from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1849, base_ring=CyclotomicField(602))
M = H._module
chi = DirichletCharacter(H, M([424]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,1849))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1849\) | |
| Conductor: | \(1849\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(301\) | 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: | $\Q(\zeta_{301})$ |
| Fixed field: | Number field defined by a degree 301 polynomial (not computed) |
First 31 of 252 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1849}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{285}{301}\right)\) | \(e\left(\frac{212}{301}\right)\) | \(e\left(\frac{269}{301}\right)\) | \(e\left(\frac{281}{301}\right)\) | \(e\left(\frac{28}{43}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{253}{301}\right)\) | \(e\left(\frac{123}{301}\right)\) | \(e\left(\frac{265}{301}\right)\) | \(e\left(\frac{270}{301}\right)\) |
| \(\chi_{1849}(11,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{135}{301}\right)\) | \(e\left(\frac{243}{301}\right)\) | \(e\left(\frac{270}{301}\right)\) | \(e\left(\frac{244}{301}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{104}{301}\right)\) | \(e\left(\frac{185}{301}\right)\) | \(e\left(\frac{78}{301}\right)\) | \(e\left(\frac{17}{301}\right)\) |
| \(\chi_{1849}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{269}{301}\right)\) | \(e\left(\frac{123}{301}\right)\) | \(e\left(\frac{237}{301}\right)\) | \(e\left(\frac{261}{301}\right)\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{205}{301}\right)\) | \(e\left(\frac{246}{301}\right)\) | \(e\left(\frac{229}{301}\right)\) | \(e\left(\frac{239}{301}\right)\) |
| \(\chi_{1849}(21,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{190}{301}\right)\) | \(e\left(\frac{41}{301}\right)\) | \(e\left(\frac{79}{301}\right)\) | \(e\left(\frac{87}{301}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{269}{301}\right)\) | \(e\left(\frac{82}{301}\right)\) | \(e\left(\frac{277}{301}\right)\) | \(e\left(\frac{180}{301}\right)\) |
| \(\chi_{1849}(35,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{74}{301}\right)\) | \(e\left(\frac{73}{301}\right)\) | \(e\left(\frac{148}{301}\right)\) | \(e\left(\frac{243}{301}\right)\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{222}{301}\right)\) | \(e\left(\frac{146}{301}\right)\) | \(e\left(\frac{16}{301}\right)\) | \(e\left(\frac{181}{301}\right)\) |
| \(\chi_{1849}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{301}\right)\) | \(e\left(\frac{29}{301}\right)\) | \(e\left(\frac{166}{301}\right)\) | \(e\left(\frac{179}{301}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{249}{301}\right)\) | \(e\left(\frac{58}{301}\right)\) | \(e\left(\frac{262}{301}\right)\) | \(e\left(\frac{142}{301}\right)\) |
| \(\chi_{1849}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{301}\right)\) | \(e\left(\frac{100}{301}\right)\) | \(e\left(\frac{178}{301}\right)\) | \(e\left(\frac{36}{301}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{267}{301}\right)\) | \(e\left(\frac{200}{301}\right)\) | \(e\left(\frac{125}{301}\right)\) | \(e\left(\frac{116}{301}\right)\) |
| \(\chi_{1849}(54,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{301}\right)\) | \(e\left(\frac{257}{301}\right)\) | \(e\left(\frac{18}{301}\right)\) | \(e\left(\frac{237}{301}\right)\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{27}{301}\right)\) | \(e\left(\frac{213}{301}\right)\) | \(e\left(\frac{246}{301}\right)\) | \(e\left(\frac{262}{301}\right)\) |
| \(\chi_{1849}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{220}{301}\right)\) | \(e\left(\frac{95}{301}\right)\) | \(e\left(\frac{139}{301}\right)\) | \(e\left(\frac{275}{301}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{58}{301}\right)\) | \(e\left(\frac{190}{301}\right)\) | \(e\left(\frac{194}{301}\right)\) | \(e\left(\frac{50}{301}\right)\) |
