from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(517, base_ring=CyclotomicField(230))
M = H._module
chi = DirichletCharacter(H, M([184,100]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,517))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(517\) | |
| Conductor: | \(517\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(115\) | 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: | $\Q(\zeta_{115})$ |
| Fixed field: | Number field defined by a degree 115 polynomial (not computed) |
First 31 of 88 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{517}(3,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{72}{115}\right)\) | \(e\left(\frac{11}{115}\right)\) | \(e\left(\frac{29}{115}\right)\) | \(e\left(\frac{73}{115}\right)\) | \(e\left(\frac{83}{115}\right)\) | \(e\left(\frac{59}{115}\right)\) | \(e\left(\frac{101}{115}\right)\) | \(e\left(\frac{22}{115}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) |
| \(\chi_{517}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{115}\right)\) | \(e\left(\frac{29}{115}\right)\) | \(e\left(\frac{66}{115}\right)\) | \(e\left(\frac{67}{115}\right)\) | \(e\left(\frac{62}{115}\right)\) | \(e\left(\frac{51}{115}\right)\) | \(e\left(\frac{99}{115}\right)\) | \(e\left(\frac{58}{115}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) |
| \(\chi_{517}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{115}\right)\) | \(e\left(\frac{22}{115}\right)\) | \(e\left(\frac{58}{115}\right)\) | \(e\left(\frac{31}{115}\right)\) | \(e\left(\frac{51}{115}\right)\) | \(e\left(\frac{3}{115}\right)\) | \(e\left(\frac{87}{115}\right)\) | \(e\left(\frac{44}{115}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) |
| \(\chi_{517}(14,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{42}{115}\right)\) | \(e\left(\frac{16}{115}\right)\) | \(e\left(\frac{84}{115}\right)\) | \(e\left(\frac{33}{115}\right)\) | \(e\left(\frac{58}{115}\right)\) | \(e\left(\frac{44}{115}\right)\) | \(e\left(\frac{11}{115}\right)\) | \(e\left(\frac{32}{115}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) |
| \(\chi_{517}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{66}{115}\right)\) | \(e\left(\frac{58}{115}\right)\) | \(e\left(\frac{17}{115}\right)\) | \(e\left(\frac{19}{115}\right)\) | \(e\left(\frac{9}{115}\right)\) | \(e\left(\frac{102}{115}\right)\) | \(e\left(\frac{83}{115}\right)\) | \(e\left(\frac{1}{115}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) |
| \(\chi_{517}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{115}\right)\) | \(e\left(\frac{31}{115}\right)\) | \(e\left(\frac{19}{115}\right)\) | \(e\left(\frac{28}{115}\right)\) | \(e\left(\frac{98}{115}\right)\) | \(e\left(\frac{114}{115}\right)\) | \(e\left(\frac{86}{115}\right)\) | \(e\left(\frac{62}{115}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) |
| \(\chi_{517}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{115}\right)\) | \(e\left(\frac{33}{115}\right)\) | \(e\left(\frac{87}{115}\right)\) | \(e\left(\frac{104}{115}\right)\) | \(e\left(\frac{19}{115}\right)\) | \(e\left(\frac{62}{115}\right)\) | \(e\left(\frac{73}{115}\right)\) | \(e\left(\frac{66}{115}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) |
| \(\chi_{517}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{115}\right)\) | \(e\left(\frac{51}{115}\right)\) | \(e\left(\frac{9}{115}\right)\) | \(e\left(\frac{98}{115}\right)\) | \(e\left(\frac{113}{115}\right)\) | \(e\left(\frac{54}{115}\right)\) | \(e\left(\frac{71}{115}\right)\) | \(e\left(\frac{102}{115}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) |
| \(\chi_{517}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{115}\right)\) | \(e\left(\frac{99}{115}\right)\) | \(e\left(\frac{31}{115}\right)\) | \(e\left(\frac{82}{115}\right)\) | \(e\left(\frac{57}{115}\right)\) | \(e\left(\frac{71}{115}\right)\) | \(e\left(\frac{104}{115}\right)\) | \(e\left(\frac{83}{115}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) |
