\) \(-i\) \(i\) \(i\) \(-1\) \(-i\) \(-1\) \(-1\) \(1\) \(-i\) \(1\)
\(\chi_{78}(35,\cdot)\) 78.h 6 No \(-1\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{78}(37,\cdot)\) 78.l 12 No \(-1\) \(1\) \(i\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(i\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{78}(41,\cdot)\) 78.k 12 No \(1\) \(1\) \(i\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-i\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{78}(43,\cdot)\) 78.i 6 No \(1\) \(1\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{78}(47,\cdot)\) 78.g 4 No \(1\) \(1\) \(-i\) \(-i\) \(i\) \(1\) \(i\) \(1\) \(-1\) \(-1\) \(i\) \(-1\)
\(\chi_{78}(49,\cdot)\) 78.i 6 No \(1\) \(1\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{78}(53,\cdot)\) 78.c 2 No \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\)
\(\chi_{78}(55,\cdot)\) 78.e 3 No \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{78}(59,\cdot)\) 78.k 12 No \(1\) \(1\) \(-i\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(i\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{78}(61,\cdot)\) 78.e 3 No \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{78}(67,\cdot)\) 78.l 12 No \(-1\) \(1\) \(-i\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(-i\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{78}(71,\cdot)\) 78.k 12 No \(1\) \(1\) \(i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(-i\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{78}(73,\cdot)\) 78.f 4 No \(-1\) \(1\) \(i\) \(-i\) \(-i\) \(-1\) \(i\) \(-1\) \(-1\) \(1\) \(i\) \(1\)
\(\chi_{78}(77,\cdot)\) 78.d 2 No \(-1\) \(1\) \(1\) \(-1\) \(1\) \(-1\) \(-1\) \(-1\) \(1\) \(-1\) \(-1\) \(-1\)