from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1049, base_ring=CyclotomicField(262))
M = H._module
chi = DirichletCharacter(H, M([149]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1049))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1049\) | |
| Conductor: | \(1049\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(262\) | 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: | $\Q(\zeta_{131})$ |
| Fixed field: | Number field defined by a degree 262 polynomial (not computed) |
First 31 of 130 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1049}(2,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{124}{131}\right)\) | \(e\left(\frac{149}{262}\right)\) | \(e\left(\frac{117}{131}\right)\) | \(e\left(\frac{96}{131}\right)\) | \(e\left(\frac{135}{262}\right)\) | \(e\left(\frac{5}{262}\right)\) | \(e\left(\frac{110}{131}\right)\) | \(e\left(\frac{18}{131}\right)\) | \(e\left(\frac{89}{131}\right)\) | \(e\left(\frac{94}{131}\right)\) |
| \(\chi_{1049}(8,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{110}{131}\right)\) | \(e\left(\frac{185}{262}\right)\) | \(e\left(\frac{89}{131}\right)\) | \(e\left(\frac{26}{131}\right)\) | \(e\left(\frac{143}{262}\right)\) | \(e\left(\frac{15}{262}\right)\) | \(e\left(\frac{68}{131}\right)\) | \(e\left(\frac{54}{131}\right)\) | \(e\left(\frac{5}{131}\right)\) | \(e\left(\frac{20}{131}\right)\) |
| \(\chi_{1049}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{131}\right)\) | \(e\left(\frac{35}{262}\right)\) | \(e\left(\frac{31}{131}\right)\) | \(e\left(\frac{12}{131}\right)\) | \(e\left(\frac{197}{262}\right)\) | \(e\left(\frac{17}{262}\right)\) | \(e\left(\frac{112}{131}\right)\) | \(e\left(\frac{35}{131}\right)\) | \(e\left(\frac{93}{131}\right)\) | \(e\left(\frac{110}{131}\right)\) |
| \(\chi_{1049}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{131}\right)\) | \(e\left(\frac{49}{262}\right)\) | \(e\left(\frac{122}{131}\right)\) | \(e\left(\frac{43}{131}\right)\) | \(e\left(\frac{171}{262}\right)\) | \(e\left(\frac{181}{262}\right)\) | \(e\left(\frac{52}{131}\right)\) | \(e\left(\frac{49}{131}\right)\) | \(e\left(\frac{104}{131}\right)\) | \(e\left(\frac{23}{131}\right)\) |
| \(\chi_{1049}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{92}{131}\right)\) | \(e\left(\frac{119}{262}\right)\) | \(e\left(\frac{53}{131}\right)\) | \(e\left(\frac{67}{131}\right)\) | \(e\left(\frac{41}{262}\right)\) | \(e\left(\frac{215}{262}\right)\) | \(e\left(\frac{14}{131}\right)\) | \(e\left(\frac{119}{131}\right)\) | \(e\left(\frac{28}{131}\right)\) | \(e\left(\frac{112}{131}\right)\) |
| \(\chi_{1049}(32,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{96}{131}\right)\) | \(e\left(\frac{221}{262}\right)\) | \(e\left(\frac{61}{131}\right)\) | \(e\left(\frac{87}{131}\right)\) | \(e\left(\frac{151}{262}\right)\) | \(e\left(\frac{25}{262}\right)\) | \(e\left(\frac{26}{131}\right)\) | \(e\left(\frac{90}{131}\right)\) | \(e\left(\frac{52}{131}\right)\) | \(e\left(\frac{77}{131}\right)\) |
| \(\chi_{1049}(38,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{131}\right)\) | \(e\left(\frac{187}{262}\right)\) | \(e\left(\frac{102}{131}\right)\) | \(e\left(\frac{124}{131}\right)\) | \(e\left(\frac{27}{262}\right)\) | \(e\left(\frac{1}{262}\right)\) | \(e\left(\frac{22}{131}\right)\) | \(e\left(\frac{56}{131}\right)\) | \(e\left(\frac{44}{131}\right)\) | \(e\left(\frac{45}{131}\right)\) |
