from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1600, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,25,64]))
chi.galois_orbit()
[g,chi] = znchar(Mod(11,1600))
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 | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1600}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{9}{80}\right)\) |
| \(\chi_{1600}(91,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{61}{80}\right)\) |
| \(\chi_{1600}(131,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{47}{80}\right)\) |
| \(\chi_{1600}(171,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{33}{80}\right)\) |
| \(\chi_{1600}(211,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{19}{80}\right)\) |
| \(\chi_{1600}(291,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{71}{80}\right)\) |
| \(\chi_{1600}(331,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{57}{80}\right)\) |
| \(\chi_{1600}(371,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{43}{80}\right)\) |
| \(\chi_{1600}(411,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{29}{80}\right)\) |
| \(\chi_{1600}(491,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{1}{80}\right)\) |
| \(\chi_{1600}(531,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{67}{80}\right)\) |
| \(\chi_{1600}(571,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{53}{80}\right)\) |
| \(\chi_{1600}(611,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{39}{80}\right)\) |
| \(\chi_{1600}(691,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{11}{80}\right)\) |
| \(\chi_{1600}(731,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{77}{80}\right)\) |
| \(\chi_{1600}(771,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{63}{80}\right)\) |
| \(\chi_{1600}(811,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{49}{80}\right)\) |
| \(\chi_{1600}(891,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{21}{80}\right)\) |
| \(\chi_{1600}(931,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{7}{80}\right)\) |
| \(\chi_{1600}(971,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{73}{80}\right)\) |
| \(\chi_{1600}(1011,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{59}{80}\right)\) |
| \(\chi_{1600}(1091,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{31}{80}\right)\) |
| \(\chi_{1600}(1131,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{17}{80}\right)\) |
| \(\chi_{1600}(1171,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{3}{80}\right)\) |
| \(\chi_{1600}(1211,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{69}{80}\right)\) |
| \(\chi_{1600}(1291,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{41}{80}\right)\) |
| \(\chi_{1600}(1331,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{27}{80}\right)\) |
| \(\chi_{1600}(1371,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{13}{80}\right)\) |
| \(\chi_{1600}(1411,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{79}{80}\right)\) |
| \(\chi_{1600}(1491,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{51}{80}\right)\) |
| \(\chi_{1600}(1531,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{37}{80}\right)\) |