from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3721, base_ring=CyclotomicField(610))
M = H._module
chi = DirichletCharacter(H, M([481]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,3721))
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: | \(610\) | 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 610 polynomial (not computed) |
First 31 of 240 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3721}(3,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{481}{610}\right)\) | \(e\left(\frac{208}{305}\right)\) | \(e\left(\frac{176}{305}\right)\) | \(e\left(\frac{181}{305}\right)\) | \(e\left(\frac{287}{610}\right)\) | \(e\left(\frac{609}{610}\right)\) | \(e\left(\frac{223}{610}\right)\) | \(e\left(\frac{111}{305}\right)\) | \(e\left(\frac{233}{610}\right)\) | \(e\left(\frac{43}{122}\right)\) |
| \(\chi_{3721}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{610}\right)\) | \(e\left(\frac{14}{305}\right)\) | \(e\left(\frac{223}{305}\right)\) | \(e\left(\frac{238}{305}\right)\) | \(e\left(\frac{251}{610}\right)\) | \(e\left(\frac{607}{610}\right)\) | \(e\left(\frac{59}{610}\right)\) | \(e\left(\frac{28}{305}\right)\) | \(e\left(\frac{89}{610}\right)\) | \(e\left(\frac{7}{122}\right)\) |
| \(\chi_{3721}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{299}{610}\right)\) | \(e\left(\frac{187}{305}\right)\) | \(e\left(\frac{299}{305}\right)\) | \(e\left(\frac{129}{305}\right)\) | \(e\left(\frac{63}{610}\right)\) | \(e\left(\frac{461}{610}\right)\) | \(e\left(\frac{287}{610}\right)\) | \(e\left(\frac{69}{305}\right)\) | \(e\left(\frac{557}{610}\right)\) | \(e\left(\frac{63}{122}\right)\) |
| \(\chi_{3721}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{207}{610}\right)\) | \(e\left(\frac{106}{305}\right)\) | \(e\left(\frac{207}{305}\right)\) | \(e\left(\frac{277}{305}\right)\) | \(e\left(\frac{419}{610}\right)\) | \(e\left(\frac{413}{610}\right)\) | \(e\left(\frac{11}{610}\right)\) | \(e\left(\frac{212}{305}\right)\) | \(e\left(\frac{151}{610}\right)\) | \(e\left(\frac{53}{122}\right)\) |
| \(\chi_{3721}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{610}\right)\) | \(e\left(\frac{223}{305}\right)\) | \(e\left(\frac{1}{305}\right)\) | \(e\left(\frac{131}{305}\right)\) | \(e\left(\frac{447}{610}\right)\) | \(e\left(\frac{279}{610}\right)\) | \(e\left(\frac{3}{610}\right)\) | \(e\left(\frac{141}{305}\right)\) | \(e\left(\frac{263}{610}\right)\) | \(e\left(\frac{81}{122}\right)\) |
| \(\chi_{3721}(88,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{373}{610}\right)\) | \(e\left(\frac{219}{305}\right)\) | \(e\left(\frac{68}{305}\right)\) | \(e\left(\frac{63}{305}\right)\) | \(e\left(\frac{201}{610}\right)\) | \(e\left(\frac{367}{610}\right)\) | \(e\left(\frac{509}{610}\right)\) | \(e\left(\frac{133}{305}\right)\) | \(e\left(\frac{499}{610}\right)\) | \(e\left(\frac{79}{122}\right)\) |
| \(\chi_{3721}(102,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{249}{610}\right)\) | \(e\left(\frac{17}{305}\right)\) | \(e\left(\frac{249}{305}\right)\) | \(e\left(\frac{289}{305}\right)\) | \(e\left(\frac{283}{610}\right)\) | \(e\left(\frac{541}{610}\right)\) | \(e\left(\frac{137}{610}\right)\) | \(e\left(\frac{34}{305}\right)\) | \(e\left(\frac{217}{610}\right)\) | \(e\left(\frac{39}{122}\right)\) |
