from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(269, base_ring=CyclotomicField(134))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,269))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(269\) | |
| Conductor: | \(269\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(134\) | 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: | $\Q(\zeta_{67})$ |
| Fixed field: | Number field defined by a degree 134 polynomial (not computed) |
First 31 of 66 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{269}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{134}\right)\) | \(e\left(\frac{109}{134}\right)\) | \(e\left(\frac{1}{67}\right)\) | \(e\left(\frac{37}{67}\right)\) | \(e\left(\frac{55}{67}\right)\) | \(e\left(\frac{19}{134}\right)\) | \(e\left(\frac{3}{134}\right)\) | \(e\left(\frac{42}{67}\right)\) | \(e\left(\frac{75}{134}\right)\) | \(e\left(\frac{48}{67}\right)\) |
| \(\chi_{269}(6,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{134}\right)\) | \(e\left(\frac{99}{134}\right)\) | \(e\left(\frac{55}{67}\right)\) | \(e\left(\frac{25}{67}\right)\) | \(e\left(\frac{10}{67}\right)\) | \(e\left(\frac{107}{134}\right)\) | \(e\left(\frac{31}{134}\right)\) | \(e\left(\frac{32}{67}\right)\) | \(e\left(\frac{105}{134}\right)\) | \(e\left(\frac{27}{67}\right)\) |
| \(\chi_{269}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{134}\right)\) | \(e\left(\frac{89}{134}\right)\) | \(e\left(\frac{42}{67}\right)\) | \(e\left(\frac{13}{67}\right)\) | \(e\left(\frac{32}{67}\right)\) | \(e\left(\frac{61}{134}\right)\) | \(e\left(\frac{59}{134}\right)\) | \(e\left(\frac{22}{67}\right)\) | \(e\left(\frac{1}{134}\right)\) | \(e\left(\frac{6}{67}\right)\) |
| \(\chi_{269}(11,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{134}\right)\) | \(e\left(\frac{73}{134}\right)\) | \(e\left(\frac{48}{67}\right)\) | \(e\left(\frac{34}{67}\right)\) | \(e\left(\frac{27}{67}\right)\) | \(e\left(\frac{41}{134}\right)\) | \(e\left(\frac{77}{134}\right)\) | \(e\left(\frac{6}{67}\right)\) | \(e\left(\frac{49}{134}\right)\) | \(e\left(\frac{26}{67}\right)\) |
| \(\chi_{269}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{134}\right)\) | \(e\left(\frac{101}{134}\right)\) | \(e\left(\frac{4}{67}\right)\) | \(e\left(\frac{14}{67}\right)\) | \(e\left(\frac{19}{67}\right)\) | \(e\left(\frac{9}{134}\right)\) | \(e\left(\frac{79}{134}\right)\) | \(e\left(\frac{34}{67}\right)\) | \(e\left(\frac{99}{134}\right)\) | \(e\left(\frac{58}{67}\right)\) |
| \(\chi_{269}(20,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{105}{134}\right)\) | \(e\left(\frac{55}{134}\right)\) | \(e\left(\frac{38}{67}\right)\) | \(e\left(\frac{66}{67}\right)\) | \(e\left(\frac{13}{67}\right)\) | \(e\left(\frac{119}{134}\right)\) | \(e\left(\frac{47}{134}\right)\) | \(e\left(\frac{55}{67}\right)\) | \(e\left(\frac{103}{134}\right)\) | \(e\left(\frac{15}{67}\right)\) |
| \(\chi_{269}(30,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{134}\right)\) | \(e\left(\frac{45}{134}\right)\) | \(e\left(\frac{25}{67}\right)\) | \(e\left(\frac{54}{67}\right)\) | \(e\left(\frac{35}{67}\right)\) | \(e\left(\frac{73}{134}\right)\) | \(e\left(\frac{75}{134}\right)\) | \(e\left(\frac{45}{67}\right)\) | \(e\left(\frac{133}{134}\right)\) | \(e\left(\frac{61}{67}\right)\) |
