Properties

Modulus $1728$
Structure \(C_{144}\times C_{2}\times C_{2}\)
Order $576$

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Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(1728)
 
pari: g = idealstar(,1728,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 576
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{144}\times C_{2}\times C_{2}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{1728}(703,\cdot)$, $\chi_{1728}(325,\cdot)$, $\chi_{1728}(1217,\cdot)$

First 32 of 576 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 \(-1\) \(1\) \(5\) \(7\) \(11\) \(13\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\)
\(\chi_{1728}(1,\cdot)\) 1728.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{1728}(5,\cdot)\) 1728.cf 144 yes \(-1\) \(1\) \(e\left(\frac{65}{144}\right)\) \(e\left(\frac{5}{72}\right)\) \(e\left(\frac{133}{144}\right)\) \(e\left(\frac{23}{144}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{67}{72}\right)\) \(e\left(\frac{65}{72}\right)\) \(e\left(\frac{139}{144}\right)\) \(e\left(\frac{1}{18}\right)\)
\(\chi_{1728}(7,\cdot)\) 1728.cd 72 no \(-1\) \(1\) \(e\left(\frac{5}{72}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{13}{72}\right)\) \(e\left(\frac{35}{72}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{55}{72}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{1728}(11,\cdot)\) 1728.ch 144 yes \(1\) \(1\) \(e\left(\frac{133}{144}\right)\) \(e\left(\frac{13}{72}\right)\) \(e\left(\frac{65}{144}\right)\) \(e\left(\frac{67}{144}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{59}{72}\right)\) \(e\left(\frac{61}{72}\right)\) \(e\left(\frac{23}{144}\right)\) \(e\left(\frac{4}{9}\right)\)
\(\chi_{1728}(13,\cdot)\) 1728.ce 144 yes \(1\) \(1\) \(e\left(\frac{23}{144}\right)\) \(e\left(\frac{35}{72}\right)\) \(e\left(\frac{67}{144}\right)\) \(e\left(\frac{89}{144}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{1}{72}\right)\) \(e\left(\frac{23}{72}\right)\) \(e\left(\frac{109}{144}\right)\) \(e\left(\frac{7}{18}\right)\)
\(\chi_{1728}(17,\cdot)\) 1728.bb 12 no \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(-1\) \(i\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{1728}(19,\cdot)\) 1728.bw 48 no \(-1\) \(1\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{43}{48}\right)\) \(i\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{1728}(23,\cdot)\) 1728.cb 72 no \(1\) \(1\) \(e\left(\frac{67}{72}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{59}{72}\right)\) \(e\left(\frac{1}{72}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{17}{72}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{1728}(25,\cdot)\) 1728.cc 72 no \(1\) \(1\) \(e\left(\frac{65}{72}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{61}{72}\right)\) \(e\left(\frac{23}{72}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{67}{72}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{1728}(29,\cdot)\) 1728.cf 144 yes \(-1\) \(1\) \(e\left(\frac{139}{144}\right)\) \(e\left(\frac{55}{72}\right)\) \(e\left(\frac{23}{144}\right)\) \(e\left(\frac{109}{144}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{17}{72}\right)\) \(e\left(\frac{67}{72}\right)\) \(e\left(\frac{89}{144}\right)\) \(e\left(\frac{11}{18}\right)\)
\(\chi_{1728}(31,\cdot)\) 1728.bh 18 no \(-1\) \(1\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{1728}(35,\cdot)\) 1728.bx 48 no \(1\) \(1\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{31}{48}\right)\) \(-i\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{1728}(37,\cdot)\) 1728.by 48 no \(1\) \(1\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{5}{48}\right)\) \(-i\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{1728}(41,\cdot)\) 1728.ca 72 no \(-1\) \(1\) \(e\left(\frac{43}{72}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{47}{72}\right)\) \(e\left(\frac{49}{72}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{41}{72}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{1728}(43,\cdot)\) 1728.cg 144 yes \(-1\) \(1\) \(e\left(\frac{133}{144}\right)\) \(e\left(\frac{13}{72}\right)\) \(e\left(\frac{65}{144}\right)\) \(e\left(\frac{139}{144}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{23}{72}\right)\) \(e\left(\frac{61}{72}\right)\) \(e\left(\frac{23}{144}\right)\) \(e\left(\frac{4}{9}\right)\)
