Properties

Modulus 315
Structure \(C_{12}\times C_{6}\times C_{2}\)
Order 144

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Show commands for: Pari/GP / SageMath

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(315)
 
pari: g = idealstar(,315,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 144
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{12}\times C_{6}\times C_{2}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{315}(218,\cdot)$, $\chi_{315}(136,\cdot)$, $\chi_{315}(134,\cdot)$

First 32 of 144 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 4 8 11 13 16 17 19 22 23
\(\chi_{315}(1,\cdot)\) 315.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{315}(2,\cdot)\) 315.bx 12 yes \(1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(i\) \(-1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(i\)
\(\chi_{315}(4,\cdot)\) 315.bo 6 yes \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\)
\(\chi_{315}(8,\cdot)\) 315.m 4 no \(1\) \(1\) \(i\) \(-1\) \(-i\) \(-1\) \(i\) \(1\) \(i\) \(-1\) \(-i\) \(-i\)
\(\chi_{315}(11,\cdot)\) 315.s 6 no \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{315}(13,\cdot)\) 315.cb 12 yes \(1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(i\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{315}(16,\cdot)\) 315.k 3 no \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\)
\(\chi_{315}(17,\cdot)\) 315.cd 12 no \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(i\) \(e\left(\frac{1}{6}\right)\) \(i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(i\) \(e\left(\frac{7}{12}\right)\)
\(\chi_{315}(19,\cdot)\) 315.bi 6 no \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{315}(22,\cdot)\) 315.by 12 no \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(-i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(i\) \(-1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{315}(23,\cdot)\) 315.bv 12 yes \(1\) \(1\) \(i\) \(-1\) \(-i\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{315}(26,\cdot)\) 315.bj 6 no \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{315}(29,\cdot)\) 315.ba 6 no \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{315}(31,\cdot)\) 315.bc 6 no \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\)
\(\chi_{315}(32,\cdot)\) 315.bx 12 yes \(1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(i\) \(-1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(i\)
\(\chi_{315}(34,\cdot)\) 315.bg 6 yes \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{315}(37,\cdot)\) 315.ca 12 no \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(-i\) \(e\left(\frac{1}{3}\right)\) \(-i\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(i\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{315}(38,\cdot)\) 315.bu 12 yes \(-1\) \(1\) \(i\) \(-1\) \(-i\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{315}(41,\cdot)\) 315.bl 6 no \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{315}(43,\cdot)\) 315.by 12 no \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(i\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(-i\) \(-1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{7}{12}\right)\)
\(\chi_{315}(44,\cdot)\) 315.y 6 no \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{315}(46,\cdot)\) 315.j 3 no \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{315}(47,\cdot)\) 315.bw 12 yes \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(i\) \(-1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(i\)
\(\chi_{315}(52,\cdot)\) 315.bs 12 yes \(1\) \(1\) \(i\) \(-1\) \(-i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{315}(53,\cdot)\) 315.ce 12 no \(1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(-i\) \(e\left(\frac{1}{6}\right)\) \(i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(-i\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{315}(58,\cdot)\) 315.bt 12 yes \(-1\) \(1\) \(-i\) \(-1\) \(i\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{12}\right)\)
\(\chi_{315}(59,\cdot)\) 315.u 6 yes \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\)
\(\chi_{315}(61,\cdot)\) 315.bc 6 no \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\)
\(\chi_{315}(62,\cdot)\) 315.n 4 no \(-1\) \(1\) \(-i\) \(-1\) \(i\) \(-1\) \(i\) \(1\) \(i\) \(1\) \(i\) \(i\)
\(\chi_{315}(64,\cdot)\) 315.d 2 no \(1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(-1\)
\(\chi_{315}(67,\cdot)\) 315.ch 12 yes \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(-i\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(-i\)
\(\chi_{315}(68,\cdot)\) 315.bu 12 yes \(-1\) \(1\) \(i\) \(-1\) \(-i\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{12}\right)\)