from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4889, base_ring=CyclotomicField(94))
M = H._module
chi = DirichletCharacter(H, M([92]))
chi.galois_orbit()
[g,chi] = znchar(Mod(20,4889))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4889\) | |
| Conductor: | \(4889\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(47\) | 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: | $\Q(\zeta_{47})$ |
| Fixed field: | Number field defined by a degree 47 polynomial |
First 31 of 46 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4889}(20,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{47}\right)\) | \(e\left(\frac{46}{47}\right)\) | \(e\left(\frac{23}{47}\right)\) | \(e\left(\frac{34}{47}\right)\) | \(e\left(\frac{34}{47}\right)\) | \(e\left(\frac{4}{47}\right)\) | \(e\left(\frac{11}{47}\right)\) | \(e\left(\frac{45}{47}\right)\) | \(e\left(\frac{22}{47}\right)\) | \(e\left(\frac{5}{47}\right)\) |
| \(\chi_{4889}(342,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{47}\right)\) | \(e\left(\frac{32}{47}\right)\) | \(e\left(\frac{16}{47}\right)\) | \(e\left(\frac{40}{47}\right)\) | \(e\left(\frac{40}{47}\right)\) | \(e\left(\frac{13}{47}\right)\) | \(e\left(\frac{24}{47}\right)\) | \(e\left(\frac{17}{47}\right)\) | \(e\left(\frac{1}{47}\right)\) | \(e\left(\frac{28}{47}\right)\) |
| \(\chi_{4889}(400,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{47}\right)\) | \(e\left(\frac{45}{47}\right)\) | \(e\left(\frac{46}{47}\right)\) | \(e\left(\frac{21}{47}\right)\) | \(e\left(\frac{21}{47}\right)\) | \(e\left(\frac{8}{47}\right)\) | \(e\left(\frac{22}{47}\right)\) | \(e\left(\frac{43}{47}\right)\) | \(e\left(\frac{44}{47}\right)\) | \(e\left(\frac{10}{47}\right)\) |
| \(\chi_{4889}(506,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{47}\right)\) | \(e\left(\frac{33}{47}\right)\) | \(e\left(\frac{40}{47}\right)\) | \(e\left(\frac{6}{47}\right)\) | \(e\left(\frac{6}{47}\right)\) | \(e\left(\frac{9}{47}\right)\) | \(e\left(\frac{13}{47}\right)\) | \(e\left(\frac{19}{47}\right)\) | \(e\left(\frac{26}{47}\right)\) | \(e\left(\frac{23}{47}\right)\) |
| \(\chi_{4889}(605,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{47}\right)\) | \(e\left(\frac{5}{47}\right)\) | \(e\left(\frac{26}{47}\right)\) | \(e\left(\frac{18}{47}\right)\) | \(e\left(\frac{18}{47}\right)\) | \(e\left(\frac{27}{47}\right)\) | \(e\left(\frac{39}{47}\right)\) | \(e\left(\frac{10}{47}\right)\) | \(e\left(\frac{31}{47}\right)\) | \(e\left(\frac{22}{47}\right)\) |
| \(\chi_{4889}(779,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{30}{47}\right)\) | \(e\left(\frac{26}{47}\right)\) | \(e\left(\frac{13}{47}\right)\) | \(e\left(\frac{9}{47}\right)\) | \(e\left(\frac{9}{47}\right)\) | \(e\left(\frac{37}{47}\right)\) | \(e\left(\frac{43}{47}\right)\) | \(e\left(\frac{5}{47}\right)\) | \(e\left(\frac{39}{47}\right)\) | \(e\left(\frac{11}{47}\right)\) |
| \(\chi_{4889}(913,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{47}\right)\) | \(e\left(\frac{25}{47}\right)\) | \(e\left(\frac{36}{47}\right)\) | \(e\left(\frac{43}{47}\right)\) | \(e\left(\frac{43}{47}\right)\) | \(e\left(\frac{41}{47}\right)\) | \(e\left(\frac{7}{47}\right)\) | \(e\left(\frac{3}{47}\right)\) | \(e\left(\frac{14}{47}\right)\) | \(e\left(\frac{16}{47}\right)\) |
