from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4014, base_ring=CyclotomicField(222))
M = H._module
chi = DirichletCharacter(H, M([74,82]))
chi.galois_orbit()
[g,chi] = znchar(Mod(31,4014))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4014\) | |
| Conductor: | \(2007\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(111\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2007.z | 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: | $\Q(\zeta_{111})$ |
| Fixed field: | Number field defined by a degree 111 polynomial (not computed) |
First 31 of 72 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4014}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{37}\right)\) | \(e\left(\frac{100}{111}\right)\) | \(e\left(\frac{95}{111}\right)\) | \(e\left(\frac{107}{111}\right)\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{59}{111}\right)\) | \(e\left(\frac{97}{111}\right)\) | \(e\left(\frac{3}{37}\right)\) | \(e\left(\frac{68}{111}\right)\) | \(e\left(\frac{106}{111}\right)\) |
| \(\chi_{4014}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{37}\right)\) | \(e\left(\frac{85}{111}\right)\) | \(e\left(\frac{53}{111}\right)\) | \(e\left(\frac{41}{111}\right)\) | \(e\left(\frac{30}{37}\right)\) | \(e\left(\frac{89}{111}\right)\) | \(e\left(\frac{88}{111}\right)\) | \(e\left(\frac{34}{37}\right)\) | \(e\left(\frac{80}{111}\right)\) | \(e\left(\frac{79}{111}\right)\) |
| \(\chi_{4014}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{37}\right)\) | \(e\left(\frac{52}{111}\right)\) | \(e\left(\frac{5}{111}\right)\) | \(e\left(\frac{29}{111}\right)\) | \(e\left(\frac{14}{37}\right)\) | \(e\left(\frac{44}{111}\right)\) | \(e\left(\frac{46}{111}\right)\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{62}{111}\right)\) | \(e\left(\frac{64}{111}\right)\) |
| \(\chi_{4014}(175,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{37}\right)\) | \(e\left(\frac{40}{111}\right)\) | \(e\left(\frac{38}{111}\right)\) | \(e\left(\frac{65}{111}\right)\) | \(e\left(\frac{25}{37}\right)\) | \(e\left(\frac{68}{111}\right)\) | \(e\left(\frac{61}{111}\right)\) | \(e\left(\frac{16}{37}\right)\) | \(e\left(\frac{5}{111}\right)\) | \(e\left(\frac{109}{111}\right)\) |
| \(\chi_{4014}(211,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{37}\right)\) | \(e\left(\frac{91}{111}\right)\) | \(e\left(\frac{92}{111}\right)\) | \(e\left(\frac{23}{111}\right)\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{77}{111}\right)\) | \(e\left(\frac{25}{111}\right)\) | \(e\left(\frac{29}{37}\right)\) | \(e\left(\frac{53}{111}\right)\) | \(e\left(\frac{1}{111}\right)\) |
| \(\chi_{4014}(241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{37}\right)\) | \(e\left(\frac{92}{111}\right)\) | \(e\left(\frac{43}{111}\right)\) | \(e\left(\frac{94}{111}\right)\) | \(e\left(\frac{2}{37}\right)\) | \(e\left(\frac{1}{111}\right)\) | \(e\left(\frac{107}{111}\right)\) | \(e\left(\frac{22}{37}\right)\) | \(e\left(\frac{67}{111}\right)\) | \(e\left(\frac{62}{111}\right)\) |
| \(\chi_{4014}(259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{37}\right)\) | \(e\left(\frac{11}{111}\right)\) | \(e\left(\frac{16}{111}\right)\) | \(e\left(\frac{4}{111}\right)\) | \(e\left(\frac{30}{37}\right)\) | \(e\left(\frac{52}{111}\right)\) | \(e\left(\frac{14}{111}\right)\) | \(e\left(\frac{34}{37}\right)\) | \(e\left(\frac{43}{111}\right)\) | \(e\left(\frac{5}{111}\right)\) |
