from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1600, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,35,72]))
chi.galois_orbit()
[g,chi] = znchar(Mod(19,1600))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1600\) | |
| Conductor: | \(1600\) | 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: | odd | 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 | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1600}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{27}{80}\right)\) |
| \(\chi_{1600}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{61}{80}\right)\) |
| \(\chi_{1600}(139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{49}{80}\right)\) |
| \(\chi_{1600}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{3}{80}\right)\) |
| \(\chi_{1600}(219,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{37}{80}\right)\) |
| \(\chi_{1600}(259,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{71}{80}\right)\) |
| \(\chi_{1600}(339,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{59}{80}\right)\) |
| \(\chi_{1600}(379,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{13}{80}\right)\) |
| \(\chi_{1600}(419,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{47}{80}\right)\) |
| \(\chi_{1600}(459,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{80}\right)\) |
| \(\chi_{1600}(539,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{69}{80}\right)\) |
| \(\chi_{1600}(579,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{23}{80}\right)\) |
| \(\chi_{1600}(619,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{57}{80}\right)\) |
| \(\chi_{1600}(659,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{11}{80}\right)\) |
| \(\chi_{1600}(739,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{79}{80}\right)\) |
| \(\chi_{1600}(779,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{33}{80}\right)\) |
| \(\chi_{1600}(819,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{67}{80}\right)\) |
| \(\chi_{1600}(859,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{21}{80}\right)\) |
| \(\chi_{1600}(939,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{9}{80}\right)\) |
| \(\chi_{1600}(979,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{43}{80}\right)\) |
| \(\chi_{1600}(1019,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{77}{80}\right)\) |
| \(\chi_{1600}(1059,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{31}{80}\right)\) |
| \(\chi_{1600}(1139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{19}{80}\right)\) |
| \(\chi_{1600}(1179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{53}{80}\right)\) |
| \(\chi_{1600}(1219,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{7}{80}\right)\) |
| \(\chi_{1600}(1259,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{41}{80}\right)\) |
| \(\chi_{1600}(1339,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{29}{80}\right)\) |
| \(\chi_{1600}(1379,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{63}{80}\right)\) |
| \(\chi_{1600}(1419,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{17}{80}\right)\) |
| \(\chi_{1600}(1459,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{51}{80}\right)\) |
| \(\chi_{1600}(1539,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{39}{80}\right)\) |