from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1208, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([0,0,58]))
chi.galois_orbit()
[g,chi] = znchar(Mod(17,1208))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1208\) | |
| Conductor: | \(151\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(75\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 151.k | 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1208}(17,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{32}{75}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{75}\right)\) |
| \(\chi_{1208}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{16}{75}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{75}\right)\) |
| \(\chi_{1208}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{16}{75}\right)\) |
| \(\chi_{1208}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{11}{75}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{62}{75}\right)\) |
| \(\chi_{1208}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{37}{75}\right)\) |
| \(\chi_{1208}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{75}\right)\) |
| \(\chi_{1208}(145,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{74}{75}\right)\) |
| \(\chi_{1208}(161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{43}{75}\right)\) |
| \(\chi_{1208}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{68}{75}\right)\) |
| \(\chi_{1208}(185,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{16}{75}\right)\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{22}{75}\right)\) |
| \(\chi_{1208}(193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{75}\right)\) |
| \(\chi_{1208}(209,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{11}{75}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{41}{75}\right)\) |
| \(\chi_{1208}(225,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{16}{75}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{64}{75}\right)\) |
| \(\chi_{1208}(241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{31}{75}\right)\) |
| \(\chi_{1208}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{34}{75}\right)\) |
| \(\chi_{1208}(313,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{56}{75}\right)\) |
| \(\chi_{1208}(345,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{67}{75}\right)\) |
| \(\chi_{1208}(401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{11}{75}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{32}{75}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{44}{75}\right)\) |
| \(\chi_{1208}(441,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{16}{75}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{75}\right)\) |
| \(\chi_{1208}(489,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{73}{75}\right)\) |
| \(\chi_{1208}(553,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{75}\right)\) |
| \(\chi_{1208}(569,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{32}{75}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{16}{75}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{46}{75}\right)\) |
| \(\chi_{1208}(609,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{38}{75}\right)\) |
| \(\chi_{1208}(625,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{32}{75}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{75}\right)\) |
| \(\chi_{1208}(641,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{11}{75}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{59}{75}\right)\) |
| \(\chi_{1208}(649,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{26}{75}\right)\) |
| \(\chi_{1208}(673,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{29}{75}\right)\) |
| \(\chi_{1208}(777,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{61}{75}\right)\) |
| \(\chi_{1208}(817,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{52}{75}\right)\) |
| \(\chi_{1208}(937,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{68}{75}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{47}{75}\right)\) |
| \(\chi_{1208}(945,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{37}{75}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{28}{75}\right)\) |