| \(\chi_{4014}(295,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{37}\right)\) | \(e\left(\frac{41}{111}\right)\) | \(e\left(\frac{100}{111}\right)\) | \(e\left(\frac{25}{111}\right)\) | \(e\left(\frac{21}{37}\right)\) | \(e\left(\frac{103}{111}\right)\) | \(e\left(\frac{32}{111}\right)\) | \(e\left(\frac{9}{37}\right)\) | \(e\left(\frac{19}{111}\right)\) | \(e\left(\frac{59}{111}\right)\) |
| \(\chi_{4014}(301,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{67}{111}\right)\) | \(e\left(\frac{47}{111}\right)\) | \(e\left(\frac{95}{111}\right)\) | \(e\left(\frac{28}{37}\right)\) | \(e\left(\frac{14}{111}\right)\) | \(e\left(\frac{55}{111}\right)\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{50}{111}\right)\) | \(e\left(\frac{91}{111}\right)\) |
| \(\chi_{4014}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{37}\right)\) | \(e\left(\frac{53}{111}\right)\) | \(e\left(\frac{67}{111}\right)\) | \(e\left(\frac{100}{111}\right)\) | \(e\left(\frac{10}{37}\right)\) | \(e\left(\frac{79}{111}\right)\) | \(e\left(\frac{17}{111}\right)\) | \(e\left(\frac{36}{37}\right)\) | \(e\left(\frac{76}{111}\right)\) | \(e\left(\frac{14}{111}\right)\) |
| \(\chi_{4014}(367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{37}\right)\) | \(e\left(\frac{71}{111}\right)\) | \(e\left(\frac{73}{111}\right)\) | \(e\left(\frac{46}{111}\right)\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{43}{111}\right)\) | \(e\left(\frac{50}{111}\right)\) | \(e\left(\frac{21}{37}\right)\) | \(e\left(\frac{106}{111}\right)\) | \(e\left(\frac{2}{111}\right)\) |
| \(\chi_{4014}(475,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{37}\right)\) | \(e\left(\frac{83}{111}\right)\) | \(e\left(\frac{40}{111}\right)\) | \(e\left(\frac{10}{111}\right)\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{19}{111}\right)\) | \(e\left(\frac{35}{111}\right)\) | \(e\left(\frac{11}{37}\right)\) | \(e\left(\frac{52}{111}\right)\) | \(e\left(\frac{68}{111}\right)\) |
| \(\chi_{4014}(493,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{37}\right)\) | \(e\left(\frac{107}{111}\right)\) | \(e\left(\frac{85}{111}\right)\) | \(e\left(\frac{49}{111}\right)\) | \(e\left(\frac{16}{37}\right)\) | \(e\left(\frac{82}{111}\right)\) | \(e\left(\frac{5}{111}\right)\) | \(e\left(\frac{28}{37}\right)\) | \(e\left(\frac{55}{111}\right)\) | \(e\left(\frac{89}{111}\right)\) |
| \(\chi_{4014}(511,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{37}\right)\) | \(e\left(\frac{101}{111}\right)\) | \(e\left(\frac{46}{111}\right)\) | \(e\left(\frac{67}{111}\right)\) | \(e\left(\frac{3}{37}\right)\) | \(e\left(\frac{94}{111}\right)\) | \(e\left(\frac{68}{111}\right)\) | \(e\left(\frac{33}{37}\right)\) | \(e\left(\frac{82}{111}\right)\) | \(e\left(\frac{56}{111}\right)\) |
| \(\chi_{4014}(529,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{5}{111}\right)\) | \(e\left(\frac{88}{111}\right)\) | \(e\left(\frac{22}{111}\right)\) | \(e\left(\frac{17}{37}\right)\) | \(e\left(\frac{64}{111}\right)\) | \(e\left(\frac{77}{111}\right)\) | \(e\left(\frac{2}{37}\right)\) | \(e\left(\frac{70}{111}\right)\) | \(e\left(\frac{83}{111}\right)\) |
