from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3672, base_ring=CyclotomicField(72))
M = H._module
chi = DirichletCharacter(H, M([0,36,44,9]))
chi.galois_orbit()
[g,chi] = znchar(Mod(77,3672))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3672\) | |
| Conductor: | \(3672\) | 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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3672}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{25}{72}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3672}(365,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{72}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3672}(389,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3672}(461,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{25}{72}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3672}(797,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3672}(869,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3672}(893,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3672}(1181,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3672}(1301,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3672}(1589,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3672}(1613,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{25}{72}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3672}(1685,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3672}(2021,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{25}{72}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3672}(2093,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3672}(2117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3672}(2405,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3672}(2525,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3672}(2813,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3672}(2837,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3672}(2909,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3672}(3245,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3672}(3317,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{3672}(3341,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{3672}(3629,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{25}{72}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) |