from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(73, base_ring=CyclotomicField(72))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,73))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(73\) | |
Conductor: | \(73\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(72\) | 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: | $\Q(\zeta_{72})$ |
Fixed field: | Number field defined by a degree 72 polynomial |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{73}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{55}{72}\right)\) |
\(\chi_{73}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{72}\right)\) |
\(\chi_{73}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{72}\right)\) |
\(\chi_{73}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{23}{72}\right)\) |
\(\chi_{73}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{25}{72}\right)\) |
\(\chi_{73}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{71}{72}\right)\) |
\(\chi_{73}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{72}\right)\) |
\(\chi_{73}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{31}{72}\right)\) |
\(\chi_{73}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{53}{72}\right)\) |
\(\chi_{73}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{29}{72}\right)\) |
\(\chi_{73}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{43}{72}\right)\) |
\(\chi_{73}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{72}\right)\) |
\(\chi_{73}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{47}{72}\right)\) |
\(\chi_{73}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{25}{72}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{72}\right)\) |
\(\chi_{73}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{65}{72}\right)\) |
\(\chi_{73}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{17}{72}\right)\) |
\(\chi_{73}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{67}{72}\right)\) |
\(\chi_{73}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{49}{72}\right)\) |
\(\chi_{73}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{35}{72}\right)\) |
\(\chi_{73}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{61}{72}\right)\) |
\(\chi_{73}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{59}{72}\right)\) |
\(\chi_{73}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{41}{72}\right)\) |
\(\chi_{73}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{37}{72}\right)\) |
\(\chi_{73}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{72}\right)\) |