| \(\chi_{4889}(1132,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{47}\right)\) | \(e\left(\frac{40}{47}\right)\) | \(e\left(\frac{20}{47}\right)\) | \(e\left(\frac{3}{47}\right)\) | \(e\left(\frac{3}{47}\right)\) | \(e\left(\frac{28}{47}\right)\) | \(e\left(\frac{30}{47}\right)\) | \(e\left(\frac{33}{47}\right)\) | \(e\left(\frac{13}{47}\right)\) | \(e\left(\frac{35}{47}\right)\) |
| \(\chi_{4889}(1401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{47}\right)\) | \(e\left(\frac{14}{47}\right)\) | \(e\left(\frac{7}{47}\right)\) | \(e\left(\frac{41}{47}\right)\) | \(e\left(\frac{41}{47}\right)\) | \(e\left(\frac{38}{47}\right)\) | \(e\left(\frac{34}{47}\right)\) | \(e\left(\frac{28}{47}\right)\) | \(e\left(\frac{21}{47}\right)\) | \(e\left(\frac{24}{47}\right)\) |
| \(\chi_{4889}(1492,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{47}\right)\) | \(e\left(\frac{34}{47}\right)\) | \(e\left(\frac{17}{47}\right)\) | \(e\left(\frac{19}{47}\right)\) | \(e\left(\frac{19}{47}\right)\) | \(e\left(\frac{5}{47}\right)\) | \(e\left(\frac{2}{47}\right)\) | \(e\left(\frac{21}{47}\right)\) | \(e\left(\frac{4}{47}\right)\) | \(e\left(\frac{18}{47}\right)\) |
| \(\chi_{4889}(1569,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{47}\right)\) | \(e\left(\frac{21}{47}\right)\) | \(e\left(\frac{34}{47}\right)\) | \(e\left(\frac{38}{47}\right)\) | \(e\left(\frac{38}{47}\right)\) | \(e\left(\frac{10}{47}\right)\) | \(e\left(\frac{4}{47}\right)\) | \(e\left(\frac{42}{47}\right)\) | \(e\left(\frac{8}{47}\right)\) | \(e\left(\frac{36}{47}\right)\) |
| \(\chi_{4889}(1572,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{47}\right)\) | \(e\left(\frac{37}{47}\right)\) | \(e\left(\frac{42}{47}\right)\) | \(e\left(\frac{11}{47}\right)\) | \(e\left(\frac{11}{47}\right)\) | \(e\left(\frac{40}{47}\right)\) | \(e\left(\frac{16}{47}\right)\) | \(e\left(\frac{27}{47}\right)\) | \(e\left(\frac{32}{47}\right)\) | \(e\left(\frac{3}{47}\right)\) |
| \(\chi_{4889}(1667,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{47}\right)\) | \(e\left(\frac{9}{47}\right)\) | \(e\left(\frac{28}{47}\right)\) | \(e\left(\frac{23}{47}\right)\) | \(e\left(\frac{23}{47}\right)\) | \(e\left(\frac{11}{47}\right)\) | \(e\left(\frac{42}{47}\right)\) | \(e\left(\frac{18}{47}\right)\) | \(e\left(\frac{37}{47}\right)\) | \(e\left(\frac{2}{47}\right)\) |
| \(\chi_{4889}(1808,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{47}\right)\) | \(e\left(\frac{19}{47}\right)\) | \(e\left(\frac{33}{47}\right)\) | \(e\left(\frac{12}{47}\right)\) | \(e\left(\frac{12}{47}\right)\) | \(e\left(\frac{18}{47}\right)\) | \(e\left(\frac{26}{47}\right)\) | \(e\left(\frac{38}{47}\right)\) | \(e\left(\frac{5}{47}\right)\) | \(e\left(\frac{46}{47}\right)\) |
| \(\chi_{4889}(1896,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{47}\right)\) | \(e\left(\frac{7}{47}\right)\) | \(e\left(\frac{27}{47}\right)\) | \(e\left(\frac{44}{47}\right)\) | \(e\left(\frac{44}{47}\right)\) | \(e\left(\frac{19}{47}\right)\) | \(e\left(\frac{17}{47}\right)\) | \(e\left(\frac{14}{47}\right)\) | \(e\left(\frac{34}{47}\right)\) | \(e\left(\frac{12}{47}\right)\) |
