from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(103, base_ring=CyclotomicField(102))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,103))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(103\) | |
| Conductor: | \(103\) | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{103}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{1}{102}\right)\) | \(e\left(\frac{83}{102}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{61}{102}\right)\) |
| \(\chi_{103}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{83}{102}\right)\) | \(e\left(\frac{55}{102}\right)\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{65}{102}\right)\) |
| \(\chi_{103}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{61}{102}\right)\) | \(e\left(\frac{65}{102}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{49}{102}\right)\) |
| \(\chi_{103}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{25}{102}\right)\) | \(e\left(\frac{35}{102}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{97}{102}\right)\) |
| \(\chi_{103}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{89}{102}\right)\) | \(e\left(\frac{43}{102}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{23}{102}\right)\) |
| \(\chi_{103}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{43}{102}\right)\) | \(e\left(\frac{101}{102}\right)\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{73}{102}\right)\) |
| \(\chi_{103}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{5}{102}\right)\) | \(e\left(\frac{7}{102}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{101}{102}\right)\) |
| \(\chi_{103}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{38}{51}\right)\) | \(e\left(\frac{31}{102}\right)\) | \(e\left(\frac{23}{102}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{55}{102}\right)\) |
| \(\chi_{103}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{77}{102}\right)\) | \(e\left(\frac{67}{102}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{5}{102}\right)\) |
| \(\chi_{103}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{47}{102}\right)\) | \(e\left(\frac{25}{102}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{11}{102}\right)\) |
| \(\chi_{103}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{79}{102}\right)\) | \(e\left(\frac{29}{102}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{25}{102}\right)\) |
| \(\chi_{103}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{38}{51}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{11}{102}\right)\) | \(e\left(\frac{97}{102}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{59}{102}\right)\) |
| \(\chi_{103}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{7}{102}\right)\) | \(e\left(\frac{71}{102}\right)\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{19}{102}\right)\) |
| \(\chi_{103}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{97}{102}\right)\) | \(e\left(\frac{95}{102}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{1}{102}\right)\) |
| \(\chi_{103}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{59}{102}\right)\) | \(e\left(\frac{1}{102}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{29}{102}\right)\) |
| \(\chi_{103}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{101}{102}\right)\) | \(e\left(\frac{19}{102}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{41}{102}\right)\) |
| \(\chi_{103}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{73}{102}\right)\) | \(e\left(\frac{41}{102}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{67}{102}\right)\) |
| \(\chi_{103}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{13}{102}\right)\) | \(e\left(\frac{59}{102}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{79}{102}\right)\) |
| \(\chi_{103}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{49}{102}\right)\) | \(e\left(\frac{89}{102}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{31}{102}\right)\) |
| \(\chi_{103}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{67}{102}\right)\) | \(e\left(\frac{53}{102}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{7}{102}\right)\) |
| \(\chi_{103}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{35}{102}\right)\) | \(e\left(\frac{49}{102}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{95}{102}\right)\) |
| \(\chi_{103}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{41}{102}\right)\) | \(e\left(\frac{37}{102}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{53}{102}\right)\) |
| \(\chi_{103}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{65}{102}\right)\) | \(e\left(\frac{91}{102}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{89}{102}\right)\) |
| \(\chi_{103}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{53}{102}\right)\) | \(e\left(\frac{13}{102}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{71}{102}\right)\) |
| \(\chi_{103}(84,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{29}{102}\right)\) | \(e\left(\frac{61}{102}\right)\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{35}{102}\right)\) |
| \(\chi_{103}(85,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{71}{102}\right)\) | \(e\left(\frac{79}{102}\right)\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{47}{102}\right)\) |
| \(\chi_{103}(86,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{19}{102}\right)\) | \(e\left(\frac{47}{102}\right)\) | \(e\left(\frac{38}{51}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{37}{102}\right)\) |
| \(\chi_{103}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{23}{102}\right)\) | \(e\left(\frac{73}{102}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{77}{102}\right)\) |
| \(\chi_{103}(88,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{91}{102}\right)\) | \(e\left(\frac{5}{102}\right)\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{43}{102}\right)\) |
| \(\chi_{103}(96,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{55}{102}\right)\) | \(e\left(\frac{77}{102}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{91}{102}\right)\) |
| \(\chi_{103}(99,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{37}{102}\right)\) | \(e\left(\frac{11}{102}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{13}{102}\right)\) |