| \(\chi_{3721}(113,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{367}{610}\right)\) | \(e\left(\frac{101}{305}\right)\) | \(e\left(\frac{62}{305}\right)\) | \(e\left(\frac{192}{305}\right)\) | \(e\left(\frac{569}{610}\right)\) | \(e\left(\frac{523}{610}\right)\) | \(e\left(\frac{491}{610}\right)\) | \(e\left(\frac{202}{305}\right)\) | \(e\left(\frac{141}{610}\right)\) | \(e\left(\frac{81}{122}\right)\) |
| \(\chi_{3721}(125,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{610}\right)\) | \(e\left(\frac{238}{305}\right)\) | \(e\left(\frac{131}{305}\right)\) | \(e\left(\frac{81}{305}\right)\) | \(e\left(\frac{607}{610}\right)\) | \(e\left(\frac{559}{610}\right)\) | \(e\left(\frac{393}{610}\right)\) | \(e\left(\frac{171}{305}\right)\) | \(e\left(\frac{293}{610}\right)\) | \(e\left(\frac{119}{122}\right)\) |
| \(\chi_{3721}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{523}{610}\right)\) | \(e\left(\frac{119}{305}\right)\) | \(e\left(\frac{218}{305}\right)\) | \(e\left(\frac{193}{305}\right)\) | \(e\left(\frac{151}{610}\right)\) | \(e\left(\frac{127}{610}\right)\) | \(e\left(\frac{349}{610}\right)\) | \(e\left(\frac{238}{305}\right)\) | \(e\left(\frac{299}{610}\right)\) | \(e\left(\frac{29}{122}\right)\) |
| \(\chi_{3721}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{610}\right)\) | \(e\left(\frac{152}{305}\right)\) | \(e\left(\frac{199}{305}\right)\) | \(e\left(\frac{144}{305}\right)\) | \(e\left(\frac{503}{610}\right)\) | \(e\left(\frac{11}{610}\right)\) | \(e\left(\frac{597}{610}\right)\) | \(e\left(\frac{304}{305}\right)\) | \(e\left(\frac{487}{610}\right)\) | \(e\left(\frac{15}{122}\right)\) |
| \(\chi_{3721}(174,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{527}{610}\right)\) | \(e\left(\frac{96}{305}\right)\) | \(e\left(\frac{222}{305}\right)\) | \(e\left(\frac{107}{305}\right)\) | \(e\left(\frac{109}{610}\right)\) | \(e\left(\frac{23}{610}\right)\) | \(e\left(\frac{361}{610}\right)\) | \(e\left(\frac{192}{305}\right)\) | \(e\left(\frac{131}{610}\right)\) | \(e\left(\frac{109}{122}\right)\) |
| \(\chi_{3721}(186,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{261}{610}\right)\) | \(e\left(\frac{253}{305}\right)\) | \(e\left(\frac{261}{305}\right)\) | \(e\left(\frac{31}{305}\right)\) | \(e\left(\frac{157}{610}\right)\) | \(e\left(\frac{229}{610}\right)\) | \(e\left(\frac{173}{610}\right)\) | \(e\left(\frac{201}{305}\right)\) | \(e\left(\frac{323}{610}\right)\) | \(e\left(\frac{35}{122}\right)\) |
| \(\chi_{3721}(210,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{610}\right)\) | \(e\left(\frac{19}{305}\right)\) | \(e\left(\frac{63}{305}\right)\) | \(e\left(\frac{18}{305}\right)\) | \(e\left(\frac{101}{610}\right)\) | \(e\left(\frac{497}{610}\right)\) | \(e\left(\frac{189}{610}\right)\) | \(e\left(\frac{38}{305}\right)\) | \(e\left(\frac{99}{610}\right)\) | \(e\left(\frac{101}{122}\right)\) |
| \(\chi_{3721}(224,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{610}\right)\) | \(e\left(\frac{287}{305}\right)\) | \(e\left(\frac{149}{305}\right)\) | \(e\left(\frac{304}{305}\right)\) | \(e\left(\frac{113}{610}\right)\) | \(e\left(\frac{91}{610}\right)\) | \(e\left(\frac{447}{610}\right)\) | \(e\left(\frac{269}{305}\right)\) | \(e\left(\frac{147}{610}\right)\) | \(e\left(\frac{113}{122}\right)\) |