| \(\chi_{1849}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{253}{301}\right)\) | \(e\left(\frac{34}{301}\right)\) | \(e\left(\frac{205}{301}\right)\) | \(e\left(\frac{241}{301}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{157}{301}\right)\) | \(e\left(\frac{68}{301}\right)\) | \(e\left(\frac{193}{301}\right)\) | \(e\left(\frac{208}{301}\right)\) |
| \(\chi_{1849}(84,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{174}{301}\right)\) | \(e\left(\frac{253}{301}\right)\) | \(e\left(\frac{47}{301}\right)\) | \(e\left(\frac{67}{301}\right)\) | \(e\left(\frac{18}{43}\right)\) | \(e\left(\frac{40}{43}\right)\) | \(e\left(\frac{221}{301}\right)\) | \(e\left(\frac{205}{301}\right)\) | \(e\left(\frac{241}{301}\right)\) | \(e\left(\frac{149}{301}\right)\) |
| \(\chi_{1849}(90,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{194}{301}\right)\) | \(e\left(\frac{289}{301}\right)\) | \(e\left(\frac{87}{301}\right)\) | \(e\left(\frac{92}{301}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{281}{301}\right)\) | \(e\left(\frac{277}{301}\right)\) | \(e\left(\frac{286}{301}\right)\) | \(e\left(\frac{263}{301}\right)\) |
| \(\chi_{1849}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{184}{301}\right)\) | \(e\left(\frac{271}{301}\right)\) | \(e\left(\frac{67}{301}\right)\) | \(e\left(\frac{230}{301}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{251}{301}\right)\) | \(e\left(\frac{241}{301}\right)\) | \(e\left(\frac{113}{301}\right)\) | \(e\left(\frac{206}{301}\right)\) |
| \(\chi_{1849}(102,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{171}{301}\right)\) | \(e\left(\frac{67}{301}\right)\) | \(e\left(\frac{41}{301}\right)\) | \(e\left(\frac{289}{301}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{212}{301}\right)\) | \(e\left(\frac{134}{301}\right)\) | \(e\left(\frac{159}{301}\right)\) | \(e\left(\frac{162}{301}\right)\) |
| \(\chi_{1849}(107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{301}\right)\) | \(e\left(\frac{27}{301}\right)\) | \(e\left(\frac{30}{301}\right)\) | \(e\left(\frac{94}{301}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{45}{301}\right)\) | \(e\left(\frac{54}{301}\right)\) | \(e\left(\frac{109}{301}\right)\) | \(e\left(\frac{236}{301}\right)\) |
| \(\chi_{1849}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{270}{301}\right)\) | \(e\left(\frac{185}{301}\right)\) | \(e\left(\frac{239}{301}\right)\) | \(e\left(\frac{187}{301}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{208}{301}\right)\) | \(e\left(\frac{69}{301}\right)\) | \(e\left(\frac{156}{301}\right)\) | \(e\left(\frac{34}{301}\right)\) |
| \(\chi_{1849}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{265}{301}\right)\) | \(e\left(\frac{176}{301}\right)\) | \(e\left(\frac{229}{301}\right)\) | \(e\left(\frac{256}{301}\right)\) | \(e\left(\frac{20}{43}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{193}{301}\right)\) | \(e\left(\frac{51}{301}\right)\) | \(e\left(\frac{220}{301}\right)\) | \(e\left(\frac{156}{301}\right)\) |
| \(\chi_{1849}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{299}{301}\right)\) | \(e\left(\frac{177}{301}\right)\) | \(e\left(\frac{297}{301}\right)\) | \(e\left(\frac{148}{301}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{295}{301}\right)\) | \(e\left(\frac{53}{301}\right)\) | \(e\left(\frac{146}{301}\right)\) | \(e\left(\frac{109}{301}\right)\) |
| \(\chi_{1849}(140,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{301}\right)\) | \(e\left(\frac{285}{301}\right)\) | \(e\left(\frac{116}{301}\right)\) | \(e\left(\frac{223}{301}\right)\) | \(e\left(\frac{6}{43}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{174}{301}\right)\) | \(e\left(\frac{269}{301}\right)\) | \(e\left(\frac{281}{301}\right)\) | \(e\left(\frac{150}{301}\right)\) |
| \(\chi_{1849}(145,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{122}{301}\right)\) | \(e\left(\frac{39}{301}\right)\) | \(e\left(\frac{244}{301}\right)\) | \(e\left(\frac{2}{301}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{65}{301}\right)\) | \(e\left(\frac{78}{301}\right)\) | \(e\left(\frac{124}{301}\right)\) | \(e\left(\frac{274}{301}\right)\) |