| \(\chi_{1049}(42,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{70}{131}\right)\) | \(e\left(\frac{213}{262}\right)\) | \(e\left(\frac{9}{131}\right)\) | \(e\left(\frac{88}{131}\right)\) | \(e\left(\frac{91}{262}\right)\) | \(e\left(\frac{81}{262}\right)\) | \(e\left(\frac{79}{131}\right)\) | \(e\left(\frac{82}{131}\right)\) | \(e\left(\frac{27}{131}\right)\) | \(e\left(\frac{108}{131}\right)\) |
| \(\chi_{1049}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{131}\right)\) | \(e\left(\frac{135}{262}\right)\) | \(e\left(\frac{26}{131}\right)\) | \(e\left(\frac{65}{131}\right)\) | \(e\left(\frac{161}{262}\right)\) | \(e\left(\frac{103}{262}\right)\) | \(e\left(\frac{39}{131}\right)\) | \(e\left(\frac{4}{131}\right)\) | \(e\left(\frac{78}{131}\right)\) | \(e\left(\frac{50}{131}\right)\) |
| \(\chi_{1049}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{114}{131}\right)\) | \(e\left(\frac{25}{262}\right)\) | \(e\left(\frac{97}{131}\right)\) | \(e\left(\frac{46}{131}\right)\) | \(e\left(\frac{253}{262}\right)\) | \(e\left(\frac{87}{262}\right)\) | \(e\left(\frac{80}{131}\right)\) | \(e\left(\frac{25}{131}\right)\) | \(e\left(\frac{29}{131}\right)\) | \(e\left(\frac{116}{131}\right)\) |
| \(\chi_{1049}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{131}\right)\) | \(e\left(\frac{71}{262}\right)\) | \(e\left(\frac{3}{131}\right)\) | \(e\left(\frac{73}{131}\right)\) | \(e\left(\frac{205}{262}\right)\) | \(e\left(\frac{27}{262}\right)\) | \(e\left(\frac{70}{131}\right)\) | \(e\left(\frac{71}{131}\right)\) | \(e\left(\frac{9}{131}\right)\) | \(e\left(\frac{36}{131}\right)\) |
| \(\chi_{1049}(55,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{131}\right)\) | \(e\left(\frac{129}{262}\right)\) | \(e\left(\frac{118}{131}\right)\) | \(e\left(\frac{33}{131}\right)\) | \(e\left(\frac{247}{262}\right)\) | \(e\left(\frac{145}{262}\right)\) | \(e\left(\frac{46}{131}\right)\) | \(e\left(\frac{129}{131}\right)\) | \(e\left(\frac{92}{131}\right)\) | \(e\left(\frac{106}{131}\right)\) |
| \(\chi_{1049}(69,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{87}{131}\right)\) | \(e\left(\frac{57}{262}\right)\) | \(e\left(\frac{43}{131}\right)\) | \(e\left(\frac{42}{131}\right)\) | \(e\left(\frac{231}{262}\right)\) | \(e\left(\frac{125}{262}\right)\) | \(e\left(\frac{130}{131}\right)\) | \(e\left(\frac{57}{131}\right)\) | \(e\left(\frac{129}{131}\right)\) | \(e\left(\frac{123}{131}\right)\) |
| \(\chi_{1049}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{131}\right)\) | \(e\left(\frac{181}{262}\right)\) | \(e\left(\frac{63}{131}\right)\) | \(e\left(\frac{92}{131}\right)\) | \(e\left(\frac{113}{262}\right)\) | \(e\left(\frac{43}{262}\right)\) | \(e\left(\frac{29}{131}\right)\) | \(e\left(\frac{50}{131}\right)\) | \(e\left(\frac{58}{131}\right)\) | \(e\left(\frac{101}{131}\right)\) |
| \(\chi_{1049}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{36}{131}\right)\) | \(e\left(\frac{1}{262}\right)\) | \(e\left(\frac{72}{131}\right)\) | \(e\left(\frac{49}{131}\right)\) | \(e\left(\frac{73}{262}\right)\) | \(e\left(\frac{255}{262}\right)\) | \(e\left(\frac{108}{131}\right)\) | \(e\left(\frac{1}{131}\right)\) | \(e\left(\frac{85}{131}\right)\) | \(e\left(\frac{78}{131}\right)\) |
