from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(571, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([104]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,571))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(571\) | |
| Conductor: | \(571\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(57\) | 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: | $\Q(\zeta_{57})$ |
| Fixed field: | Number field defined by a degree 57 polynomial |
First 31 of 36 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{571}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) |
| \(\chi_{571}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) |
| \(\chi_{571}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) |
| \(\chi_{571}(82,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{49}{57}\right)\) |
| \(\chi_{571}(124,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) |
| \(\chi_{571}(143,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) |
| \(\chi_{571}(150,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) |
| \(\chi_{571}(220,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) |
| \(\chi_{571}(231,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) |
| \(\chi_{571}(236,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) |
| \(\chi_{571}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{55}{57}\right)\) |
| \(\chi_{571}(258,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) |
| \(\chi_{571}(270,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) |
| \(\chi_{571}(285,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) |
| \(\chi_{571}(309,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) |
| \(\chi_{571}(328,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) |
| \(\chi_{571}(339,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) |
| \(\chi_{571}(362,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) |
| \(\chi_{571}(373,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) |
| \(\chi_{571}(376,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) |
| \(\chi_{571}(383,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) |
| \(\chi_{571}(396,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) |
| \(\chi_{571}(418,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) |
| \(\chi_{571}(436,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) |
| \(\chi_{571}(442,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) |
| \(\chi_{571}(443,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) |
| \(\chi_{571}(453,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) |
| \(\chi_{571}(464,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) |
| \(\chi_{571}(486,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) |
| \(\chi_{571}(496,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) |
| \(\chi_{571}(509,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{32}{57}\right)\) |