from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(36000, base_ring=CyclotomicField(600))
M = H._module
chi = DirichletCharacter(H, M([0,225,100,372]))
chi.galois_orbit()
[g,chi] = znchar(Mod(29,36000))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(36000\) | |
| Conductor: | \(36000\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(600\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{600})$ |
| Fixed field: | Number field defined by a degree 600 polynomial (not computed) |
First 31 of 160 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{36000}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{97}{600}\right)\) | \(e\left(\frac{83}{600}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{157}{200}\right)\) | \(e\left(\frac{91}{300}\right)\) | \(e\left(\frac{439}{600}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{71}{200}\right)\) | \(e\left(\frac{109}{300}\right)\) |
| \(\chi_{36000}(389,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{91}{600}\right)\) | \(e\left(\frac{449}{600}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{71}{200}\right)\) | \(e\left(\frac{73}{300}\right)\) | \(e\left(\frac{517}{600}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{13}{200}\right)\) | \(e\left(\frac{127}{300}\right)\) |
| \(\chi_{36000}(509,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{209}{600}\right)\) | \(e\left(\frac{451}{600}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{29}{200}\right)\) | \(e\left(\frac{227}{300}\right)\) | \(e\left(\frac{383}{600}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{87}{200}\right)\) | \(e\left(\frac{173}{300}\right)\) |
| \(\chi_{36000}(869,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{323}{600}\right)\) | \(e\left(\frac{97}{600}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{63}{200}\right)\) | \(e\left(\frac{269}{300}\right)\) | \(e\left(\frac{101}{600}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{189}{200}\right)\) | \(e\left(\frac{131}{300}\right)\) |
| \(\chi_{36000}(1109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{199}{600}\right)\) | \(e\left(\frac{461}{600}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{19}{200}\right)\) | \(e\left(\frac{97}{300}\right)\) | \(e\left(\frac{313}{600}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{57}{200}\right)\) | \(e\left(\frac{103}{300}\right)\) |
| \(\chi_{36000}(1229,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{77}{600}\right)\) | \(e\left(\frac{103}{600}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{137}{200}\right)\) | \(e\left(\frac{131}{300}\right)\) | \(e\left(\frac{299}{600}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{11}{200}\right)\) | \(e\left(\frac{269}{300}\right)\) |
| \(\chi_{36000}(1469,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{313}{600}\right)\) | \(e\left(\frac{107}{600}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{53}{200}\right)\) | \(e\left(\frac{139}{300}\right)\) | \(e\left(\frac{31}{600}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{159}{200}\right)\) | \(e\left(\frac{61}{300}\right)\) |
| \(\chi_{36000}(1589,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{71}{600}\right)\) | \(e\left(\frac{469}{600}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{51}{200}\right)\) | \(e\left(\frac{113}{300}\right)\) | \(e\left(\frac{377}{600}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{153}{200}\right)\) | \(e\left(\frac{287}{300}\right)\) |
| \(\chi_{36000}(1829,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{67}{600}\right)\) | \(e\left(\frac{113}{600}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{127}{200}\right)\) | \(e\left(\frac{1}{300}\right)\) | \(e\left(\frac{229}{600}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{181}{200}\right)\) | \(e\left(\frac{199}{300}\right)\) |
| \(\chi_{36000}(2189,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{61}{600}\right)\) | \(e\left(\frac{479}{600}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{41}{200}\right)\) | \(e\left(\frac{283}{300}\right)\) | \(e\left(\frac{307}{600}\right)\) | \(e\left(\frac{16}{75}\right)\) | \(e\left(\frac{123}{200}\right)\) | \(e\left(\frac{217}{300}\right)\) |
