from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8003, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([75,38]))
chi.galois_orbit()
[g,chi] = znchar(Mod(423,8003))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8003\) | |
| Conductor: | \(8003\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(150\) | 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: | $\Q(\zeta_{75})$ |
| Fixed field: | Number field defined by a degree 150 polynomial (not computed) |
First 31 of 40 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8003}(423,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{131}{150}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{61}{75}\right)\) |
| \(\chi_{8003}(635,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{61}{150}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{41}{75}\right)\) |
| \(\chi_{8003}(741,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{119}{150}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{64}{75}\right)\) |
| \(\chi_{8003}(794,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{89}{150}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{34}{75}\right)\) |
| \(\chi_{8003}(900,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{37}{150}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{16}{75}\right)\) | \(e\left(\frac{47}{75}\right)\) |
| \(\chi_{8003}(953,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{137}{150}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{22}{75}\right)\) |
| \(\chi_{8003}(1006,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{43}{150}\right)\) | \(e\left(\frac{32}{75}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{8}{75}\right)\) |
| \(\chi_{8003}(1112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{47}{150}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{7}{75}\right)\) |
| \(\chi_{8003}(1218,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{59}{150}\right)\) | \(e\left(\frac{16}{75}\right)\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{4}{75}\right)\) |
| \(\chi_{8003}(1324,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{23}{150}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{13}{75}\right)\) |
| \(\chi_{8003}(1377,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{109}{150}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{29}{75}\right)\) |
| \(\chi_{8003}(1695,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{11}{150}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{16}{75}\right)\) |
| \(\chi_{8003}(1854,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{41}{150}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{46}{75}\right)\) |
| \(\chi_{8003}(1907,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{49}{150}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{44}{75}\right)\) |
| \(\chi_{8003}(2066,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{150}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{17}{75}\right)\) |
| \(\chi_{8003}(2119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{19}{150}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{14}{75}\right)\) |
| \(\chi_{8003}(2172,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{133}{150}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{23}{75}\right)\) |
| \(\chi_{8003}(2437,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{150}\right)\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{56}{75}\right)\) |
| \(\chi_{8003}(2490,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{107}{150}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{67}{75}\right)\) |
| \(\chi_{8003}(2755,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{67}{150}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{11}{75}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{2}{75}\right)\) |
| \(\chi_{8003}(2808,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{53}{150}\right)\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{43}{75}\right)\) |
| \(\chi_{8003}(2914,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{150}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{53}{75}\right)\) |
| \(\chi_{8003}(3391,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{127}{150}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{62}{75}\right)\) |
| \(\chi_{8003}(3762,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{17}{150}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{52}{75}\right)\) |
| \(\chi_{8003}(3815,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{139}{150}\right)\) | \(e\left(\frac{11}{75}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{59}{75}\right)\) |
| \(\chi_{8003}(4239,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{103}{150}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{68}{75}\right)\) |
| \(\chi_{8003}(4610,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{29}{150}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{49}{75}\right)\) |
| \(\chi_{8003}(4769,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{73}{150}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{38}{75}\right)\) |
| \(\chi_{8003}(4875,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{71}{150}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{1}{75}\right)\) |
| \(\chi_{8003}(5458,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{143}{150}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{58}{75}\right)\) |
| \(\chi_{8003}(5723,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{91}{150}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{71}{75}\right)\) |