from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2601, base_ring=CyclotomicField(102))
M = H._module
chi = DirichletCharacter(H, M([85,39]))
chi.galois_orbit()
[g,chi] = znchar(Mod(50,2601))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2601\) | |
| Conductor: | \(2601\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(102\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{51})$ |
| Fixed field: | Number field defined by a degree 102 polynomial (not computed) |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2601}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{102}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{61}{102}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{47}{51}\right)\) |
| \(\chi_{2601}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{102}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{71}{102}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{43}{51}\right)\) |
| \(\chi_{2601}(203,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{102}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{91}{102}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{35}{51}\right)\) |
| \(\chi_{2601}(254,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{102}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{101}{102}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{31}{51}\right)\) |
| \(\chi_{2601}(356,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{102}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{19}{102}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{23}{51}\right)\) |
| \(\chi_{2601}(407,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{102}\right)\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{29}{102}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{19}{51}\right)\) |
| \(\chi_{2601}(509,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{102}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{49}{102}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{11}{51}\right)\) |
| \(\chi_{2601}(560,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{102}\right)\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{59}{102}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{7}{51}\right)\) |
| \(\chi_{2601}(662,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{102}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{79}{102}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{50}{51}\right)\) |
| \(\chi_{2601}(713,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{102}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{89}{102}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{46}{51}\right)\) |
| \(\chi_{2601}(815,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{102}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{7}{102}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{38}{51}\right)\) |
| \(\chi_{2601}(968,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{102}\right)\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{37}{102}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{26}{51}\right)\) |
| \(\chi_{2601}(1019,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{102}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{47}{102}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{22}{51}\right)\) |
| \(\chi_{2601}(1121,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{102}\right)\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{67}{102}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{14}{51}\right)\) |
| \(\chi_{2601}(1172,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{102}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{77}{102}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{10}{51}\right)\) |
| \(\chi_{2601}(1274,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{102}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{97}{102}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{2}{51}\right)\) |
| \(\chi_{2601}(1325,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{102}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{5}{102}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{49}{51}\right)\) |
| \(\chi_{2601}(1427,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{102}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{25}{102}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{41}{51}\right)\) |
| \(\chi_{2601}(1478,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{102}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{35}{102}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{37}{51}\right)\) |
| \(\chi_{2601}(1580,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{102}\right)\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{55}{102}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{29}{51}\right)\) |
| \(\chi_{2601}(1631,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{102}\right)\) | \(e\left(\frac{38}{51}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{65}{102}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{25}{51}\right)\) |
| \(\chi_{2601}(1784,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{102}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{95}{102}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{38}{51}\right)\) | \(e\left(\frac{13}{51}\right)\) |
| \(\chi_{2601}(1886,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{102}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{13}{102}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{5}{51}\right)\) |
| \(\chi_{2601}(1937,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{102}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{23}{102}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{1}{51}\right)\) |
| \(\chi_{2601}(2039,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{102}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{43}{102}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{44}{51}\right)\) |
| \(\chi_{2601}(2090,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{102}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{38}{51}\right)\) | \(e\left(\frac{53}{102}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{40}{51}\right)\) |
| \(\chi_{2601}(2192,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{102}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{73}{102}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{32}{51}\right)\) |
| \(\chi_{2601}(2243,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{102}\right)\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{83}{102}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{38}{51}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{28}{51}\right)\) |
| \(\chi_{2601}(2345,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{102}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{1}{102}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{20}{51}\right)\) |
| \(\chi_{2601}(2396,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{102}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{11}{102}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{16}{51}\right)\) |
| \(\chi_{2601}(2498,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{102}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{31}{102}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{38}{51}\right)\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{8}{51}\right)\) |