from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1444, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([0,56]))
chi.galois_orbit()
[g,chi] = znchar(Mod(45,1444))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1444\) | |
| 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1444}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) |
| \(\chi_{1444}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) |
| \(\chi_{1444}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) |
| \(\chi_{1444}(125,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) |
| \(\chi_{1444}(197,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) |
| \(\chi_{1444}(201,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) |
| \(\chi_{1444}(273,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) |
| \(\chi_{1444}(277,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) |
| \(\chi_{1444}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) |
| \(\chi_{1444}(353,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{16}{57}\right)\) |
| \(\chi_{1444}(425,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{32}{57}\right)\) |
| \(\chi_{1444}(501,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) |
| \(\chi_{1444}(505,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) |
| \(\chi_{1444}(577,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) |
| \(\chi_{1444}(581,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) |
| \(\chi_{1444}(657,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) |
| \(\chi_{1444}(729,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) |
| \(\chi_{1444}(733,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) |
| \(\chi_{1444}(805,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) |
| \(\chi_{1444}(809,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) |
| \(\chi_{1444}(881,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) |
| \(\chi_{1444}(885,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) |
| \(\chi_{1444}(957,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) |
| \(\chi_{1444}(961,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) |
| \(\chi_{1444}(1033,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) |
| \(\chi_{1444}(1037,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) |
| \(\chi_{1444}(1109,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) |
| \(\chi_{1444}(1113,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) |
| \(\chi_{1444}(1185,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{14}{57}\right)\) |
| \(\chi_{1444}(1189,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{49}{57}\right)\) |
| \(\chi_{1444}(1261,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) |