| \(\chi_{517}(42,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{114}{115}\right)\) | \(e\left(\frac{27}{115}\right)\) | \(e\left(\frac{113}{115}\right)\) | \(e\left(\frac{106}{115}\right)\) | \(e\left(\frac{26}{115}\right)\) | \(e\left(\frac{103}{115}\right)\) | \(e\left(\frac{112}{115}\right)\) | \(e\left(\frac{54}{115}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) |
| \(\chi_{517}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{115}\right)\) | \(e\left(\frac{3}{115}\right)\) | \(e\left(\frac{102}{115}\right)\) | \(e\left(\frac{114}{115}\right)\) | \(e\left(\frac{54}{115}\right)\) | \(e\left(\frac{37}{115}\right)\) | \(e\left(\frac{38}{115}\right)\) | \(e\left(\frac{6}{115}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) |
| \(\chi_{517}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{54}{115}\right)\) | \(e\left(\frac{37}{115}\right)\) | \(e\left(\frac{108}{115}\right)\) | \(e\left(\frac{26}{115}\right)\) | \(e\left(\frac{91}{115}\right)\) | \(e\left(\frac{73}{115}\right)\) | \(e\left(\frac{47}{115}\right)\) | \(e\left(\frac{74}{115}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) |
| \(\chi_{517}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{115}\right)\) | \(e\left(\frac{109}{115}\right)\) | \(e\left(\frac{26}{115}\right)\) | \(e\left(\frac{2}{115}\right)\) | \(e\left(\frac{7}{115}\right)\) | \(e\left(\frac{41}{115}\right)\) | \(e\left(\frac{39}{115}\right)\) | \(e\left(\frac{103}{115}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) |
| \(\chi_{517}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{115}\right)\) | \(e\left(\frac{87}{115}\right)\) | \(e\left(\frac{83}{115}\right)\) | \(e\left(\frac{86}{115}\right)\) | \(e\left(\frac{71}{115}\right)\) | \(e\left(\frac{38}{115}\right)\) | \(e\left(\frac{67}{115}\right)\) | \(e\left(\frac{59}{115}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) |
| \(\chi_{517}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{115}\right)\) | \(e\left(\frac{43}{115}\right)\) | \(e\left(\frac{82}{115}\right)\) | \(e\left(\frac{24}{115}\right)\) | \(e\left(\frac{84}{115}\right)\) | \(e\left(\frac{32}{115}\right)\) | \(e\left(\frac{8}{115}\right)\) | \(e\left(\frac{86}{115}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) |
| \(\chi_{517}(75,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{115}\right)\) | \(e\left(\frac{42}{115}\right)\) | \(e\left(\frac{48}{115}\right)\) | \(e\left(\frac{101}{115}\right)\) | \(e\left(\frac{66}{115}\right)\) | \(e\left(\frac{58}{115}\right)\) | \(e\left(\frac{72}{115}\right)\) | \(e\left(\frac{84}{115}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) |
| \(\chi_{517}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{115}\right)\) | \(e\left(\frac{44}{115}\right)\) | \(e\left(\frac{1}{115}\right)\) | \(e\left(\frac{62}{115}\right)\) | \(e\left(\frac{102}{115}\right)\) | \(e\left(\frac{6}{115}\right)\) | \(e\left(\frac{59}{115}\right)\) | \(e\left(\frac{88}{115}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) |
| \(\chi_{517}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{115}\right)\) | \(e\left(\frac{57}{115}\right)\) | \(e\left(\frac{98}{115}\right)\) | \(e\left(\frac{96}{115}\right)\) | \(e\left(\frac{106}{115}\right)\) | \(e\left(\frac{13}{115}\right)\) | \(e\left(\frac{32}{115}\right)\) | \(e\left(\frac{114}{115}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) |
| \(\chi_{517}(102,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{115}\right)\) | \(e\left(\frac{101}{115}\right)\) | \(e\left(\frac{99}{115}\right)\) | \(e\left(\frac{43}{115}\right)\) | \(e\left(\frac{93}{115}\right)\) | \(e\left(\frac{19}{115}\right)\) | \(e\left(\frac{91}{115}\right)\) | \(e\left(\frac{87}{115}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) |
| \(\chi_{517}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{98}{115}\right)\) | \(e\left(\frac{114}{115}\right)\) | \(e\left(\frac{81}{115}\right)\) | \(e\left(\frac{77}{115}\right)\) | \(e\left(\frac{97}{115}\right)\) | \(e\left(\frac{26}{115}\right)\) | \(e\left(\frac{64}{115}\right)\) | \(e\left(\frac{113}{115}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) |