| \(\chi_{4014}(535,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{37}\right)\) | \(e\left(\frac{16}{111}\right)\) | \(e\left(\frac{104}{111}\right)\) | \(e\left(\frac{26}{111}\right)\) | \(e\left(\frac{10}{37}\right)\) | \(e\left(\frac{5}{111}\right)\) | \(e\left(\frac{91}{111}\right)\) | \(e\left(\frac{36}{37}\right)\) | \(e\left(\frac{2}{111}\right)\) | \(e\left(\frac{88}{111}\right)\) |
| \(\chi_{4014}(727,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{30}{37}\right)\) | \(e\left(\frac{2}{111}\right)\) | \(e\left(\frac{13}{111}\right)\) | \(e\left(\frac{31}{111}\right)\) | \(e\left(\frac{29}{37}\right)\) | \(e\left(\frac{70}{111}\right)\) | \(e\left(\frac{53}{111}\right)\) | \(e\left(\frac{23}{37}\right)\) | \(e\left(\frac{28}{111}\right)\) | \(e\left(\frac{11}{111}\right)\) |
| \(\chi_{4014}(763,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{37}\right)\) | \(e\left(\frac{26}{111}\right)\) | \(e\left(\frac{58}{111}\right)\) | \(e\left(\frac{70}{111}\right)\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{22}{111}\right)\) | \(e\left(\frac{23}{111}\right)\) | \(e\left(\frac{3}{37}\right)\) | \(e\left(\frac{31}{111}\right)\) | \(e\left(\frac{32}{111}\right)\) |
| \(\chi_{4014}(769,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{37}\right)\) | \(e\left(\frac{28}{111}\right)\) | \(e\left(\frac{71}{111}\right)\) | \(e\left(\frac{101}{111}\right)\) | \(e\left(\frac{36}{37}\right)\) | \(e\left(\frac{92}{111}\right)\) | \(e\left(\frac{76}{111}\right)\) | \(e\left(\frac{26}{37}\right)\) | \(e\left(\frac{59}{111}\right)\) | \(e\left(\frac{43}{111}\right)\) |
| \(\chi_{4014}(799,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{37}\right)\) | \(e\left(\frac{20}{111}\right)\) | \(e\left(\frac{19}{111}\right)\) | \(e\left(\frac{88}{111}\right)\) | \(e\left(\frac{31}{37}\right)\) | \(e\left(\frac{34}{111}\right)\) | \(e\left(\frac{86}{111}\right)\) | \(e\left(\frac{8}{37}\right)\) | \(e\left(\frac{58}{111}\right)\) | \(e\left(\frac{110}{111}\right)\) |
| \(\chi_{4014}(817,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{37}\right)\) | \(e\left(\frac{110}{111}\right)\) | \(e\left(\frac{49}{111}\right)\) | \(e\left(\frac{40}{111}\right)\) | \(e\left(\frac{4}{37}\right)\) | \(e\left(\frac{76}{111}\right)\) | \(e\left(\frac{29}{111}\right)\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{97}{111}\right)\) | \(e\left(\frac{50}{111}\right)\) |
| \(\chi_{4014}(835,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{35}{111}\right)\) | \(e\left(\frac{61}{111}\right)\) | \(e\left(\frac{43}{111}\right)\) | \(e\left(\frac{8}{37}\right)\) | \(e\left(\frac{4}{111}\right)\) | \(e\left(\frac{95}{111}\right)\) | \(e\left(\frac{14}{37}\right)\) | \(e\left(\frac{46}{111}\right)\) | \(e\left(\frac{26}{111}\right)\) |
| \(\chi_{4014}(841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{37}\right)\) | \(e\left(\frac{55}{111}\right)\) | \(e\left(\frac{80}{111}\right)\) | \(e\left(\frac{20}{111}\right)\) | \(e\left(\frac{2}{37}\right)\) | \(e\left(\frac{38}{111}\right)\) | \(e\left(\frac{70}{111}\right)\) | \(e\left(\frac{22}{37}\right)\) | \(e\left(\frac{104}{111}\right)\) | \(e\left(\frac{25}{111}\right)\) |