\(\chi_{1728}(47,\cdot)\) 1728.bv 36 no \(1\) \(1\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{1728}(49,\cdot)\) 1728.bs 36 no \(1\) \(1\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{5}{9}\right)\)
\(\chi_{1728}(53,\cdot)\) 1728.bd 16 no \(-1\) \(1\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(i\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{15}{16}\right)\) \(-1\)
\(\chi_{1728}(55,\cdot)\) 1728.u 8 no \(-1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(i\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(-1\) \(e\left(\frac{1}{8}\right)\) \(-i\) \(-i\) \(e\left(\frac{1}{8}\right)\) \(-1\)
\(\chi_{1728}(59,\cdot)\) 1728.ch 144 yes \(1\) \(1\) \(e\left(\frac{65}{144}\right)\) \(e\left(\frac{41}{72}\right)\) \(e\left(\frac{61}{144}\right)\) \(e\left(\frac{23}{144}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{31}{72}\right)\) \(e\left(\frac{65}{72}\right)\) \(e\left(\frac{139}{144}\right)\) \(e\left(\frac{5}{9}\right)\)
\(\chi_{1728}(61,\cdot)\) 1728.ce 144 yes \(1\) \(1\) \(e\left(\frac{91}{144}\right)\) \(e\left(\frac{7}{72}\right)\) \(e\left(\frac{71}{144}\right)\) \(e\left(\frac{133}{144}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{47}{48}\right)\) \(e\left(\frac{29}{72}\right)\) \(e\left(\frac{19}{72}\right)\) \(e\left(\frac{137}{144}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{1728}(65,\cdot)\) 1728.bk 18 no \(-1\) \(1\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{4}{9}\right)\)
\(\chi_{1728}(67,\cdot)\) 1728.cg 144 yes \(-1\) \(1\) \(e\left(\frac{59}{144}\right)\) \(e\left(\frac{35}{72}\right)\) \(e\left(\frac{31}{144}\right)\) \(e\left(\frac{53}{144}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{1}{72}\right)\) \(e\left(\frac{59}{72}\right)\) \(e\left(\frac{73}{144}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{1728}(71,\cdot)\) 1728.br 24 no \(1\) \(1\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{1}{24}\right)\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{1728}(73,\cdot)\) 1728.bo 24 no \(1\) \(1\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{23}{24}\right)\) \(-1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{1728}(77,\cdot)\) 1728.cf 144 yes \(-1\) \(1\) \(e\left(\frac{143}{144}\right)\) \(e\left(\frac{11}{72}\right)\) \(e\left(\frac{91}{144}\right)\) \(e\left(\frac{137}{144}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{61}{72}\right)\) \(e\left(\frac{71}{72}\right)\) \(e\left(\frac{133}{144}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{1728}(79,\cdot)\) 1728.bt 36 no \(-1\) \(1\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{11}{18}\right)\)
\(\chi_{1728}(83,\cdot)\) 1728.ch 144 yes \(1\) \(1\) \(e\left(\frac{103}{144}\right)\) \(e\left(\frac{55}{72}\right)\) \(e\left(\frac{59}{144}\right)\) \(e\left(\frac{1}{144}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{17}{72}\right)\) \(e\left(\frac{31}{72}\right)\) \(e\left(\frac{125}{144}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{1728}(85,\cdot)\) 1728.ce 144 yes \(1\) \(1\) \(e\left(\frac{53}{144}\right)\) \(e\left(\frac{65}{72}\right)\) \(e\left(\frac{73}{144}\right)\) \(e\left(\frac{11}{144}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{43}{72}\right)\) \(e\left(\frac{53}{72}\right)\) \(e\left(\frac{7}{144}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{1728}(89,\cdot)\) 1728.bq 24 no \(-1\) \(1\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{5}{24}\right)\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{1728}(91,\cdot)\) 1728.bw 48 no \(-1\) \(1\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{5}{48}\right)\) \(-i\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{1728}(95,\cdot)\) 1728.bl 18 no \(1\) \(1\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{7}{18}\right)\)