from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3721, base_ring=CyclotomicField(610))
M = H._module
chi = DirichletCharacter(H, M([352]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,3721))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3721\) | |
| Conductor: | \(3721\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(305\) | 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: | $\Q(\zeta_{305})$ |
| Fixed field: | Number field defined by a degree 305 polynomial (not computed) |
First 31 of 240 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3721}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{176}{305}\right)\) | \(e\left(\frac{111}{305}\right)\) | \(e\left(\frac{47}{305}\right)\) | \(e\left(\frac{57}{305}\right)\) | \(e\left(\frac{287}{305}\right)\) | \(e\left(\frac{304}{305}\right)\) | \(e\left(\frac{223}{305}\right)\) | \(e\left(\frac{222}{305}\right)\) | \(e\left(\frac{233}{305}\right)\) | \(e\left(\frac{43}{61}\right)\) |
| \(\chi_{3721}(20,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{305}\right)\) | \(e\left(\frac{52}{305}\right)\) | \(e\left(\frac{44}{305}\right)\) | \(e\left(\frac{274}{305}\right)\) | \(e\left(\frac{74}{305}\right)\) | \(e\left(\frac{38}{305}\right)\) | \(e\left(\frac{66}{305}\right)\) | \(e\left(\frac{104}{305}\right)\) | \(e\left(\frac{296}{305}\right)\) | \(e\left(\frac{13}{61}\right)\) |
| \(\chi_{3721}(34,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{189}{305}\right)\) | \(e\left(\frac{114}{305}\right)\) | \(e\left(\frac{73}{305}\right)\) | \(e\left(\frac{108}{305}\right)\) | \(e\left(\frac{303}{305}\right)\) | \(e\left(\frac{271}{305}\right)\) | \(e\left(\frac{262}{305}\right)\) | \(e\left(\frac{228}{305}\right)\) | \(e\left(\frac{297}{305}\right)\) | \(e\left(\frac{59}{61}\right)\) |
| \(\chi_{3721}(58,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{305}\right)\) | \(e\left(\frac{193}{305}\right)\) | \(e\left(\frac{46}{305}\right)\) | \(e\left(\frac{231}{305}\right)\) | \(e\left(\frac{216}{305}\right)\) | \(e\left(\frac{12}{305}\right)\) | \(e\left(\frac{69}{305}\right)\) | \(e\left(\frac{81}{305}\right)\) | \(e\left(\frac{254}{305}\right)\) | \(e\left(\frac{33}{61}\right)\) |
| \(\chi_{3721}(70,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{96}{305}\right)\) | \(e\left(\frac{116}{305}\right)\) | \(e\left(\frac{192}{305}\right)\) | \(e\left(\frac{142}{305}\right)\) | \(e\left(\frac{212}{305}\right)\) | \(e\left(\frac{249}{305}\right)\) | \(e\left(\frac{288}{305}\right)\) | \(e\left(\frac{232}{305}\right)\) | \(e\left(\frac{238}{305}\right)\) | \(e\left(\frac{29}{61}\right)\) |
| \(\chi_{3721}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{305}\right)\) | \(e\left(\frac{222}{305}\right)\) | \(e\left(\frac{94}{305}\right)\) | \(e\left(\frac{114}{305}\right)\) | \(e\left(\frac{269}{305}\right)\) | \(e\left(\frac{303}{305}\right)\) | \(e\left(\frac{141}{305}\right)\) | \(e\left(\frac{139}{305}\right)\) | \(e\left(\frac{161}{305}\right)\) | \(e\left(\frac{25}{61}\right)\) |
| \(\chi_{3721}(95,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{114}{305}\right)\) | \(e\left(\frac{214}{305}\right)\) | \(e\left(\frac{228}{305}\right)\) | \(e\left(\frac{283}{305}\right)\) | \(e\left(\frac{23}{305}\right)\) | \(e\left(\frac{86}{305}\right)\) | \(e\left(\frac{37}{305}\right)\) | \(e\left(\frac{123}{305}\right)\) | \(e\left(\frac{92}{305}\right)\) | \(e\left(\frac{23}{61}\right)\) |