| \(\chi_{4014}(961,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{37}\right)\) | \(e\left(\frac{89}{111}\right)\) | \(e\left(\frac{79}{111}\right)\) | \(e\left(\frac{103}{111}\right)\) | \(e\left(\frac{14}{37}\right)\) | \(e\left(\frac{7}{111}\right)\) | \(e\left(\frac{83}{111}\right)\) | \(e\left(\frac{6}{37}\right)\) | \(e\left(\frac{25}{111}\right)\) | \(e\left(\frac{101}{111}\right)\) |
| \(\chi_{4014}(1105,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{37}\right)\) | \(e\left(\frac{14}{111}\right)\) | \(e\left(\frac{91}{111}\right)\) | \(e\left(\frac{106}{111}\right)\) | \(e\left(\frac{18}{37}\right)\) | \(e\left(\frac{46}{111}\right)\) | \(e\left(\frac{38}{111}\right)\) | \(e\left(\frac{13}{37}\right)\) | \(e\left(\frac{85}{111}\right)\) | \(e\left(\frac{77}{111}\right)\) |
| \(\chi_{4014}(1165,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{37}\right)\) | \(e\left(\frac{109}{111}\right)\) | \(e\left(\frac{98}{111}\right)\) | \(e\left(\frac{80}{111}\right)\) | \(e\left(\frac{8}{37}\right)\) | \(e\left(\frac{41}{111}\right)\) | \(e\left(\frac{58}{111}\right)\) | \(e\left(\frac{14}{37}\right)\) | \(e\left(\frac{83}{111}\right)\) | \(e\left(\frac{100}{111}\right)\) |
| \(\chi_{4014}(1201,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{37}\right)\) | \(e\left(\frac{25}{111}\right)\) | \(e\left(\frac{107}{111}\right)\) | \(e\left(\frac{110}{111}\right)\) | \(e\left(\frac{11}{37}\right)\) | \(e\left(\frac{98}{111}\right)\) | \(e\left(\frac{52}{111}\right)\) | \(e\left(\frac{10}{37}\right)\) | \(e\left(\frac{17}{111}\right)\) | \(e\left(\frac{82}{111}\right)\) |
| \(\chi_{4014}(1231,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{36}{37}\right)\) | \(e\left(\frac{32}{111}\right)\) | \(e\left(\frac{97}{111}\right)\) | \(e\left(\frac{52}{111}\right)\) | \(e\left(\frac{20}{37}\right)\) | \(e\left(\frac{10}{111}\right)\) | \(e\left(\frac{71}{111}\right)\) | \(e\left(\frac{35}{37}\right)\) | \(e\left(\frac{4}{111}\right)\) | \(e\left(\frac{65}{111}\right)\) |
| \(\chi_{4014}(1303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{37}\right)\) | \(e\left(\frac{56}{111}\right)\) | \(e\left(\frac{31}{111}\right)\) | \(e\left(\frac{91}{111}\right)\) | \(e\left(\frac{35}{37}\right)\) | \(e\left(\frac{73}{111}\right)\) | \(e\left(\frac{41}{111}\right)\) | \(e\left(\frac{15}{37}\right)\) | \(e\left(\frac{7}{111}\right)\) | \(e\left(\frac{86}{111}\right)\) |
| \(\chi_{4014}(1363,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{79}{111}\right)\) | \(e\left(\frac{14}{111}\right)\) | \(e\left(\frac{59}{111}\right)\) | \(e\left(\frac{17}{37}\right)\) | \(e\left(\frac{101}{111}\right)\) | \(e\left(\frac{40}{111}\right)\) | \(e\left(\frac{2}{37}\right)\) | \(e\left(\frac{107}{111}\right)\) | \(e\left(\frac{46}{111}\right)\) |
| \(\chi_{4014}(1375,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{37}\right)\) | \(e\left(\frac{50}{111}\right)\) | \(e\left(\frac{103}{111}\right)\) | \(e\left(\frac{109}{111}\right)\) | \(e\left(\frac{22}{37}\right)\) | \(e\left(\frac{85}{111}\right)\) | \(e\left(\frac{104}{111}\right)\) | \(e\left(\frac{20}{37}\right)\) | \(e\left(\frac{34}{111}\right)\) | \(e\left(\frac{53}{111}\right)\) |