from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1208, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([0,75,112]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,1208))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1208\) | |
| Conductor: | \(1208\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(150\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{75})$ |
| Fixed field: | Number field defined by a degree 150 polynomial (not computed) |
First 31 of 40 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1208}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{19}{150}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{103}{150}\right)\) | \(e\left(\frac{17}{150}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{150}\right)\) |
| \(\chi_{1208}(21,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{1}{150}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{37}{150}\right)\) | \(e\left(\frac{143}{150}\right)\) | \(e\left(\frac{32}{75}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{79}{150}\right)\) |
| \(\chi_{1208}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{67}{150}\right)\) | \(e\left(\frac{11}{75}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{79}{150}\right)\) | \(e\left(\frac{131}{150}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{43}{150}\right)\) |
| \(\chi_{1208}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{13}{150}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{31}{150}\right)\) | \(e\left(\frac{59}{150}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{127}{150}\right)\) |
| \(\chi_{1208}(69,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{127}{150}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{49}{150}\right)\) | \(e\left(\frac{11}{150}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{133}{150}\right)\) |
| \(\chi_{1208}(173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{143}{150}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{41}{150}\right)\) | \(e\left(\frac{49}{150}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{47}{150}\right)\) |
| \(\chi_{1208}(213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{101}{150}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{137}{150}\right)\) | \(e\left(\frac{43}{150}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{29}{150}\right)\) |
| \(\chi_{1208}(333,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{61}{150}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{7}{150}\right)\) | \(e\left(\frac{23}{150}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{150}\right)\) |
| \(\chi_{1208}(341,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{89}{150}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{143}{150}\right)\) | \(e\left(\frac{127}{150}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{131}{150}\right)\) |
| \(\chi_{1208}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{137}{150}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{119}{150}\right)\) | \(e\left(\frac{91}{150}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{23}{150}\right)\) |
| \(\chi_{1208}(357,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{47}{150}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{89}{150}\right)\) | \(e\left(\frac{121}{150}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{113}{150}\right)\) |
| \(\chi_{1208}(397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{49}{150}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{13}{150}\right)\) | \(e\left(\frac{107}{150}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{121}{150}\right)\) |
| \(\chi_{1208}(405,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{7}{150}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{109}{150}\right)\) | \(e\left(\frac{101}{150}\right)\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{103}{150}\right)\) |
| \(\chi_{1208}(493,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{139}{150}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{43}{150}\right)\) | \(e\left(\frac{77}{150}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{31}{150}\right)\) |
| \(\chi_{1208}(533,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{29}{150}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{23}{150}\right)\) | \(e\left(\frac{97}{150}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{41}{150}\right)\) |
| \(\chi_{1208}(541,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{73}{150}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{1}{150}\right)\) | \(e\left(\frac{89}{150}\right)\) | \(e\left(\frac{11}{75}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{67}{150}\right)\) |
| \(\chi_{1208}(589,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{91}{150}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{67}{150}\right)\) | \(e\left(\frac{113}{150}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{139}{150}\right)\) |
| \(\chi_{1208}(597,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{79}{150}\right)\) | \(e\left(\frac{32}{75}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{73}{150}\right)\) | \(e\left(\frac{47}{150}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{91}{150}\right)\) |
| \(\chi_{1208}(621,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{121}{150}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{127}{150}\right)\) | \(e\left(\frac{53}{150}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{32}{75}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{109}{150}\right)\) |
| \(\chi_{1208}(629,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{113}{150}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{131}{150}\right)\) | \(e\left(\frac{109}{150}\right)\) | \(e\left(\frac{16}{75}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{77}{150}\right)\) |
| \(\chi_{1208}(653,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{83}{150}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{71}{150}\right)\) | \(e\left(\frac{19}{150}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{107}{150}\right)\) |
| \(\chi_{1208}(701,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{31}{150}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{97}{150}\right)\) | \(e\left(\frac{83}{150}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{49}{150}\right)\) |
| \(\chi_{1208}(725,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{131}{150}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{47}{150}\right)\) | \(e\left(\frac{133}{150}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{149}{150}\right)\) |
| \(\chi_{1208}(741,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{119}{150}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{53}{150}\right)\) | \(e\left(\frac{67}{150}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{101}{150}\right)\) |
| \(\chi_{1208}(749,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{37}{150}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{19}{150}\right)\) | \(e\left(\frac{41}{150}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{73}{150}\right)\) |
| \(\chi_{1208}(765,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{59}{150}\right)\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{83}{150}\right)\) | \(e\left(\frac{37}{150}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{150}\right)\) |
| \(\chi_{1208}(773,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{109}{150}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{133}{150}\right)\) | \(e\left(\frac{137}{150}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{61}{150}\right)\) |
| \(\chi_{1208}(789,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{11}{150}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{107}{150}\right)\) | \(e\left(\frac{73}{150}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{119}{150}\right)\) |
| \(\chi_{1208}(797,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{41}{150}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{17}{150}\right)\) | \(e\left(\frac{13}{150}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{89}{150}\right)\) |
| \(\chi_{1208}(813,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{133}{150}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{121}{150}\right)\) | \(e\left(\frac{119}{150}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{11}{75}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{150}\right)\) |
| \(\chi_{1208}(829,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{107}{150}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{59}{150}\right)\) | \(e\left(\frac{1}{150}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{53}{150}\right)\) |