from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1125, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([100,6]))
chi.galois_orbit()
[g,chi] = znchar(Mod(16,1125))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1125\) | |
| Conductor: | \(1125\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(75\) | 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 75 polynomial |
First 31 of 40 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1125}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) |
| \(\chi_{1125}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) |
| \(\chi_{1125}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) |
| \(\chi_{1125}(106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) |
| \(\chi_{1125}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) |
| \(\chi_{1125}(166,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) |
| \(\chi_{1125}(196,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) |
| \(\chi_{1125}(211,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) |
| \(\chi_{1125}(241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) |
| \(\chi_{1125}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) |
| \(\chi_{1125}(286,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) |
| \(\chi_{1125}(331,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) |
| \(\chi_{1125}(346,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) |
| \(\chi_{1125}(391,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) |
| \(\chi_{1125}(421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) |
| \(\chi_{1125}(436,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) |
| \(\chi_{1125}(466,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) |
| \(\chi_{1125}(481,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{11}{75}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) |
| \(\chi_{1125}(511,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{75}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{32}{75}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) |
| \(\chi_{1125}(556,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) |
| \(\chi_{1125}(571,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{16}{75}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) |
| \(\chi_{1125}(616,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) |
| \(\chi_{1125}(646,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{16}{75}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) |
| \(\chi_{1125}(661,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{32}{75}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) |
| \(\chi_{1125}(691,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) |
| \(\chi_{1125}(706,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) |
| \(\chi_{1125}(736,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) |
| \(\chi_{1125}(781,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{75}\right)\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{11}{75}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) |
| \(\chi_{1125}(796,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{11}{75}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) |
| \(\chi_{1125}(841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{11}{75}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) |
| \(\chi_{1125}(871,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) |