| \(\chi_{3721}(235,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{610}\right)\) | \(e\left(\frac{91}{305}\right)\) | \(e\left(\frac{77}{305}\right)\) | \(e\left(\frac{22}{305}\right)\) | \(e\left(\frac{259}{610}\right)\) | \(e\left(\frac{133}{610}\right)\) | \(e\left(\frac{231}{610}\right)\) | \(e\left(\frac{182}{305}\right)\) | \(e\left(\frac{121}{610}\right)\) | \(e\left(\frac{15}{122}\right)\) |
| \(\chi_{3721}(247,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{391}{610}\right)\) | \(e\left(\frac{268}{305}\right)\) | \(e\left(\frac{86}{305}\right)\) | \(e\left(\frac{286}{305}\right)\) | \(e\left(\frac{317}{610}\right)\) | \(e\left(\frac{509}{610}\right)\) | \(e\left(\frac{563}{610}\right)\) | \(e\left(\frac{231}{305}\right)\) | \(e\left(\frac{353}{610}\right)\) | \(e\left(\frac{73}{122}\right)\) |
| \(\chi_{3721}(271,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{213}{610}\right)\) | \(e\left(\frac{224}{305}\right)\) | \(e\left(\frac{213}{305}\right)\) | \(e\left(\frac{148}{305}\right)\) | \(e\left(\frac{51}{610}\right)\) | \(e\left(\frac{257}{610}\right)\) | \(e\left(\frac{29}{610}\right)\) | \(e\left(\frac{143}{305}\right)\) | \(e\left(\frac{509}{610}\right)\) | \(e\left(\frac{51}{122}\right)\) |
| \(\chi_{3721}(285,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{610}\right)\) | \(e\left(\frac{117}{305}\right)\) | \(e\left(\frac{99}{305}\right)\) | \(e\left(\frac{159}{305}\right)\) | \(e\left(\frac{333}{610}\right)\) | \(e\left(\frac{171}{610}\right)\) | \(e\left(\frac{297}{610}\right)\) | \(e\left(\frac{234}{305}\right)\) | \(e\left(\frac{417}{610}\right)\) | \(e\left(\frac{89}{122}\right)\) |
| \(\chi_{3721}(296,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{237}{610}\right)\) | \(e\left(\frac{86}{305}\right)\) | \(e\left(\frac{237}{305}\right)\) | \(e\left(\frac{242}{305}\right)\) | \(e\left(\frac{409}{610}\right)\) | \(e\left(\frac{243}{610}\right)\) | \(e\left(\frac{101}{610}\right)\) | \(e\left(\frac{172}{305}\right)\) | \(e\left(\frac{111}{610}\right)\) | \(e\left(\frac{43}{122}\right)\) |
| \(\chi_{3721}(308,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{521}{610}\right)\) | \(e\left(\frac{283}{305}\right)\) | \(e\left(\frac{216}{305}\right)\) | \(e\left(\frac{236}{305}\right)\) | \(e\left(\frac{477}{610}\right)\) | \(e\left(\frac{179}{610}\right)\) | \(e\left(\frac{343}{610}\right)\) | \(e\left(\frac{261}{305}\right)\) | \(e\left(\frac{383}{610}\right)\) | \(e\left(\frac{111}{122}\right)\) |
| \(\chi_{3721}(332,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{363}{610}\right)\) | \(e\left(\frac{124}{305}\right)\) | \(e\left(\frac{58}{305}\right)\) | \(e\left(\frac{278}{305}\right)\) | \(e\left(\frac{1}{610}\right)\) | \(e\left(\frac{17}{610}\right)\) | \(e\left(\frac{479}{610}\right)\) | \(e\left(\frac{248}{305}\right)\) | \(e\left(\frac{309}{610}\right)\) | \(e\left(\frac{1}{122}\right)\) |
| \(\chi_{3721}(346,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{610}\right)\) | \(e\left(\frac{252}{305}\right)\) | \(e\left(\frac{49}{305}\right)\) | \(e\left(\frac{14}{305}\right)\) | \(e\left(\frac{553}{610}\right)\) | \(e\left(\frac{251}{610}\right)\) | \(e\left(\frac{147}{610}\right)\) | \(e\left(\frac{199}{305}\right)\) | \(e\left(\frac{77}{610}\right)\) | \(e\left(\frac{65}{122}\right)\) |
