from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2025, base_ring=CyclotomicField(270))
M = H._module
chi = DirichletCharacter(H, M([85,81]))
chi.galois_orbit()
[g,chi] = znchar(Mod(14,2025))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2025\) | |
| Conductor: | \(2025\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(270\) | 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: | $\Q(\zeta_{135})$ |
| Fixed field: | Number field defined by a degree 270 polynomial (not computed) |
First 31 of 72 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2025}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{135}\right)\) | \(e\left(\frac{31}{135}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{241}{270}\right)\) | \(e\left(\frac{59}{270}\right)\) | \(e\left(\frac{41}{270}\right)\) | \(e\left(\frac{62}{135}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) |
| \(\chi_{2025}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{106}{135}\right)\) | \(e\left(\frac{77}{135}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{137}{270}\right)\) | \(e\left(\frac{103}{270}\right)\) | \(e\left(\frac{67}{270}\right)\) | \(e\left(\frac{19}{135}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) |
| \(\chi_{2025}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{62}{135}\right)\) | \(e\left(\frac{124}{135}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{19}{270}\right)\) | \(e\left(\frac{101}{270}\right)\) | \(e\left(\frac{29}{270}\right)\) | \(e\left(\frac{113}{135}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) |
| \(\chi_{2025}(104,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{135}\right)\) | \(e\left(\frac{82}{135}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{67}{270}\right)\) | \(e\left(\frac{143}{270}\right)\) | \(e\left(\frac{17}{270}\right)\) | \(e\left(\frac{29}{135}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) |
| \(\chi_{2025}(119,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{135}\right)\) | \(e\left(\frac{83}{135}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{53}{270}\right)\) | \(e\left(\frac{97}{270}\right)\) | \(e\left(\frac{223}{270}\right)\) | \(e\left(\frac{31}{135}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) |
| \(\chi_{2025}(164,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{135}\right)\) | \(e\left(\frac{86}{135}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{11}{270}\right)\) | \(e\left(\frac{229}{270}\right)\) | \(e\left(\frac{31}{270}\right)\) | \(e\left(\frac{37}{135}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) |
| \(\chi_{2025}(194,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{134}{135}\right)\) | \(e\left(\frac{133}{135}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{163}{270}\right)\) | \(e\left(\frac{227}{270}\right)\) | \(e\left(\frac{263}{270}\right)\) | \(e\left(\frac{131}{135}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) |
| \(\chi_{2025}(209,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{112}{135}\right)\) | \(e\left(\frac{89}{135}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{239}{270}\right)\) | \(e\left(\frac{91}{270}\right)\) | \(e\left(\frac{109}{270}\right)\) | \(e\left(\frac{43}{135}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) |
| \(\chi_{2025}(239,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{135}\right)\) | \(e\left(\frac{91}{135}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{211}{270}\right)\) | \(e\left(\frac{269}{270}\right)\) | \(e\left(\frac{251}{270}\right)\) | \(e\left(\frac{47}{135}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) |
| \(\chi_{2025}(254,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{46}{135}\right)\) | \(e\left(\frac{92}{135}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{197}{270}\right)\) | \(e\left(\frac{223}{270}\right)\) | \(e\left(\frac{187}{270}\right)\) | \(e\left(\frac{49}{135}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) |
| \(\chi_{2025}(284,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{92}{135}\right)\) | \(e\left(\frac{49}{135}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{259}{270}\right)\) | \(e\left(\frac{41}{270}\right)\) | \(e\left(\frac{239}{270}\right)\) | \(e\left(\frac{98}{135}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) |
| \(\chi_{2025}(329,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{135}\right)\) | \(e\left(\frac{7}{135}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{37}{270}\right)\) | \(e\left(\frac{83}{270}\right)\) | \(e\left(\frac{227}{270}\right)\) | \(e\left(\frac{14}{135}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{11}{45}\right)\) |
| \(\chi_{2025}(344,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{135}\right)\) | \(e\left(\frac{98}{135}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{113}{270}\right)\) | \(e\left(\frac{217}{270}\right)\) | \(e\left(\frac{73}{270}\right)\) | \(e\left(\frac{61}{135}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{19}{45}\right)\) |
| \(\chi_{2025}(389,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{118}{135}\right)\) | \(e\left(\frac{101}{135}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{71}{270}\right)\) | \(e\left(\frac{79}{270}\right)\) | \(e\left(\frac{151}{270}\right)\) | \(e\left(\frac{67}{135}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{43}{45}\right)\) |
| \(\chi_{2025}(419,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{135}\right)\) | \(e\left(\frac{58}{135}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{133}{270}\right)\) | \(e\left(\frac{167}{270}\right)\) | \(e\left(\frac{203}{270}\right)\) | \(e\left(\frac{116}{135}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{14}{45}\right)\) |