| \(\chi_{1049}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{48}{131}\right)\) | \(e\left(\frac{45}{262}\right)\) | \(e\left(\frac{96}{131}\right)\) | \(e\left(\frac{109}{131}\right)\) | \(e\left(\frac{141}{262}\right)\) | \(e\left(\frac{209}{262}\right)\) | \(e\left(\frac{13}{131}\right)\) | \(e\left(\frac{45}{131}\right)\) | \(e\left(\frac{26}{131}\right)\) | \(e\left(\frac{104}{131}\right)\) |
| \(\chi_{1049}(99,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{112}{131}\right)\) | \(e\left(\frac{105}{262}\right)\) | \(e\left(\frac{93}{131}\right)\) | \(e\left(\frac{36}{131}\right)\) | \(e\left(\frac{67}{262}\right)\) | \(e\left(\frac{51}{262}\right)\) | \(e\left(\frac{74}{131}\right)\) | \(e\left(\frac{105}{131}\right)\) | \(e\left(\frac{17}{131}\right)\) | \(e\left(\frac{68}{131}\right)\) |
| \(\chi_{1049}(100,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{131}\right)\) | \(e\left(\frac{85}{262}\right)\) | \(e\left(\frac{94}{131}\right)\) | \(e\left(\frac{104}{131}\right)\) | \(e\left(\frac{179}{262}\right)\) | \(e\left(\frac{191}{262}\right)\) | \(e\left(\frac{10}{131}\right)\) | \(e\left(\frac{85}{131}\right)\) | \(e\left(\frac{20}{131}\right)\) | \(e\left(\frac{80}{131}\right)\) |
| \(\chi_{1049}(106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{131}\right)\) | \(e\left(\frac{193}{262}\right)\) | \(e\left(\frac{10}{131}\right)\) | \(e\left(\frac{25}{131}\right)\) | \(e\left(\frac{203}{262}\right)\) | \(e\left(\frac{221}{262}\right)\) | \(e\left(\frac{15}{131}\right)\) | \(e\left(\frac{62}{131}\right)\) | \(e\left(\frac{30}{131}\right)\) | \(e\left(\frac{120}{131}\right)\) |
| \(\chi_{1049}(116,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{78}{131}\right)\) | \(e\left(\frac{155}{262}\right)\) | \(e\left(\frac{25}{131}\right)\) | \(e\left(\frac{128}{131}\right)\) | \(e\left(\frac{49}{262}\right)\) | \(e\left(\frac{225}{262}\right)\) | \(e\left(\frac{103}{131}\right)\) | \(e\left(\frac{24}{131}\right)\) | \(e\left(\frac{75}{131}\right)\) | \(e\left(\frac{38}{131}\right)\) |
| \(\chi_{1049}(119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{131}\right)\) | \(e\left(\frac{253}{262}\right)\) | \(e\left(\frac{7}{131}\right)\) | \(e\left(\frac{83}{131}\right)\) | \(e\left(\frac{129}{262}\right)\) | \(e\left(\frac{63}{262}\right)\) | \(e\left(\frac{76}{131}\right)\) | \(e\left(\frac{122}{131}\right)\) | \(e\left(\frac{21}{131}\right)\) | \(e\left(\frac{84}{131}\right)\) |
| \(\chi_{1049}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{131}\right)\) | \(e\left(\frac{209}{262}\right)\) | \(e\left(\frac{114}{131}\right)\) | \(e\left(\frac{23}{131}\right)\) | \(e\left(\frac{61}{262}\right)\) | \(e\left(\frac{109}{262}\right)\) | \(e\left(\frac{40}{131}\right)\) | \(e\left(\frac{78}{131}\right)\) | \(e\left(\frac{80}{131}\right)\) | \(e\left(\frac{58}{131}\right)\) |
| \(\chi_{1049}(122,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{118}{131}\right)\) | \(e\left(\frac{127}{262}\right)\) | \(e\left(\frac{105}{131}\right)\) | \(e\left(\frac{66}{131}\right)\) | \(e\left(\frac{101}{262}\right)\) | \(e\left(\frac{159}{262}\right)\) | \(e\left(\frac{92}{131}\right)\) | \(e\left(\frac{127}{131}\right)\) | \(e\left(\frac{53}{131}\right)\) | \(e\left(\frac{81}{131}\right)\) |
