from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(187, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([72,75]))
chi.galois_orbit()
[g,chi] = znchar(Mod(6,187))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(187\) | |
| Conductor: | \(187\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(80\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{80})$ |
| Fixed field: | Number field defined by a degree 80 polynomial |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{187}(6,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) |
| \(\chi_{187}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) |
| \(\chi_{187}(24,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) |
| \(\chi_{187}(28,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) |
| \(\chi_{187}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) |
| \(\chi_{187}(39,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) |
| \(\chi_{187}(40,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) |
| \(\chi_{187}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) |
| \(\chi_{187}(46,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) |
| \(\chi_{187}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) |
| \(\chi_{187}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) |
| \(\chi_{187}(62,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) |
| \(\chi_{187}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) |
| \(\chi_{187}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) |
| \(\chi_{187}(74,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) |
| \(\chi_{187}(79,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) |
| \(\chi_{187}(90,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) |
| \(\chi_{187}(95,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) |
| \(\chi_{187}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) |
| \(\chi_{187}(105,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) |
| \(\chi_{187}(107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) |
| \(\chi_{187}(112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) |
| \(\chi_{187}(116,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) |
| \(\chi_{187}(129,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) |
| \(\chi_{187}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) |
| \(\chi_{187}(150,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) |
| \(\chi_{187}(156,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) |
| \(\chi_{187}(160,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) |
| \(\chi_{187}(167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) |
| \(\chi_{187}(173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) |
| \(\chi_{187}(182,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) |