| \(\chi_{2025}(434,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{52}{135}\right)\) | \(e\left(\frac{104}{135}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{29}{270}\right)\) | \(e\left(\frac{211}{270}\right)\) | \(e\left(\frac{229}{270}\right)\) | \(e\left(\frac{73}{135}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{22}{45}\right)\) |
| \(\chi_{2025}(464,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{135}\right)\) | \(e\left(\frac{16}{135}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{181}{270}\right)\) | \(e\left(\frac{209}{270}\right)\) | \(e\left(\frac{191}{270}\right)\) | \(e\left(\frac{32}{135}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) |
| \(\chi_{2025}(479,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{135}\right)\) | \(e\left(\frac{107}{135}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{257}{270}\right)\) | \(e\left(\frac{73}{270}\right)\) | \(e\left(\frac{37}{270}\right)\) | \(e\left(\frac{79}{135}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{1}{45}\right)\) |
| \(\chi_{2025}(509,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{122}{135}\right)\) | \(e\left(\frac{109}{135}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{229}{270}\right)\) | \(e\left(\frac{251}{270}\right)\) | \(e\left(\frac{179}{270}\right)\) | \(e\left(\frac{83}{135}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{17}{45}\right)\) |
| \(\chi_{2025}(554,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{135}\right)\) | \(e\left(\frac{67}{135}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{7}{270}\right)\) | \(e\left(\frac{23}{270}\right)\) | \(e\left(\frac{167}{270}\right)\) | \(e\left(\frac{134}{135}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{41}{45}\right)\) |
| \(\chi_{2025}(569,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{124}{135}\right)\) | \(e\left(\frac{113}{135}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{173}{270}\right)\) | \(e\left(\frac{67}{270}\right)\) | \(e\left(\frac{193}{270}\right)\) | \(e\left(\frac{91}{135}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{4}{45}\right)\) |
| \(\chi_{2025}(614,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{58}{135}\right)\) | \(e\left(\frac{116}{135}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{131}{270}\right)\) | \(e\left(\frac{199}{270}\right)\) | \(e\left(\frac{1}{270}\right)\) | \(e\left(\frac{97}{135}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) |
| \(\chi_{2025}(644,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{135}\right)\) | \(e\left(\frac{118}{135}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{103}{270}\right)\) | \(e\left(\frac{107}{270}\right)\) | \(e\left(\frac{143}{270}\right)\) | \(e\left(\frac{101}{135}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) |
| \(\chi_{2025}(659,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{135}\right)\) | \(e\left(\frac{119}{135}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{89}{270}\right)\) | \(e\left(\frac{61}{270}\right)\) | \(e\left(\frac{79}{270}\right)\) | \(e\left(\frac{103}{135}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) |
| \(\chi_{2025}(689,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{38}{135}\right)\) | \(e\left(\frac{76}{135}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{151}{270}\right)\) | \(e\left(\frac{149}{270}\right)\) | \(e\left(\frac{131}{270}\right)\) | \(e\left(\frac{17}{135}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{23}{45}\right)\) |
| \(\chi_{2025}(704,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{135}\right)\) | \(e\left(\frac{122}{135}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{47}{270}\right)\) | \(e\left(\frac{193}{270}\right)\) | \(e\left(\frac{157}{270}\right)\) | \(e\left(\frac{109}{135}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{31}{45}\right)\) |
| \(\chi_{2025}(734,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{135}\right)\) | \(e\left(\frac{34}{135}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{199}{270}\right)\) | \(e\left(\frac{191}{270}\right)\) | \(e\left(\frac{119}{270}\right)\) | \(e\left(\frac{68}{135}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{2}{45}\right)\) |
| \(\chi_{2025}(779,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{135}\right)\) | \(e\left(\frac{127}{135}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{247}{270}\right)\) | \(e\left(\frac{233}{270}\right)\) | \(e\left(\frac{107}{270}\right)\) | \(e\left(\frac{119}{135}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{26}{45}\right)\) |
| \(\chi_{2025}(794,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{64}{135}\right)\) | \(e\left(\frac{128}{135}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{233}{270}\right)\) | \(e\left(\frac{187}{270}\right)\) | \(e\left(\frac{43}{270}\right)\) | \(e\left(\frac{121}{135}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) |
| \(\chi_{2025}(839,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{135}\right)\) | \(e\left(\frac{131}{135}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{191}{270}\right)\) | \(e\left(\frac{49}{270}\right)\) | \(e\left(\frac{121}{270}\right)\) | \(e\left(\frac{127}{135}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{13}{45}\right)\) |
| \(\chi_{2025}(869,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{135}\right)\) | \(e\left(\frac{43}{135}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{73}{270}\right)\) | \(e\left(\frac{47}{270}\right)\) | \(e\left(\frac{83}{270}\right)\) | \(e\left(\frac{86}{135}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{29}{45}\right)\) |