| \(\chi_{3721}(119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{263}{305}\right)\) | \(e\left(\frac{178}{305}\right)\) | \(e\left(\frac{221}{305}\right)\) | \(e\left(\frac{281}{305}\right)\) | \(e\left(\frac{136}{305}\right)\) | \(e\left(\frac{177}{305}\right)\) | \(e\left(\frac{179}{305}\right)\) | \(e\left(\frac{51}{305}\right)\) | \(e\left(\frac{239}{305}\right)\) | \(e\left(\frac{14}{61}\right)\) |
| \(\chi_{3721}(131,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{305}\right)\) | \(e\left(\frac{121}{305}\right)\) | \(e\left(\frac{32}{305}\right)\) | \(e\left(\frac{227}{305}\right)\) | \(e\left(\frac{137}{305}\right)\) | \(e\left(\frac{194}{305}\right)\) | \(e\left(\frac{48}{305}\right)\) | \(e\left(\frac{242}{305}\right)\) | \(e\left(\frac{243}{305}\right)\) | \(e\left(\frac{15}{61}\right)\) |
| \(\chi_{3721}(142,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{72}{305}\right)\) | \(e\left(\frac{87}{305}\right)\) | \(e\left(\frac{144}{305}\right)\) | \(e\left(\frac{259}{305}\right)\) | \(e\left(\frac{159}{305}\right)\) | \(e\left(\frac{263}{305}\right)\) | \(e\left(\frac{216}{305}\right)\) | \(e\left(\frac{174}{305}\right)\) | \(e\left(\frac{26}{305}\right)\) | \(e\left(\frac{37}{61}\right)\) |
| \(\chi_{3721}(156,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{305}\right)\) | \(e\left(\frac{9}{305}\right)\) | \(e\left(\frac{78}{305}\right)\) | \(e\left(\frac{153}{305}\right)\) | \(e\left(\frac{48}{305}\right)\) | \(e\left(\frac{206}{305}\right)\) | \(e\left(\frac{117}{305}\right)\) | \(e\left(\frac{18}{305}\right)\) | \(e\left(\frac{192}{305}\right)\) | \(e\left(\frac{48}{61}\right)\) |
| \(\chi_{3721}(180,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{198}{305}\right)\) | \(e\left(\frac{163}{305}\right)\) | \(e\left(\frac{91}{305}\right)\) | \(e\left(\frac{26}{305}\right)\) | \(e\left(\frac{56}{305}\right)\) | \(e\left(\frac{37}{305}\right)\) | \(e\left(\frac{289}{305}\right)\) | \(e\left(\frac{21}{305}\right)\) | \(e\left(\frac{224}{305}\right)\) | \(e\left(\frac{56}{61}\right)\) |
| \(\chi_{3721}(192,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{241}{305}\right)\) | \(e\left(\frac{126}{305}\right)\) | \(e\left(\frac{177}{305}\right)\) | \(e\left(\frac{7}{305}\right)\) | \(e\left(\frac{62}{305}\right)\) | \(e\left(\frac{139}{305}\right)\) | \(e\left(\frac{113}{305}\right)\) | \(e\left(\frac{252}{305}\right)\) | \(e\left(\frac{248}{305}\right)\) | \(e\left(\frac{1}{61}\right)\) |
| \(\chi_{3721}(203,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{305}\right)\) | \(e\left(\frac{257}{305}\right)\) | \(e\left(\frac{194}{305}\right)\) | \(e\left(\frac{99}{305}\right)\) | \(e\left(\frac{49}{305}\right)\) | \(e\left(\frac{223}{305}\right)\) | \(e\left(\frac{291}{305}\right)\) | \(e\left(\frac{209}{305}\right)\) | \(e\left(\frac{196}{305}\right)\) | \(e\left(\frac{49}{61}\right)\) |
| \(\chi_{3721}(217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{269}{305}\right)\) | \(e\left(\frac{109}{305}\right)\) | \(e\left(\frac{233}{305}\right)\) | \(e\left(\frac{23}{305}\right)\) | \(e\left(\frac{73}{305}\right)\) | \(e\left(\frac{21}{305}\right)\) | \(e\left(\frac{197}{305}\right)\) | \(e\left(\frac{218}{305}\right)\) | \(e\left(\frac{292}{305}\right)\) | \(e\left(\frac{12}{61}\right)\) |
