from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4205, base_ring=CyclotomicField(116))
M = H._module
chi = DirichletCharacter(H, M([29,77]))
chi.galois_orbit()
[g,chi] = znchar(Mod(12,4205))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4205\) | |
| Conductor: | \(4205\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(116\) | 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: | $\Q(\zeta_{116})$ |
| Fixed field: | Number field defined by a degree 116 polynomial (not computed) |
First 31 of 56 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4205}(12,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{97}{116}\right)\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{89}{116}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{21}{116}\right)\) |
| \(\chi_{4205}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{13}{58}\right)\) | \(e\left(\frac{31}{116}\right)\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{99}{116}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{39}{116}\right)\) |
| \(\chi_{4205}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{58}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{105}{116}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{21}{116}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{61}{116}\right)\) |
| \(\chi_{4205}(278,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{23}{116}\right)\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{51}{116}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{115}{116}\right)\) |
| \(\chi_{4205}(302,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{113}{116}\right)\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{69}{116}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{101}{116}\right)\) |
| \(\chi_{4205}(423,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{15}{116}\right)\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{3}{116}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{75}{116}\right)\) |
| \(\chi_{4205}(447,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{47}{58}\right)\) | \(e\left(\frac{5}{116}\right)\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{1}{116}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{25}{116}\right)\) |
| \(\chi_{4205}(568,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{58}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{7}{116}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{71}{116}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{35}{116}\right)\) |
| \(\chi_{4205}(592,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{13}{116}\right)\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{49}{116}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{65}{116}\right)\) |
| \(\chi_{4205}(713,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{115}{116}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{23}{116}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{111}{116}\right)\) |
| \(\chi_{4205}(737,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{21}{116}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{97}{116}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{105}{116}\right)\) |
| \(\chi_{4205}(858,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{107}{116}\right)\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{91}{116}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{71}{116}\right)\) |
| \(\chi_{4205}(1003,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{49}{58}\right)\) | \(e\left(\frac{99}{116}\right)\) | \(e\left(\frac{11}{58}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{43}{116}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{31}{116}\right)\) |
| \(\chi_{4205}(1027,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{37}{116}\right)\) | \(e\left(\frac{17}{58}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{77}{116}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{69}{116}\right)\) |
| \(\chi_{4205}(1148,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{91}{116}\right)\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{111}{116}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{107}{116}\right)\) |
| \(\chi_{4205}(1172,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{17}{58}\right)\) | \(e\left(\frac{45}{116}\right)\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{9}{116}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{109}{116}\right)\) |
| \(\chi_{4205}(1293,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{83}{116}\right)\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{63}{116}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{67}{116}\right)\) |
| \(\chi_{4205}(1317,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{58}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{11}{58}\right)\) | \(e\left(\frac{53}{116}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{57}{116}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{33}{116}\right)\) |
| \(\chi_{4205}(1438,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{75}{116}\right)\) | \(e\left(\frac{47}{58}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{15}{116}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{27}{116}\right)\) |
| \(\chi_{4205}(1462,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{58}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{61}{116}\right)\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{105}{116}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{73}{116}\right)\) |
| \(\chi_{4205}(1583,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{67}{116}\right)\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{83}{116}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{103}{116}\right)\) |
| \(\chi_{4205}(1607,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{69}{116}\right)\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{37}{116}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{113}{116}\right)\) |
| \(\chi_{4205}(1728,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{59}{116}\right)\) | \(e\left(\frac{13}{58}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{35}{116}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{63}{116}\right)\) |
| \(\chi_{4205}(1752,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{77}{116}\right)\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{85}{116}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{37}{116}\right)\) |
| \(\chi_{4205}(1873,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{58}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{51}{116}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{103}{116}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{23}{116}\right)\) |
| \(\chi_{4205}(1897,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{85}{116}\right)\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{17}{116}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{77}{116}\right)\) |
| \(\chi_{4205}(2018,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{43}{116}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{55}{116}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{99}{116}\right)\) |
| \(\chi_{4205}(2042,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{93}{116}\right)\) | \(e\left(\frac{49}{58}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{65}{116}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{1}{116}\right)\) |
| \(\chi_{4205}(2163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{35}{116}\right)\) | \(e\left(\frac{49}{58}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{7}{116}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{59}{116}\right)\) |
| \(\chi_{4205}(2187,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{101}{116}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{113}{116}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{41}{116}\right)\) |
| \(\chi_{4205}(2308,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{27}{116}\right)\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{75}{116}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{19}{116}\right)\) |