| \(\chi_{4889}(1937,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{47}\right)\) | \(e\left(\frac{18}{47}\right)\) | \(e\left(\frac{9}{47}\right)\) | \(e\left(\frac{46}{47}\right)\) | \(e\left(\frac{46}{47}\right)\) | \(e\left(\frac{22}{47}\right)\) | \(e\left(\frac{37}{47}\right)\) | \(e\left(\frac{36}{47}\right)\) | \(e\left(\frac{27}{47}\right)\) | \(e\left(\frac{4}{47}\right)\) |
| \(\chi_{4889}(1951,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{47}\right)\) | \(e\left(\frac{31}{47}\right)\) | \(e\left(\frac{39}{47}\right)\) | \(e\left(\frac{27}{47}\right)\) | \(e\left(\frac{27}{47}\right)\) | \(e\left(\frac{17}{47}\right)\) | \(e\left(\frac{35}{47}\right)\) | \(e\left(\frac{15}{47}\right)\) | \(e\left(\frac{23}{47}\right)\) | \(e\left(\frac{33}{47}\right)\) |
| \(\chi_{4889}(2046,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{47}\right)\) | \(e\left(\frac{20}{47}\right)\) | \(e\left(\frac{10}{47}\right)\) | \(e\left(\frac{25}{47}\right)\) | \(e\left(\frac{25}{47}\right)\) | \(e\left(\frac{14}{47}\right)\) | \(e\left(\frac{15}{47}\right)\) | \(e\left(\frac{40}{47}\right)\) | \(e\left(\frac{30}{47}\right)\) | \(e\left(\frac{41}{47}\right)\) |
| \(\chi_{4889}(2106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{47}\right)\) | \(e\left(\frac{36}{47}\right)\) | \(e\left(\frac{18}{47}\right)\) | \(e\left(\frac{45}{47}\right)\) | \(e\left(\frac{45}{47}\right)\) | \(e\left(\frac{44}{47}\right)\) | \(e\left(\frac{27}{47}\right)\) | \(e\left(\frac{25}{47}\right)\) | \(e\left(\frac{7}{47}\right)\) | \(e\left(\frac{8}{47}\right)\) |
| \(\chi_{4889}(2239,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{42}{47}\right)\) | \(e\left(\frac{27}{47}\right)\) | \(e\left(\frac{37}{47}\right)\) | \(e\left(\frac{22}{47}\right)\) | \(e\left(\frac{22}{47}\right)\) | \(e\left(\frac{33}{47}\right)\) | \(e\left(\frac{32}{47}\right)\) | \(e\left(\frac{7}{47}\right)\) | \(e\left(\frac{17}{47}\right)\) | \(e\left(\frac{6}{47}\right)\) |
| \(\chi_{4889}(2312,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{47}\right)\) | \(e\left(\frac{28}{47}\right)\) | \(e\left(\frac{14}{47}\right)\) | \(e\left(\frac{35}{47}\right)\) | \(e\left(\frac{35}{47}\right)\) | \(e\left(\frac{29}{47}\right)\) | \(e\left(\frac{21}{47}\right)\) | \(e\left(\frac{9}{47}\right)\) | \(e\left(\frac{42}{47}\right)\) | \(e\left(\frac{1}{47}\right)\) |
| \(\chi_{4889}(2322,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{47}\right)\) | \(e\left(\frac{4}{47}\right)\) | \(e\left(\frac{2}{47}\right)\) | \(e\left(\frac{5}{47}\right)\) | \(e\left(\frac{5}{47}\right)\) | \(e\left(\frac{31}{47}\right)\) | \(e\left(\frac{3}{47}\right)\) | \(e\left(\frac{8}{47}\right)\) | \(e\left(\frac{6}{47}\right)\) | \(e\left(\frac{27}{47}\right)\) |
| \(\chi_{4889}(2338,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{47}\right)\) | \(e\left(\frac{16}{47}\right)\) | \(e\left(\frac{8}{47}\right)\) | \(e\left(\frac{20}{47}\right)\) | \(e\left(\frac{20}{47}\right)\) | \(e\left(\frac{30}{47}\right)\) | \(e\left(\frac{12}{47}\right)\) | \(e\left(\frac{32}{47}\right)\) | \(e\left(\frac{24}{47}\right)\) | \(e\left(\frac{14}{47}\right)\) |