| \(\chi_{3721}(241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{305}\right)\) | \(e\left(\frac{148}{305}\right)\) | \(e\left(\frac{266}{305}\right)\) | \(e\left(\frac{76}{305}\right)\) | \(e\left(\frac{281}{305}\right)\) | \(e\left(\frac{202}{305}\right)\) | \(e\left(\frac{94}{305}\right)\) | \(e\left(\frac{296}{305}\right)\) | \(e\left(\frac{209}{305}\right)\) | \(e\left(\frac{37}{61}\right)\) |
| \(\chi_{3721}(253,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{305}\right)\) | \(e\left(\frac{131}{305}\right)\) | \(e\left(\frac{17}{305}\right)\) | \(e\left(\frac{92}{305}\right)\) | \(e\left(\frac{292}{305}\right)\) | \(e\left(\frac{84}{305}\right)\) | \(e\left(\frac{178}{305}\right)\) | \(e\left(\frac{262}{305}\right)\) | \(e\left(\frac{253}{305}\right)\) | \(e\left(\frac{48}{61}\right)\) |
| \(\chi_{3721}(278,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{194}{305}\right)\) | \(e\left(\frac{209}{305}\right)\) | \(e\left(\frac{83}{305}\right)\) | \(e\left(\frac{198}{305}\right)\) | \(e\left(\frac{98}{305}\right)\) | \(e\left(\frac{141}{305}\right)\) | \(e\left(\frac{277}{305}\right)\) | \(e\left(\frac{113}{305}\right)\) | \(e\left(\frac{87}{305}\right)\) | \(e\left(\frac{37}{61}\right)\) |
| \(\chi_{3721}(302,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{305}\right)\) | \(e\left(\frac{133}{305}\right)\) | \(e\left(\frac{136}{305}\right)\) | \(e\left(\frac{126}{305}\right)\) | \(e\left(\frac{201}{305}\right)\) | \(e\left(\frac{62}{305}\right)\) | \(e\left(\frac{204}{305}\right)\) | \(e\left(\frac{266}{305}\right)\) | \(e\left(\frac{194}{305}\right)\) | \(e\left(\frac{18}{61}\right)\) |
| \(\chi_{3721}(314,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{305}\right)\) | \(e\left(\frac{136}{305}\right)\) | \(e\left(\frac{162}{305}\right)\) | \(e\left(\frac{177}{305}\right)\) | \(e\left(\frac{217}{305}\right)\) | \(e\left(\frac{29}{305}\right)\) | \(e\left(\frac{243}{305}\right)\) | \(e\left(\frac{272}{305}\right)\) | \(e\left(\frac{258}{305}\right)\) | \(e\left(\frac{34}{61}\right)\) |
| \(\chi_{3721}(325,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{147}{305}\right)\) | \(e\left(\frac{292}{305}\right)\) | \(e\left(\frac{294}{305}\right)\) | \(e\left(\frac{84}{305}\right)\) | \(e\left(\frac{134}{305}\right)\) | \(e\left(\frac{143}{305}\right)\) | \(e\left(\frac{136}{305}\right)\) | \(e\left(\frac{279}{305}\right)\) | \(e\left(\frac{231}{305}\right)\) | \(e\left(\frac{12}{61}\right)\) |
| \(\chi_{3721}(339,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{305}\right)\) | \(e\left(\frac{4}{305}\right)\) | \(e\left(\frac{238}{305}\right)\) | \(e\left(\frac{68}{305}\right)\) | \(e\left(\frac{123}{305}\right)\) | \(e\left(\frac{261}{305}\right)\) | \(e\left(\frac{52}{305}\right)\) | \(e\left(\frac{8}{305}\right)\) | \(e\left(\frac{187}{305}\right)\) | \(e\left(\frac{1}{61}\right)\) |
| \(\chi_{3721}(363,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{305}\right)\) | \(e\left(\frac{118}{305}\right)\) | \(e\left(\frac{6}{305}\right)\) | \(e\left(\frac{176}{305}\right)\) | \(e\left(\frac{121}{305}\right)\) | \(e\left(\frac{227}{305}\right)\) | \(e\left(\frac{9}{305}\right)\) | \(e\left(\frac{236}{305}\right)\) | \(e\left(\frac{179}{305}\right)\) | \(e\left(\frac{60}{61}\right)\) |