| \(\chi_{517}(108,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{115}\right)\) | \(e\left(\frac{62}{115}\right)\) | \(e\left(\frac{38}{115}\right)\) | \(e\left(\frac{56}{115}\right)\) | \(e\left(\frac{81}{115}\right)\) | \(e\left(\frac{113}{115}\right)\) | \(e\left(\frac{57}{115}\right)\) | \(e\left(\frac{9}{115}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) |
| \(\chi_{517}(115,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{86}{115}\right)\) | \(e\left(\frac{93}{115}\right)\) | \(e\left(\frac{57}{115}\right)\) | \(e\left(\frac{84}{115}\right)\) | \(e\left(\frac{64}{115}\right)\) | \(e\left(\frac{112}{115}\right)\) | \(e\left(\frac{28}{115}\right)\) | \(e\left(\frac{71}{115}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) |
| \(\chi_{517}(119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{115}\right)\) | \(e\left(\frac{77}{115}\right)\) | \(e\left(\frac{88}{115}\right)\) | \(e\left(\frac{51}{115}\right)\) | \(e\left(\frac{6}{115}\right)\) | \(e\left(\frac{68}{115}\right)\) | \(e\left(\frac{17}{115}\right)\) | \(e\left(\frac{39}{115}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) |
| \(\chi_{517}(126,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{115}\right)\) | \(e\left(\frac{38}{115}\right)\) | \(e\left(\frac{27}{115}\right)\) | \(e\left(\frac{64}{115}\right)\) | \(e\left(\frac{109}{115}\right)\) | \(e\left(\frac{47}{115}\right)\) | \(e\left(\frac{98}{115}\right)\) | \(e\left(\frac{76}{115}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) |
| \(\chi_{517}(130,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{115}\right)\) | \(e\left(\frac{97}{115}\right)\) | \(e\left(\frac{78}{115}\right)\) | \(e\left(\frac{6}{115}\right)\) | \(e\left(\frac{21}{115}\right)\) | \(e\left(\frac{8}{115}\right)\) | \(e\left(\frac{2}{115}\right)\) | \(e\left(\frac{79}{115}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) |
| \(\chi_{517}(136,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{115}\right)\) | \(e\left(\frac{4}{115}\right)\) | \(e\left(\frac{21}{115}\right)\) | \(e\left(\frac{37}{115}\right)\) | \(e\left(\frac{72}{115}\right)\) | \(e\left(\frac{11}{115}\right)\) | \(e\left(\frac{89}{115}\right)\) | \(e\left(\frac{8}{115}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) |
| \(\chi_{517}(147,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{115}\right)\) | \(e\left(\frac{14}{115}\right)\) | \(e\left(\frac{16}{115}\right)\) | \(e\left(\frac{72}{115}\right)\) | \(e\left(\frac{22}{115}\right)\) | \(e\left(\frac{96}{115}\right)\) | \(e\left(\frac{24}{115}\right)\) | \(e\left(\frac{28}{115}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) |
| \(\chi_{517}(148,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{106}{115}\right)\) | \(e\left(\frac{13}{115}\right)\) | \(e\left(\frac{97}{115}\right)\) | \(e\left(\frac{34}{115}\right)\) | \(e\left(\frac{4}{115}\right)\) | \(e\left(\frac{7}{115}\right)\) | \(e\left(\frac{88}{115}\right)\) | \(e\left(\frac{26}{115}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) |
| \(\chi_{517}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{112}{115}\right)\) | \(e\left(\frac{81}{115}\right)\) | \(e\left(\frac{109}{115}\right)\) | \(e\left(\frac{88}{115}\right)\) | \(e\left(\frac{78}{115}\right)\) | \(e\left(\frac{79}{115}\right)\) | \(e\left(\frac{106}{115}\right)\) | \(e\left(\frac{47}{115}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) |
| \(\chi_{517}(158,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{115}\right)\) | \(e\left(\frac{64}{115}\right)\) | \(e\left(\frac{106}{115}\right)\) | \(e\left(\frac{17}{115}\right)\) | \(e\left(\frac{2}{115}\right)\) | \(e\left(\frac{61}{115}\right)\) | \(e\left(\frac{44}{115}\right)\) | \(e\left(\frac{13}{115}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) |
| \(\chi_{517}(159,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{115}\right)\) | \(e\left(\frac{48}{115}\right)\) | \(e\left(\frac{22}{115}\right)\) | \(e\left(\frac{99}{115}\right)\) | \(e\left(\frac{59}{115}\right)\) | \(e\left(\frac{17}{115}\right)\) | \(e\left(\frac{33}{115}\right)\) | \(e\left(\frac{96}{115}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) |