| \(\chi_{269}(34,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{134}\right)\) | \(e\left(\frac{15}{134}\right)\) | \(e\left(\frac{53}{67}\right)\) | \(e\left(\frac{18}{67}\right)\) | \(e\left(\frac{34}{67}\right)\) | \(e\left(\frac{69}{134}\right)\) | \(e\left(\frac{25}{134}\right)\) | \(e\left(\frac{15}{67}\right)\) | \(e\left(\frac{89}{134}\right)\) | \(e\left(\frac{65}{67}\right)\) |
| \(\chi_{269}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{134}\right)\) | \(e\left(\frac{41}{134}\right)\) | \(e\left(\frac{60}{67}\right)\) | \(e\left(\frac{9}{67}\right)\) | \(e\left(\frac{17}{67}\right)\) | \(e\left(\frac{1}{134}\right)\) | \(e\left(\frac{113}{134}\right)\) | \(e\left(\frac{41}{67}\right)\) | \(e\left(\frac{11}{134}\right)\) | \(e\left(\frac{66}{67}\right)\) |
| \(\chi_{269}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{134}\right)\) | \(e\left(\frac{35}{134}\right)\) | \(e\left(\frac{12}{67}\right)\) | \(e\left(\frac{42}{67}\right)\) | \(e\left(\frac{57}{67}\right)\) | \(e\left(\frac{27}{134}\right)\) | \(e\left(\frac{103}{134}\right)\) | \(e\left(\frac{35}{67}\right)\) | \(e\left(\frac{29}{134}\right)\) | \(e\left(\frac{40}{67}\right)\) |
| \(\chi_{269}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{134}\right)\) | \(e\left(\frac{61}{134}\right)\) | \(e\left(\frac{19}{67}\right)\) | \(e\left(\frac{33}{67}\right)\) | \(e\left(\frac{40}{67}\right)\) | \(e\left(\frac{93}{134}\right)\) | \(e\left(\frac{57}{134}\right)\) | \(e\left(\frac{61}{67}\right)\) | \(e\left(\frac{85}{134}\right)\) | \(e\left(\frac{41}{67}\right)\) |
| \(\chi_{269}(51,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{134}\right)\) | \(e\left(\frac{5}{134}\right)\) | \(e\left(\frac{40}{67}\right)\) | \(e\left(\frac{6}{67}\right)\) | \(e\left(\frac{56}{67}\right)\) | \(e\left(\frac{23}{134}\right)\) | \(e\left(\frac{53}{134}\right)\) | \(e\left(\frac{5}{67}\right)\) | \(e\left(\frac{119}{134}\right)\) | \(e\left(\frac{44}{67}\right)\) |
| \(\chi_{269}(55,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{134}\right)\) | \(e\left(\frac{19}{134}\right)\) | \(e\left(\frac{18}{67}\right)\) | \(e\left(\frac{63}{67}\right)\) | \(e\left(\frac{52}{67}\right)\) | \(e\left(\frac{7}{134}\right)\) | \(e\left(\frac{121}{134}\right)\) | \(e\left(\frac{19}{67}\right)\) | \(e\left(\frac{77}{134}\right)\) | \(e\left(\frac{60}{67}\right)\) |
| \(\chi_{269}(56,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{134}\right)\) | \(e\left(\frac{127}{134}\right)\) | \(e\left(\frac{11}{67}\right)\) | \(e\left(\frac{5}{67}\right)\) | \(e\left(\frac{2}{67}\right)\) | \(e\left(\frac{75}{134}\right)\) | \(e\left(\frac{33}{134}\right)\) | \(e\left(\frac{60}{67}\right)\) | \(e\left(\frac{21}{134}\right)\) | \(e\left(\frac{59}{67}\right)\) |
| \(\chi_{269}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{134}\right)\) | \(e\left(\frac{59}{134}\right)\) | \(e\left(\frac{3}{67}\right)\) | \(e\left(\frac{44}{67}\right)\) | \(e\left(\frac{31}{67}\right)\) | \(e\left(\frac{57}{134}\right)\) | \(e\left(\frac{9}{134}\right)\) | \(e\left(\frac{59}{67}\right)\) | \(e\left(\frac{91}{134}\right)\) | \(e\left(\frac{10}{67}\right)\) |