| \(\chi_{3721}(375,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{305}\right)\) | \(e\left(\frac{141}{305}\right)\) | \(e\left(\frac{2}{305}\right)\) | \(e\left(\frac{262}{305}\right)\) | \(e\left(\frac{142}{305}\right)\) | \(e\left(\frac{279}{305}\right)\) | \(e\left(\frac{3}{305}\right)\) | \(e\left(\frac{282}{305}\right)\) | \(e\left(\frac{263}{305}\right)\) | \(e\left(\frac{20}{61}\right)\) |
| \(\chi_{3721}(386,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{172}{305}\right)\) | \(e\left(\frac{157}{305}\right)\) | \(e\left(\frac{39}{305}\right)\) | \(e\left(\frac{229}{305}\right)\) | \(e\left(\frac{24}{305}\right)\) | \(e\left(\frac{103}{305}\right)\) | \(e\left(\frac{211}{305}\right)\) | \(e\left(\frac{9}{305}\right)\) | \(e\left(\frac{96}{305}\right)\) | \(e\left(\frac{24}{61}\right)\) |
| \(\chi_{3721}(400,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{305}\right)\) | \(e\left(\frac{104}{305}\right)\) | \(e\left(\frac{88}{305}\right)\) | \(e\left(\frac{243}{305}\right)\) | \(e\left(\frac{148}{305}\right)\) | \(e\left(\frac{76}{305}\right)\) | \(e\left(\frac{132}{305}\right)\) | \(e\left(\frac{208}{305}\right)\) | \(e\left(\frac{287}{305}\right)\) | \(e\left(\frac{26}{61}\right)\) |
| \(\chi_{3721}(424,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{243}{305}\right)\) | \(e\left(\frac{103}{305}\right)\) | \(e\left(\frac{181}{305}\right)\) | \(e\left(\frac{226}{305}\right)\) | \(e\left(\frac{41}{305}\right)\) | \(e\left(\frac{87}{305}\right)\) | \(e\left(\frac{119}{305}\right)\) | \(e\left(\frac{206}{305}\right)\) | \(e\left(\frac{164}{305}\right)\) | \(e\left(\frac{41}{61}\right)\) |
| \(\chi_{3721}(436,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{226}{305}\right)\) | \(e\left(\frac{146}{305}\right)\) | \(e\left(\frac{147}{305}\right)\) | \(e\left(\frac{42}{305}\right)\) | \(e\left(\frac{67}{305}\right)\) | \(e\left(\frac{224}{305}\right)\) | \(e\left(\frac{68}{305}\right)\) | \(e\left(\frac{292}{305}\right)\) | \(e\left(\frac{268}{305}\right)\) | \(e\left(\frac{6}{61}\right)\) |
| \(\chi_{3721}(447,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{305}\right)\) | \(e\left(\frac{22}{305}\right)\) | \(e\left(\frac{89}{305}\right)\) | \(e\left(\frac{69}{305}\right)\) | \(e\left(\frac{219}{305}\right)\) | \(e\left(\frac{63}{305}\right)\) | \(e\left(\frac{286}{305}\right)\) | \(e\left(\frac{44}{305}\right)\) | \(e\left(\frac{266}{305}\right)\) | \(e\left(\frac{36}{61}\right)\) |
| \(\chi_{3721}(461,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{274}{305}\right)\) | \(e\left(\frac{204}{305}\right)\) | \(e\left(\frac{243}{305}\right)\) | \(e\left(\frac{113}{305}\right)\) | \(e\left(\frac{173}{305}\right)\) | \(e\left(\frac{196}{305}\right)\) | \(e\left(\frac{212}{305}\right)\) | \(e\left(\frac{103}{305}\right)\) | \(e\left(\frac{82}{305}\right)\) | \(e\left(\frac{51}{61}\right)\) |
| \(\chi_{3721}(485,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{178}{305}\right)\) | \(e\left(\frac{88}{305}\right)\) | \(e\left(\frac{51}{305}\right)\) | \(e\left(\frac{276}{305}\right)\) | \(e\left(\frac{266}{305}\right)\) | \(e\left(\frac{252}{305}\right)\) | \(e\left(\frac{229}{305}\right)\) | \(e\left(\frac{176}{305}\right)\) | \(e\left(\frac{149}{305}\right)\) | \(e\left(\frac{22}{61}\right)\) |