| \(\chi_{3721}(357,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{397}{610}\right)\) | \(e\left(\frac{81}{305}\right)\) | \(e\left(\frac{92}{305}\right)\) | \(e\left(\frac{157}{305}\right)\) | \(e\left(\frac{559}{610}\right)\) | \(e\left(\frac{353}{610}\right)\) | \(e\left(\frac{581}{610}\right)\) | \(e\left(\frac{162}{305}\right)\) | \(e\left(\frac{101}{610}\right)\) | \(e\left(\frac{71}{122}\right)\) |
| \(\chi_{3721}(369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{610}\right)\) | \(e\left(\frac{298}{305}\right)\) | \(e\left(\frac{41}{305}\right)\) | \(e\left(\frac{186}{305}\right)\) | \(e\left(\frac{27}{610}\right)\) | \(e\left(\frac{459}{610}\right)\) | \(e\left(\frac{123}{610}\right)\) | \(e\left(\frac{291}{305}\right)\) | \(e\left(\frac{413}{610}\right)\) | \(e\left(\frac{27}{122}\right)\) |
| \(\chi_{3721}(393,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{513}{610}\right)\) | \(e\left(\frac{24}{305}\right)\) | \(e\left(\frac{208}{305}\right)\) | \(e\left(\frac{103}{305}\right)\) | \(e\left(\frac{561}{610}\right)\) | \(e\left(\frac{387}{610}\right)\) | \(e\left(\frac{319}{610}\right)\) | \(e\left(\frac{48}{305}\right)\) | \(e\left(\frac{109}{610}\right)\) | \(e\left(\frac{73}{122}\right)\) |
| \(\chi_{3721}(407,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{609}{610}\right)\) | \(e\left(\frac{82}{305}\right)\) | \(e\left(\frac{304}{305}\right)\) | \(e\left(\frac{174}{305}\right)\) | \(e\left(\frac{163}{610}\right)\) | \(e\left(\frac{331}{610}\right)\) | \(e\left(\frac{607}{610}\right)\) | \(e\left(\frac{164}{305}\right)\) | \(e\left(\frac{347}{610}\right)\) | \(e\left(\frac{41}{122}\right)\) |
| \(\chi_{3721}(418,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{557}{610}\right)\) | \(e\left(\frac{76}{305}\right)\) | \(e\left(\frac{252}{305}\right)\) | \(e\left(\frac{72}{305}\right)\) | \(e\left(\frac{99}{610}\right)\) | \(e\left(\frac{463}{610}\right)\) | \(e\left(\frac{451}{610}\right)\) | \(e\left(\frac{152}{305}\right)\) | \(e\left(\frac{91}{610}\right)\) | \(e\left(\frac{99}{122}\right)\) |
| \(\chi_{3721}(430,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{171}{610}\right)\) | \(e\left(\frac{8}{305}\right)\) | \(e\left(\frac{171}{305}\right)\) | \(e\left(\frac{136}{305}\right)\) | \(e\left(\frac{187}{610}\right)\) | \(e\left(\frac{129}{610}\right)\) | \(e\left(\frac{513}{610}\right)\) | \(e\left(\frac{16}{305}\right)\) | \(e\left(\frac{443}{610}\right)\) | \(e\left(\frac{65}{122}\right)\) |
| \(\chi_{3721}(454,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{610}\right)\) | \(e\left(\frac{229}{305}\right)\) | \(e\left(\frac{53}{305}\right)\) | \(e\left(\frac{233}{305}\right)\) | \(e\left(\frac{511}{610}\right)\) | \(e\left(\frac{147}{610}\right)\) | \(e\left(\frac{159}{610}\right)\) | \(e\left(\frac{153}{305}\right)\) | \(e\left(\frac{519}{610}\right)\) | \(e\left(\frac{23}{122}\right)\) |
| \(\chi_{3721}(468,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{559}{610}\right)\) | \(e\left(\frac{217}{305}\right)\) | \(e\left(\frac{254}{305}\right)\) | \(e\left(\frac{29}{305}\right)\) | \(e\left(\frac{383}{610}\right)\) | \(e\left(\frac{411}{610}\right)\) | \(e\left(\frac{457}{610}\right)\) | \(e\left(\frac{129}{305}\right)\) | \(e\left(\frac{7}{610}\right)\) | \(e\left(\frac{17}{122}\right)\) |