| \(\chi_{269}(65,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{134}\right)\) | \(e\left(\frac{47}{134}\right)\) | \(e\left(\frac{41}{67}\right)\) | \(e\left(\frac{43}{67}\right)\) | \(e\left(\frac{44}{67}\right)\) | \(e\left(\frac{109}{134}\right)\) | \(e\left(\frac{123}{134}\right)\) | \(e\left(\frac{47}{67}\right)\) | \(e\left(\frac{127}{134}\right)\) | \(e\left(\frac{25}{67}\right)\) |
| \(\chi_{269}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{87}{134}\right)\) | \(e\left(\frac{103}{134}\right)\) | \(e\left(\frac{20}{67}\right)\) | \(e\left(\frac{3}{67}\right)\) | \(e\left(\frac{28}{67}\right)\) | \(e\left(\frac{45}{134}\right)\) | \(e\left(\frac{127}{134}\right)\) | \(e\left(\frac{36}{67}\right)\) | \(e\left(\frac{93}{134}\right)\) | \(e\left(\frac{22}{67}\right)\) |
| \(\chi_{269}(79,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{134}\right)\) | \(e\left(\frac{27}{134}\right)\) | \(e\left(\frac{15}{67}\right)\) | \(e\left(\frac{19}{67}\right)\) | \(e\left(\frac{21}{67}\right)\) | \(e\left(\frac{17}{134}\right)\) | \(e\left(\frac{45}{134}\right)\) | \(e\left(\frac{27}{67}\right)\) | \(e\left(\frac{53}{134}\right)\) | \(e\left(\frac{50}{67}\right)\) |
| \(\chi_{269}(84,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{134}\right)\) | \(e\left(\frac{117}{134}\right)\) | \(e\left(\frac{65}{67}\right)\) | \(e\left(\frac{60}{67}\right)\) | \(e\left(\frac{24}{67}\right)\) | \(e\left(\frac{29}{134}\right)\) | \(e\left(\frac{61}{134}\right)\) | \(e\left(\frac{50}{67}\right)\) | \(e\left(\frac{51}{134}\right)\) | \(e\left(\frac{38}{67}\right)\) |
| \(\chi_{269}(89,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{134}\right)\) | \(e\left(\frac{77}{134}\right)\) | \(e\left(\frac{13}{67}\right)\) | \(e\left(\frac{12}{67}\right)\) | \(e\left(\frac{45}{67}\right)\) | \(e\left(\frac{113}{134}\right)\) | \(e\left(\frac{39}{134}\right)\) | \(e\left(\frac{10}{67}\right)\) | \(e\left(\frac{37}{134}\right)\) | \(e\left(\frac{21}{67}\right)\) |
| \(\chi_{269}(92,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{134}\right)\) | \(e\left(\frac{53}{134}\right)\) | \(e\left(\frac{22}{67}\right)\) | \(e\left(\frac{10}{67}\right)\) | \(e\left(\frac{4}{67}\right)\) | \(e\left(\frac{83}{134}\right)\) | \(e\left(\frac{133}{134}\right)\) | \(e\left(\frac{53}{67}\right)\) | \(e\left(\frac{109}{134}\right)\) | \(e\left(\frac{51}{67}\right)\) |
| \(\chi_{269}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{134}\right)\) | \(e\left(\frac{49}{134}\right)\) | \(e\left(\frac{57}{67}\right)\) | \(e\left(\frac{32}{67}\right)\) | \(e\left(\frac{53}{67}\right)\) | \(e\left(\frac{11}{134}\right)\) | \(e\left(\frac{37}{134}\right)\) | \(e\left(\frac{49}{67}\right)\) | \(e\left(\frac{121}{134}\right)\) | \(e\left(\frac{56}{67}\right)\) |
| \(\chi_{269}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{134}\right)\) | \(e\left(\frac{83}{134}\right)\) | \(e\left(\frac{61}{67}\right)\) | \(e\left(\frac{46}{67}\right)\) | \(e\left(\frac{5}{67}\right)\) | \(e\left(\frac{87}{134}\right)\) | \(e\left(\frac{49}{134}\right)\) | \(e\left(\frac{16}{67}\right)\) | \(e\left(\frac{19}{134}\right)\) | \(e\left(\frac{47}{67}\right)\) |