| \(\chi_{1849}(150,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{78}{301}\right)\) | \(e\left(\frac{20}{301}\right)\) | \(e\left(\frac{156}{301}\right)\) | \(e\left(\frac{248}{301}\right)\) | \(e\left(\frac{14}{43}\right)\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{234}{301}\right)\) | \(e\left(\frac{40}{301}\right)\) | \(e\left(\frac{25}{301}\right)\) | \(e\left(\frac{264}{301}\right)\) |
| \(\chi_{1849}(164,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{301}\right)\) | \(e\left(\frac{241}{301}\right)\) | \(e\left(\frac{134}{301}\right)\) | \(e\left(\frac{159}{301}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{201}{301}\right)\) | \(e\left(\frac{181}{301}\right)\) | \(e\left(\frac{226}{301}\right)\) | \(e\left(\frac{111}{301}\right)\) |
| \(\chi_{1849}(170,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{301}\right)\) | \(e\left(\frac{99}{301}\right)\) | \(e\left(\frac{110}{301}\right)\) | \(e\left(\frac{144}{301}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{165}{301}\right)\) | \(e\left(\frac{198}{301}\right)\) | \(e\left(\frac{199}{301}\right)\) | \(e\left(\frac{163}{301}\right)\) |
| \(\chi_{1849}(176,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{301}\right)\) | \(e\left(\frac{65}{301}\right)\) | \(e\left(\frac{206}{301}\right)\) | \(e\left(\frac{204}{301}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{8}{301}\right)\) | \(e\left(\frac{130}{301}\right)\) | \(e\left(\frac{6}{301}\right)\) | \(e\left(\frac{256}{301}\right)\) |
| \(\chi_{1849}(183,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{233}{301}\right)\) | \(e\left(\frac{299}{301}\right)\) | \(e\left(\frac{165}{301}\right)\) | \(e\left(\frac{216}{301}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{97}{301}\right)\) | \(e\left(\frac{297}{301}\right)\) | \(e\left(\frac{148}{301}\right)\) | \(e\left(\frac{94}{301}\right)\) |
| \(\chi_{1849}(188,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{301}\right)\) | \(e\left(\frac{11}{301}\right)\) | \(e\left(\frac{146}{301}\right)\) | \(e\left(\frac{16}{301}\right)\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{219}{301}\right)\) | \(e\left(\frac{22}{301}\right)\) | \(e\left(\frac{89}{301}\right)\) | \(e\left(\frac{85}{301}\right)\) |
| \(\chi_{1849}(193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{141}{301}\right)\) | \(e\left(\frac{13}{301}\right)\) | \(e\left(\frac{282}{301}\right)\) | \(e\left(\frac{101}{301}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{122}{301}\right)\) | \(e\left(\frac{26}{301}\right)\) | \(e\left(\frac{242}{301}\right)\) | \(e\left(\frac{292}{301}\right)\) |
| \(\chi_{1849}(207,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{165}{301}\right)\) | \(e\left(\frac{297}{301}\right)\) | \(e\left(\frac{29}{301}\right)\) | \(e\left(\frac{131}{301}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{194}{301}\right)\) | \(e\left(\frac{293}{301}\right)\) | \(e\left(\frac{296}{301}\right)\) | \(e\left(\frac{188}{301}\right)\) |
| \(\chi_{1849}(213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{146}{301}\right)\) | \(e\left(\frac{22}{301}\right)\) | \(e\left(\frac{292}{301}\right)\) | \(e\left(\frac{32}{301}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{39}{43}\right)\) | \(e\left(\frac{137}{301}\right)\) | \(e\left(\frac{44}{301}\right)\) | \(e\left(\frac{178}{301}\right)\) | \(e\left(\frac{170}{301}\right)\) |
| \(\chi_{1849}(219,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{208}{301}\right)\) | \(e\left(\frac{254}{301}\right)\) | \(e\left(\frac{115}{301}\right)\) | \(e\left(\frac{260}{301}\right)\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{22}{301}\right)\) | \(e\left(\frac{207}{301}\right)\) | \(e\left(\frac{167}{301}\right)\) | \(e\left(\frac{102}{301}\right)\) |
| \(\chi_{1849}(226,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{301}\right)\) | \(e\left(\frac{12}{301}\right)\) | \(e\left(\frac{214}{301}\right)\) | \(e\left(\frac{209}{301}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{20}{301}\right)\) | \(e\left(\frac{24}{301}\right)\) | \(e\left(\frac{15}{301}\right)\) | \(e\left(\frac{38}{301}\right)\) |