from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1600, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,15,28]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,1600))
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: | 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 | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{1600}(3,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{43}{80}\right)\) |
\(\chi_{1600}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{49}{80}\right)\) |
\(\chi_{1600}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{47}{80}\right)\) |
\(\chi_{1600}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{51}{80}\right)\) |
\(\chi_{1600}(187,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{41}{80}\right)\) |
\(\chi_{1600}(267,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{77}{80}\right)\) |
\(\chi_{1600}(323,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{59}{80}\right)\) |
\(\chi_{1600}(347,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{33}{80}\right)\) |
\(\chi_{1600}(403,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{63}{80}\right)\) |
\(\chi_{1600}(427,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{69}{80}\right)\) |
\(\chi_{1600}(483,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{67}{80}\right)\) |
\(\chi_{1600}(563,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{71}{80}\right)\) |
\(\chi_{1600}(587,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{61}{80}\right)\) |
\(\chi_{1600}(667,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{17}{80}\right)\) |
\(\chi_{1600}(723,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{79}{80}\right)\) |
\(\chi_{1600}(747,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{53}{80}\right)\) |
\(\chi_{1600}(803,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{3}{80}\right)\) |
\(\chi_{1600}(827,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{9}{80}\right)\) |
\(\chi_{1600}(883,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{80}\right)\) |
\(\chi_{1600}(963,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{11}{80}\right)\) |
\(\chi_{1600}(987,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{1}{80}\right)\) |
\(\chi_{1600}(1067,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{37}{80}\right)\) |
\(\chi_{1600}(1123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{19}{80}\right)\) |
\(\chi_{1600}(1147,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{73}{80}\right)\) |
\(\chi_{1600}(1203,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{23}{80}\right)\) |
\(\chi_{1600}(1227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{29}{80}\right)\) |
\(\chi_{1600}(1283,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{27}{80}\right)\) |
\(\chi_{1600}(1363,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{31}{80}\right)\) |
\(\chi_{1600}(1387,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{21}{80}\right)\) |
\(\chi_{1600}(1467,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{57}{80}\right)\) |
\(\chi_{1600}(1523,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{39}{80}\right)\) |