from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(722, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([50]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,722))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(722\) | |
Conductor: | \(361\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(57\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 361.i | 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{722}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) |
\(\chi_{722}(11,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) |
\(\chi_{722}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) |
\(\chi_{722}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) |
\(\chi_{722}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) |
\(\chi_{722}(87,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) |
\(\chi_{722}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) |
\(\chi_{722}(125,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) |
\(\chi_{722}(159,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) |
\(\chi_{722}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) |
\(\chi_{722}(197,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) |
\(\chi_{722}(201,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) |
\(\chi_{722}(235,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) |
\(\chi_{722}(239,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) |
\(\chi_{722}(273,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) |
\(\chi_{722}(277,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) |
\(\chi_{722}(311,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) |
\(\chi_{722}(315,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) |
\(\chi_{722}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) |
\(\chi_{722}(353,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{16}{57}\right)\) |
\(\chi_{722}(387,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) |
\(\chi_{722}(391,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) |
\(\chi_{722}(425,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{32}{57}\right)\) |
\(\chi_{722}(463,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{14}{57}\right)\) |
\(\chi_{722}(467,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{49}{57}\right)\) |
\(\chi_{722}(501,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) |
\(\chi_{722}(505,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) |
\(\chi_{722}(539,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) |
\(\chi_{722}(543,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) |
\(\chi_{722}(577,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) |
\(\chi_{722}(581,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) |