| \(\chi_{269}(100,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{75}{134}\right)\) | \(e\left(\frac{1}{134}\right)\) | \(e\left(\frac{8}{67}\right)\) | \(e\left(\frac{28}{67}\right)\) | \(e\left(\frac{38}{67}\right)\) | \(e\left(\frac{85}{134}\right)\) | \(e\left(\frac{91}{134}\right)\) | \(e\left(\frac{1}{67}\right)\) | \(e\left(\frac{131}{134}\right)\) | \(e\left(\frac{49}{67}\right)\) |
| \(\chi_{269}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{134}\right)\) | \(e\left(\frac{65}{134}\right)\) | \(e\left(\frac{51}{67}\right)\) | \(e\left(\frac{11}{67}\right)\) | \(e\left(\frac{58}{67}\right)\) | \(e\left(\frac{31}{134}\right)\) | \(e\left(\frac{19}{134}\right)\) | \(e\left(\frac{65}{67}\right)\) | \(e\left(\frac{73}{134}\right)\) | \(e\left(\frac{36}{67}\right)\) |
| \(\chi_{269}(126,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{134}\right)\) | \(e\left(\frac{107}{134}\right)\) | \(e\left(\frac{52}{67}\right)\) | \(e\left(\frac{48}{67}\right)\) | \(e\left(\frac{46}{67}\right)\) | \(e\left(\frac{117}{134}\right)\) | \(e\left(\frac{89}{134}\right)\) | \(e\left(\frac{40}{67}\right)\) | \(e\left(\frac{81}{134}\right)\) | \(e\left(\frac{17}{67}\right)\) |
| \(\chi_{269}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{134}\right)\) | \(e\left(\frac{3}{134}\right)\) | \(e\left(\frac{24}{67}\right)\) | \(e\left(\frac{17}{67}\right)\) | \(e\left(\frac{47}{67}\right)\) | \(e\left(\frac{121}{134}\right)\) | \(e\left(\frac{5}{134}\right)\) | \(e\left(\frac{3}{67}\right)\) | \(e\left(\frac{125}{134}\right)\) | \(e\left(\frac{13}{67}\right)\) |
| \(\chi_{269}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{134}\right)\) | \(e\left(\frac{57}{134}\right)\) | \(e\left(\frac{54}{67}\right)\) | \(e\left(\frac{55}{67}\right)\) | \(e\left(\frac{22}{67}\right)\) | \(e\left(\frac{21}{134}\right)\) | \(e\left(\frac{95}{134}\right)\) | \(e\left(\frac{57}{67}\right)\) | \(e\left(\frac{97}{134}\right)\) | \(e\left(\frac{46}{67}\right)\) |
| \(\chi_{269}(138,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{134}\right)\) | \(e\left(\frac{43}{134}\right)\) | \(e\left(\frac{9}{67}\right)\) | \(e\left(\frac{65}{67}\right)\) | \(e\left(\frac{26}{67}\right)\) | \(e\left(\frac{37}{134}\right)\) | \(e\left(\frac{27}{134}\right)\) | \(e\left(\frac{43}{67}\right)\) | \(e\left(\frac{5}{134}\right)\) | \(e\left(\frac{30}{67}\right)\) |
| \(\chi_{269}(144,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{111}{134}\right)\) | \(e\left(\frac{39}{134}\right)\) | \(e\left(\frac{44}{67}\right)\) | \(e\left(\frac{20}{67}\right)\) | \(e\left(\frac{8}{67}\right)\) | \(e\left(\frac{99}{134}\right)\) | \(e\left(\frac{65}{134}\right)\) | \(e\left(\frac{39}{67}\right)\) | \(e\left(\frac{17}{134}\right)\) | \(e\left(\frac{35}{67}\right)\) |
| \(\chi_{269}(148,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{134}\right)\) | \(e\left(\frac{79}{134}\right)\) | \(e\left(\frac{29}{67}\right)\) | \(e\left(\frac{1}{67}\right)\) | \(e\left(\frac{54}{67}\right)\) | \(e\left(\frac{15}{134}\right)\) | \(e\left(\frac{87}{134}\right)\) | \(e\left(\frac{12}{67}\right)\) | \(e\left(\frac{31}{134}\right)\) | \(e\left(\frac{52}{67}\right)\) |