from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1728, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([0,9,40]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,1728))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1728\) | |
| Conductor: | \(1728\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(144\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{144})$ |
| Fixed field: | Number field defined by a degree 144 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1728}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{144}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{139}{144}\right)\) | \(e\left(\frac{1}{18}\right)\) |
| \(\chi_{1728}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{144}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{109}{144}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{89}{144}\right)\) | \(e\left(\frac{11}{18}\right)\) |
| \(\chi_{1728}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{144}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{91}{144}\right)\) | \(e\left(\frac{137}{144}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{13}{18}\right)\) |
| \(\chi_{1728}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{144}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{125}{144}\right)\) | \(e\left(\frac{79}{144}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{83}{144}\right)\) | \(e\left(\frac{5}{18}\right)\) |
| \(\chi_{1728}(149,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{144}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{49}{144}\right)\) | \(e\left(\frac{107}{144}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{127}{144}\right)\) | \(e\left(\frac{7}{18}\right)\) |
| \(\chi_{1728}(173,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{144}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{83}{144}\right)\) | \(e\left(\frac{49}{144}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{77}{144}\right)\) | \(e\left(\frac{17}{18}\right)\) |
| \(\chi_{1728}(221,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{144}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{7}{144}\right)\) | \(e\left(\frac{77}{144}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{121}{144}\right)\) | \(e\left(\frac{1}{18}\right)\) |
| \(\chi_{1728}(245,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{144}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{41}{144}\right)\) | \(e\left(\frac{19}{144}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{71}{144}\right)\) | \(e\left(\frac{11}{18}\right)\) |
| \(\chi_{1728}(293,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{144}\right)\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{109}{144}\right)\) | \(e\left(\frac{47}{144}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{115}{144}\right)\) | \(e\left(\frac{13}{18}\right)\) |
| \(\chi_{1728}(317,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{144}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{143}{144}\right)\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{65}{144}\right)\) | \(e\left(\frac{5}{18}\right)\) |
| \(\chi_{1728}(365,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{67}{144}\right)\) | \(e\left(\frac{17}{144}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{109}{144}\right)\) | \(e\left(\frac{7}{18}\right)\) |
| \(\chi_{1728}(389,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{144}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{101}{144}\right)\) | \(e\left(\frac{103}{144}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{25}{72}\right)\) | \(e\left(\frac{59}{144}\right)\) | \(e\left(\frac{17}{18}\right)\) |
| \(\chi_{1728}(437,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{144}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{25}{144}\right)\) | \(e\left(\frac{131}{144}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{103}{144}\right)\) | \(e\left(\frac{1}{18}\right)\) |
| \(\chi_{1728}(461,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{144}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{59}{144}\right)\) | \(e\left(\frac{73}{144}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{53}{144}\right)\) | \(e\left(\frac{11}{18}\right)\) |
| \(\chi_{1728}(509,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{144}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{127}{144}\right)\) | \(e\left(\frac{101}{144}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{25}{72}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{97}{144}\right)\) | \(e\left(\frac{13}{18}\right)\) |
| \(\chi_{1728}(533,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{144}\right)\) | \(e\left(\frac{25}{72}\right)\) | \(e\left(\frac{17}{144}\right)\) | \(e\left(\frac{43}{144}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{47}{144}\right)\) | \(e\left(\frac{5}{18}\right)\) |
| \(\chi_{1728}(581,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{144}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{85}{144}\right)\) | \(e\left(\frac{71}{144}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{91}{144}\right)\) | \(e\left(\frac{7}{18}\right)\) |
| \(\chi_{1728}(605,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{144}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{119}{144}\right)\) | \(e\left(\frac{13}{144}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{41}{144}\right)\) | \(e\left(\frac{17}{18}\right)\) |
| \(\chi_{1728}(653,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{144}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{43}{144}\right)\) | \(e\left(\frac{41}{144}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{85}{144}\right)\) | \(e\left(\frac{1}{18}\right)\) |
| \(\chi_{1728}(677,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{144}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{77}{144}\right)\) | \(e\left(\frac{127}{144}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{35}{144}\right)\) | \(e\left(\frac{11}{18}\right)\) |
| \(\chi_{1728}(725,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{144}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{1}{144}\right)\) | \(e\left(\frac{11}{144}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{79}{144}\right)\) | \(e\left(\frac{13}{18}\right)\) |
| \(\chi_{1728}(749,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{144}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{35}{144}\right)\) | \(e\left(\frac{97}{144}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{29}{144}\right)\) | \(e\left(\frac{5}{18}\right)\) |
| \(\chi_{1728}(797,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{144}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{103}{144}\right)\) | \(e\left(\frac{125}{144}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{73}{144}\right)\) | \(e\left(\frac{7}{18}\right)\) |
| \(\chi_{1728}(821,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{137}{144}\right)\) | \(e\left(\frac{67}{144}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{17}{18}\right)\) |
| \(\chi_{1728}(869,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{144}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{61}{144}\right)\) | \(e\left(\frac{95}{144}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{67}{144}\right)\) | \(e\left(\frac{1}{18}\right)\) |
| \(\chi_{1728}(893,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{144}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{95}{144}\right)\) | \(e\left(\frac{37}{144}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{17}{144}\right)\) | \(e\left(\frac{11}{18}\right)\) |
| \(\chi_{1728}(941,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{144}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{19}{144}\right)\) | \(e\left(\frac{65}{144}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{61}{144}\right)\) | \(e\left(\frac{13}{18}\right)\) |
| \(\chi_{1728}(965,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{144}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{53}{144}\right)\) | \(e\left(\frac{7}{144}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{11}{144}\right)\) | \(e\left(\frac{5}{18}\right)\) |
| \(\chi_{1728}(1013,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{144}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{121}{144}\right)\) | \(e\left(\frac{35}{144}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{55}{144}\right)\) | \(e\left(\frac{7}{18}\right)\) |
| \(\chi_{1728}(1037,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{144}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{11}{144}\right)\) | \(e\left(\frac{121}{144}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{5}{144}\right)\) | \(e\left(\frac{17}{18}\right)\) |
| \(\chi_{1728}(1085,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{144}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{79}{144}\right)\) | \(e\left(\frac{5}{144}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{49}{144}\right)\) | \(e\left(\frac{1}{18}\right)\) |