from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(407, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([72,115]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,407))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(407\) | |
| Conductor: | \(407\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(180\) | 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: | $\Q(\zeta_{180})$ |
| Fixed field: | Number field defined by a degree 180 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{407}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{180}\right)\) | \(e\left(\frac{73}{90}\right)\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{53}{180}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) |
| \(\chi_{407}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{180}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{19}{180}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) |
| \(\chi_{407}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{180}\right)\) | \(e\left(\frac{77}{90}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{67}{180}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{9}\right)\) |
| \(\chi_{407}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{180}\right)\) | \(e\left(\frac{37}{90}\right)\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{17}{180}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) |
| \(\chi_{407}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{180}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{109}{180}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) |
| \(\chi_{407}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{180}\right)\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{79}{90}\right)\) | \(e\left(\frac{71}{180}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) |
| \(\chi_{407}(91,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{180}\right)\) | \(e\left(\frac{41}{90}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{121}{180}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{9}\right)\) |
| \(\chi_{407}(92,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{180}\right)\) | \(e\left(\frac{79}{90}\right)\) | \(e\left(\frac{31}{90}\right)\) | \(e\left(\frac{119}{180}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) |
| \(\chi_{407}(93,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{180}\right)\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{173}{180}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) |
| \(\chi_{407}(113,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{180}\right)\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{151}{180}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) |
| \(\chi_{407}(124,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{180}\right)\) | \(e\left(\frac{31}{90}\right)\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{41}{180}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) |
| \(\chi_{407}(126,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{180}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{47}{90}\right)\) | \(e\left(\frac{163}{180}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) |
| \(\chi_{407}(130,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{180}\right)\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{13}{90}\right)\) | \(e\left(\frac{137}{180}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) |
| \(\chi_{407}(135,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{180}\right)\) | \(e\left(\frac{31}{90}\right)\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{131}{180}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) |
| \(\chi_{407}(146,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{180}\right)\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{61}{180}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) |
| \(\chi_{407}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{180}\right)\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{83}{90}\right)\) | \(e\left(\frac{127}{180}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) |
| \(\chi_{407}(168,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{180}\right)\) | \(e\left(\frac{41}{90}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{31}{180}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{9}\right)\) |
| \(\chi_{407}(170,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{180}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{47}{90}\right)\) | \(e\left(\frac{73}{180}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) |
| \(\chi_{407}(180,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{180}\right)\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{61}{90}\right)\) | \(e\left(\frac{179}{180}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) |
| \(\chi_{407}(190,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{180}\right)\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{79}{90}\right)\) | \(e\left(\frac{161}{180}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) |
| \(\chi_{407}(202,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{180}\right)\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{71}{90}\right)\) | \(e\left(\frac{49}{180}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{9}\right)\) |
| \(\chi_{407}(203,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{180}\right)\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{83}{180}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) |
| \(\chi_{407}(207,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{180}\right)\) | \(e\left(\frac{17}{90}\right)\) | \(e\left(\frac{83}{90}\right)\) | \(e\left(\frac{37}{180}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) |
| \(\chi_{407}(224,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{180}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{41}{90}\right)\) | \(e\left(\frac{79}{180}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) |
| \(\chi_{407}(235,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{180}\right)\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{149}{180}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) |
| \(\chi_{407}(240,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{180}\right)\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{13}{90}\right)\) | \(e\left(\frac{47}{180}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) |
| \(\chi_{407}(246,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{180}\right)\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{1}{90}\right)\) | \(e\left(\frac{59}{180}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) |
| \(\chi_{407}(257,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{180}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{41}{90}\right)\) | \(e\left(\frac{169}{180}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) |
| \(\chi_{407}(278,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{180}\right)\) | \(e\left(\frac{61}{90}\right)\) | \(e\left(\frac{49}{90}\right)\) | \(e\left(\frac{101}{180}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) |
| \(\chi_{407}(279,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{180}\right)\) | \(e\left(\frac{59}{90}\right)\) | \(e\left(\frac{71}{90}\right)\) | \(e\left(\frac{139}{180}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{9}\right)\) |
| \(\chi_{407}(291,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{180}\right)\) | \(e\left(\frac{73}{90}\right)\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{143}{180}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) |