| \(\chi_{1049}(128,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{82}{131}\right)\) | \(e\left(\frac{257}{262}\right)\) | \(e\left(\frac{33}{131}\right)\) | \(e\left(\frac{17}{131}\right)\) | \(e\left(\frac{159}{262}\right)\) | \(e\left(\frac{35}{262}\right)\) | \(e\left(\frac{115}{131}\right)\) | \(e\left(\frac{126}{131}\right)\) | \(e\left(\frac{99}{131}\right)\) | \(e\left(\frac{3}{131}\right)\) |
| \(\chi_{1049}(152,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{131}\right)\) | \(e\left(\frac{223}{262}\right)\) | \(e\left(\frac{74}{131}\right)\) | \(e\left(\frac{54}{131}\right)\) | \(e\left(\frac{35}{262}\right)\) | \(e\left(\frac{11}{262}\right)\) | \(e\left(\frac{111}{131}\right)\) | \(e\left(\frac{92}{131}\right)\) | \(e\left(\frac{91}{131}\right)\) | \(e\left(\frac{102}{131}\right)\) |
| \(\chi_{1049}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{131}\right)\) | \(e\left(\frac{29}{262}\right)\) | \(e\left(\frac{123}{131}\right)\) | \(e\left(\frac{111}{131}\right)\) | \(e\left(\frac{21}{262}\right)\) | \(e\left(\frac{59}{262}\right)\) | \(e\left(\frac{119}{131}\right)\) | \(e\left(\frac{29}{131}\right)\) | \(e\left(\frac{107}{131}\right)\) | \(e\left(\frac{35}{131}\right)\) |
| \(\chi_{1049}(168,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{56}{131}\right)\) | \(e\left(\frac{249}{262}\right)\) | \(e\left(\frac{112}{131}\right)\) | \(e\left(\frac{18}{131}\right)\) | \(e\left(\frac{99}{262}\right)\) | \(e\left(\frac{91}{262}\right)\) | \(e\left(\frac{37}{131}\right)\) | \(e\left(\frac{118}{131}\right)\) | \(e\left(\frac{74}{131}\right)\) | \(e\left(\frac{34}{131}\right)\) |
| \(\chi_{1049}(172,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{130}{131}\right)\) | \(e\left(\frac{171}{262}\right)\) | \(e\left(\frac{129}{131}\right)\) | \(e\left(\frac{126}{131}\right)\) | \(e\left(\frac{169}{262}\right)\) | \(e\left(\frac{113}{262}\right)\) | \(e\left(\frac{128}{131}\right)\) | \(e\left(\frac{40}{131}\right)\) | \(e\left(\frac{125}{131}\right)\) | \(e\left(\frac{107}{131}\right)\) |
| \(\chi_{1049}(180,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{100}{131}\right)\) | \(e\left(\frac{61}{262}\right)\) | \(e\left(\frac{69}{131}\right)\) | \(e\left(\frac{107}{131}\right)\) | \(e\left(\frac{261}{262}\right)\) | \(e\left(\frac{97}{262}\right)\) | \(e\left(\frac{38}{131}\right)\) | \(e\left(\frac{61}{131}\right)\) | \(e\left(\frac{76}{131}\right)\) | \(e\left(\frac{42}{131}\right)\) |
| \(\chi_{1049}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{131}\right)\) | \(e\left(\frac{179}{262}\right)\) | \(e\left(\frac{50}{131}\right)\) | \(e\left(\frac{125}{131}\right)\) | \(e\left(\frac{229}{262}\right)\) | \(e\left(\frac{57}{262}\right)\) | \(e\left(\frac{75}{131}\right)\) | \(e\left(\frac{48}{131}\right)\) | \(e\left(\frac{19}{131}\right)\) | \(e\left(\frac{76}{131}\right)\) |
| \(\chi_{1049}(201,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{131}\right)\) | \(e\left(\frac{247}{262}\right)\) | \(e\left(\frac{99}{131}\right)\) | \(e\left(\frac{51}{131}\right)\) | \(e\left(\frac{215}{262}\right)\) | \(e\left(\frac{105}{262}\right)\) | \(e\left(\frac{83}{131}\right)\) | \(e\left(\frac{116}{131}\right)\) | \(e\left(\frac{35}{131}\right)\) | \(e\left(\frac{9}{131}\right)\) |