Properties

Modulus $680$
Structure \(C_{2}\times C_{2}\times C_{4}\times C_{16}\)
Order $256$

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Show commands: PariGP / SageMath

sage: H = DirichletGroup(680)
 
pari: g = idealstar(,680,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 256
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{2}\times C_{2}\times C_{4}\times C_{16}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{680}(511,\cdot)$, $\chi_{680}(341,\cdot)$, $\chi_{680}(137,\cdot)$, $\chi_{680}(241,\cdot)$

First 32 of 256 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 \(-1\) \(1\) \(3\) \(7\) \(9\) \(11\) \(13\) \(19\) \(21\) \(23\) \(27\) \(29\)
\(\chi_{680}(1,\cdot)\) 680.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{680}(3,\cdot)\) 680.cr 16 yes \(-1\) \(1\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{16}\right)\) \(1\) \(e\left(\frac{3}{8}\right)\) \(i\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{13}{16}\right)\)
\(\chi_{680}(7,\cdot)\) 680.ce 16 no \(-1\) \(1\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{16}\right)\) \(-1\) \(e\left(\frac{5}{8}\right)\) \(i\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{7}{16}\right)\)
\(\chi_{680}(9,\cdot)\) 680.cb 8 no \(1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(i\) \(e\left(\frac{7}{8}\right)\) \(1\) \(-i\) \(-1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{8}\right)\)
\(\chi_{680}(11,\cdot)\) 680.ci 16 no \(1\) \(1\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{16}\right)\) \(i\) \(e\left(\frac{1}{8}\right)\) \(-i\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{3}{16}\right)\)
\(\chi_{680}(13,\cdot)\) 680.bk 4 yes \(-1\) \(1\) \(1\) \(-1\) \(1\) \(i\) \(-i\) \(-1\) \(-1\) \(1\) \(1\) \(i\)
\(\chi_{680}(19,\cdot)\) 680.bq 8 yes \(-1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(-i\) \(e\left(\frac{1}{8}\right)\) \(-1\) \(i\) \(1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\)
\(\chi_{680}(21,\cdot)\) 680.ba 4 no \(1\) \(1\) \(i\) \(i\) \(-1\) \(-i\) \(-1\) \(1\) \(-1\) \(i\) \(-i\) \(i\)
\(\chi_{680}(23,\cdot)\) 680.cs 16 no \(-1\) \(1\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{16}\right)\) \(1\) \(e\left(\frac{1}{8}\right)\) \(i\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{11}{16}\right)\)
\(\chi_{680}(27,\cdot)\) 680.cr 16 yes \(-1\) \(1\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{16}\right)\) \(1\) \(e\left(\frac{1}{8}\right)\) \(-i\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{7}{16}\right)\)
\(\chi_{680}(29,\cdot)\) 680.cm 16 yes \(-1\) \(1\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{16}\right)\) \(i\) \(e\left(\frac{7}{8}\right)\) \(i\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{1}{16}\right)\)
\(\chi_{680}(31,\cdot)\) 680.cl 16 no \(1\) \(1\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{16}\right)\) \(i\) \(e\left(\frac{3}{8}\right)\) \(-i\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{5}{16}\right)\)
\(\chi_{680}(33,\cdot)\) 680.v 4 no \(-1\) \(1\) \(-i\) \(i\) \(-1\) \(-1\) \(i\) \(-1\) \(1\) \(-i\) \(i\) \(1\)
\(\chi_{680}(37,\cdot)\) 680.cf 16 yes \(1\) \(1\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{15}{16}\right)\) \(-1\) \(e\left(\frac{7}{8}\right)\) \(i\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{13}{16}\right)\)
\(\chi_{680}(39,\cdot)\) 680.cn 16 no \(1\) \(1\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{11}{16}\right)\) \(-i\) \(e\left(\frac{7}{8}\right)\) \(-i\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{1}{16}\right)\)
\(\chi_{680}(41,\cdot)\) 680.cj 16 no \(-1\) \(1\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{13}{16}\right)\) \(-i\) \(e\left(\frac{5}{8}\right)\) \(i\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{15}{16}\right)\)
\(\chi_{680}(43,\cdot)\) 680.bw 8 yes \(1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(-i\) \(e\left(\frac{7}{8}\right)\) \(i\) \(i\) \(1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{5}{8}\right)\)
\(\chi_{680}(47,\cdot)\) 680.q 4 no \(1\) \(1\) \(-1\) \(-1\) \(1\) \(i\) \(-i\) \(-1\) \(1\) \(1\) \(-1\) \(-i\)
\(\chi_{680}(49,\cdot)\) 680.cb 8 no \(1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(-i\) \(e\left(\frac{5}{8}\right)\) \(1\) \(i\) \(-1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{8}\right)\)
\(\chi_{680}(53,\cdot)\) 680.bx 8 yes \(-1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(i\) \(e\left(\frac{5}{8}\right)\) \(i\) \(i\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{8}\right)\)
\(\chi_{680}(57,\cdot)\) 680.cg 16 no \(1\) \(1\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{9}{16}\right)\) \(-1\) \(e\left(\frac{5}{8}\right)\) \(i\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{11}{16}\right)\)
\(\chi_{680}(59,\cdot)\) 680.bq 8 yes \(-1\) \(1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(i\) \(e\left(\frac{3}{8}\right)\) \(-1\) \(-i\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\)
\(\chi_{680}(61,\cdot)\) 680.ck 16 no \(-1\) \(1\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{13}{16}\right)\) \(i\) \(e\left(\frac{1}{8}\right)\) \(-i\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{15}{16}\right)\)
\(\chi_{680}(63,\cdot)\) 680.ce 16 no \(-1\) \(1\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{16}\right)\) \(-1\) \(e\left(\frac{3}{8}\right)\) \(-i\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{1}{16}\right)\)
\(\chi_{680}(67,\cdot)\) 680.u 4 yes \(1\) \(1\) \(i\) \(i\) \(-1\) \(-1\) \(i\) \(-1\) \(-1\) \(-i\) \(-i\) \(-1\)
\(\chi_{680}(69,\cdot)\) 680.j 2 no \(1\) \(1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(-1\) \(-1\) \(1\) \(-1\)
\(\chi_{680}(71,\cdot)\) 680.cl 16 no \(1\) \(1\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{15}{16}\right)\) \(i\) \(e\left(\frac{3}{8}\right)\) \(-i\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{13}{16}\right)\)
\(\chi_{680}(73,\cdot)\) 680.cg 16 no \(1\) \(1\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{16}\right)\) \(-1\) \(e\left(\frac{7}{8}\right)\) \(-i\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{9}{16}\right)\)
\(\chi_{680}(77,\cdot)\) 680.bx 8 yes \(-1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(-i\) \(e\left(\frac{3}{8}\right)\) \(-i\) \(-i\) \(1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{5}{8}\right)\)
\(\chi_{680}(79,\cdot)\) 680.cn 16 no \(1\) \(1\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{9}{16}\right)\) \(i\) \(e\left(\frac{5}{8}\right)\) \(i\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{11}{16}\right)\)
\(\chi_{680}(81,\cdot)\) 680.bd 4 no \(1\) \(1\) \(i\) \(-i\) \(-1\) \(-i\) \(1\) \(-1\) \(1\) \(-i\) \(-i\) \(i\)
\(\chi_{680}(83,\cdot)\) 680.bz 8 yes \(1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(i\) \(e\left(\frac{5}{8}\right)\) \(i\) \(-i\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{7}{8}\right)\)
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