| \(\chi_{4889}(2412,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{47}\right)\) | \(e\left(\frac{11}{47}\right)\) | \(e\left(\frac{29}{47}\right)\) | \(e\left(\frac{2}{47}\right)\) | \(e\left(\frac{2}{47}\right)\) | \(e\left(\frac{3}{47}\right)\) | \(e\left(\frac{20}{47}\right)\) | \(e\left(\frac{22}{47}\right)\) | \(e\left(\frac{40}{47}\right)\) | \(e\left(\frac{39}{47}\right)\) |
| \(\chi_{4889}(2439,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{36}{47}\right)\) | \(e\left(\frac{3}{47}\right)\) | \(e\left(\frac{25}{47}\right)\) | \(e\left(\frac{39}{47}\right)\) | \(e\left(\frac{39}{47}\right)\) | \(e\left(\frac{35}{47}\right)\) | \(e\left(\frac{14}{47}\right)\) | \(e\left(\frac{6}{47}\right)\) | \(e\left(\frac{28}{47}\right)\) | \(e\left(\frac{32}{47}\right)\) |
| \(\chi_{4889}(2594,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{47}\right)\) | \(e\left(\frac{42}{47}\right)\) | \(e\left(\frac{21}{47}\right)\) | \(e\left(\frac{29}{47}\right)\) | \(e\left(\frac{29}{47}\right)\) | \(e\left(\frac{20}{47}\right)\) | \(e\left(\frac{8}{47}\right)\) | \(e\left(\frac{37}{47}\right)\) | \(e\left(\frac{16}{47}\right)\) | \(e\left(\frac{25}{47}\right)\) |
| \(\chi_{4889}(2689,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{47}\right)\) | \(e\left(\frac{1}{47}\right)\) | \(e\left(\frac{24}{47}\right)\) | \(e\left(\frac{13}{47}\right)\) | \(e\left(\frac{13}{47}\right)\) | \(e\left(\frac{43}{47}\right)\) | \(e\left(\frac{36}{47}\right)\) | \(e\left(\frac{2}{47}\right)\) | \(e\left(\frac{25}{47}\right)\) | \(e\left(\frac{42}{47}\right)\) |
| \(\chi_{4889}(2759,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{47}\right)\) | \(e\left(\frac{15}{47}\right)\) | \(e\left(\frac{31}{47}\right)\) | \(e\left(\frac{7}{47}\right)\) | \(e\left(\frac{7}{47}\right)\) | \(e\left(\frac{34}{47}\right)\) | \(e\left(\frac{23}{47}\right)\) | \(e\left(\frac{30}{47}\right)\) | \(e\left(\frac{46}{47}\right)\) | \(e\left(\frac{19}{47}\right)\) |
| \(\chi_{4889}(2990,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{47}\right)\) | \(e\left(\frac{41}{47}\right)\) | \(e\left(\frac{44}{47}\right)\) | \(e\left(\frac{16}{47}\right)\) | \(e\left(\frac{16}{47}\right)\) | \(e\left(\frac{24}{47}\right)\) | \(e\left(\frac{19}{47}\right)\) | \(e\left(\frac{35}{47}\right)\) | \(e\left(\frac{38}{47}\right)\) | \(e\left(\frac{30}{47}\right)\) |
| \(\chi_{4889}(3008,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{47}\right)\) | \(e\left(\frac{35}{47}\right)\) | \(e\left(\frac{41}{47}\right)\) | \(e\left(\frac{32}{47}\right)\) | \(e\left(\frac{32}{47}\right)\) | \(e\left(\frac{1}{47}\right)\) | \(e\left(\frac{38}{47}\right)\) | \(e\left(\frac{23}{47}\right)\) | \(e\left(\frac{29}{47}\right)\) | \(e\left(\frac{13}{47}\right)\) |
| \(\chi_{4889}(3012,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{47}\right)\) | \(e\left(\frac{38}{47}\right)\) | \(e\left(\frac{19}{47}\right)\) | \(e\left(\frac{24}{47}\right)\) | \(e\left(\frac{24}{47}\right)\) | \(e\left(\frac{36}{47}\right)\) | \(e\left(\frac{5}{47}\right)\) | \(e\left(\frac{29}{47}\right)\) | \(e\left(\frac{10}{47}\right)\) | \(e\left(\frac{45}{47}\right)\) |