from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1728, base_ring=CyclotomicField(72))
M = H._module
chi = DirichletCharacter(H, M([36,45,64]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,1728))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1728\) | |
| Conductor: | \(864\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(72\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 864.bv | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| 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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1728}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{5}{18}\right)\) |
| \(\chi_{1728}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{1}{18}\right)\) |
| \(\chi_{1728}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{17}{18}\right)\) |
| \(\chi_{1728}(247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{13}{18}\right)\) |
| \(\chi_{1728}(295,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{25}{72}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{11}{18}\right)\) |
| \(\chi_{1728}(391,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{7}{18}\right)\) |
| \(\chi_{1728}(439,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{5}{18}\right)\) |
| \(\chi_{1728}(535,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{25}{72}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{1}{18}\right)\) |
| \(\chi_{1728}(583,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{17}{18}\right)\) |
| \(\chi_{1728}(679,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{13}{18}\right)\) |
| \(\chi_{1728}(727,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{11}{18}\right)\) |
| \(\chi_{1728}(823,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{7}{18}\right)\) |
| \(\chi_{1728}(871,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{5}{18}\right)\) |
| \(\chi_{1728}(967,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{1}{18}\right)\) |
| \(\chi_{1728}(1015,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{25}{72}\right)\) | \(e\left(\frac{17}{18}\right)\) |
| \(\chi_{1728}(1111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{13}{18}\right)\) |
| \(\chi_{1728}(1159,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{11}{18}\right)\) |
| \(\chi_{1728}(1255,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{72}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{7}{18}\right)\) |
| \(\chi_{1728}(1303,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{5}{18}\right)\) |
| \(\chi_{1728}(1399,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{1}{18}\right)\) |
| \(\chi_{1728}(1447,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{17}{18}\right)\) |
| \(\chi_{1728}(1543,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{13}{18}\right)\) |
| \(\chi_{1728}(1591,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{11}{18}\right)\) |
| \(\chi_{1728}(1687,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{7}{18}\right)\) |