| \(\chi_{36000}(2309,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{179}{600}\right)\) | \(e\left(\frac{481}{600}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{199}{200}\right)\) | \(e\left(\frac{137}{300}\right)\) | \(e\left(\frac{173}{600}\right)\) | \(e\left(\frac{74}{75}\right)\) | \(e\left(\frac{197}{200}\right)\) | \(e\left(\frac{263}{300}\right)\) |
| \(\chi_{36000}(2669,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{293}{600}\right)\) | \(e\left(\frac{127}{600}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{33}{200}\right)\) | \(e\left(\frac{179}{300}\right)\) | \(e\left(\frac{491}{600}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{99}{200}\right)\) | \(e\left(\frac{221}{300}\right)\) |
| \(\chi_{36000}(2909,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{169}{600}\right)\) | \(e\left(\frac{491}{600}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{189}{200}\right)\) | \(e\left(\frac{7}{300}\right)\) | \(e\left(\frac{103}{600}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{167}{200}\right)\) | \(e\left(\frac{193}{300}\right)\) |
| \(\chi_{36000}(3029,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{47}{600}\right)\) | \(e\left(\frac{133}{600}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{107}{200}\right)\) | \(e\left(\frac{41}{300}\right)\) | \(e\left(\frac{89}{600}\right)\) | \(e\left(\frac{32}{75}\right)\) | \(e\left(\frac{121}{200}\right)\) | \(e\left(\frac{59}{300}\right)\) |
| \(\chi_{36000}(3269,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{283}{600}\right)\) | \(e\left(\frac{137}{600}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{23}{200}\right)\) | \(e\left(\frac{49}{300}\right)\) | \(e\left(\frac{421}{600}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{69}{200}\right)\) | \(e\left(\frac{151}{300}\right)\) |
| \(\chi_{36000}(3389,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{41}{600}\right)\) | \(e\left(\frac{499}{600}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{21}{200}\right)\) | \(e\left(\frac{23}{300}\right)\) | \(e\left(\frac{167}{600}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{63}{200}\right)\) | \(e\left(\frac{77}{300}\right)\) |
| \(\chi_{36000}(3629,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{37}{600}\right)\) | \(e\left(\frac{143}{600}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{97}{200}\right)\) | \(e\left(\frac{211}{300}\right)\) | \(e\left(\frac{19}{600}\right)\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{91}{200}\right)\) | \(e\left(\frac{289}{300}\right)\) |
| \(\chi_{36000}(3989,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{31}{600}\right)\) | \(e\left(\frac{509}{600}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{11}{200}\right)\) | \(e\left(\frac{193}{300}\right)\) | \(e\left(\frac{97}{600}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{33}{200}\right)\) | \(e\left(\frac{7}{300}\right)\) |
| \(\chi_{36000}(4109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{149}{600}\right)\) | \(e\left(\frac{511}{600}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{169}{200}\right)\) | \(e\left(\frac{47}{300}\right)\) | \(e\left(\frac{563}{600}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{107}{200}\right)\) | \(e\left(\frac{53}{300}\right)\) |
| \(\chi_{36000}(4469,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{263}{600}\right)\) | \(e\left(\frac{157}{600}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{3}{200}\right)\) | \(e\left(\frac{89}{300}\right)\) | \(e\left(\frac{281}{600}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{9}{200}\right)\) | \(e\left(\frac{11}{300}\right)\) |
| \(\chi_{36000}(4709,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{139}{600}\right)\) | \(e\left(\frac{521}{600}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{159}{200}\right)\) | \(e\left(\frac{217}{300}\right)\) | \(e\left(\frac{493}{600}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{77}{200}\right)\) | \(e\left(\frac{283}{300}\right)\) |
| \(\chi_{36000}(4829,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{600}\right)\) | \(e\left(\frac{163}{600}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{77}{200}\right)\) | \(e\left(\frac{251}{300}\right)\) | \(e\left(\frac{479}{600}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{31}{200}\right)\) | \(e\left(\frac{149}{300}\right)\) |
| \(\chi_{36000}(5069,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{253}{600}\right)\) | \(e\left(\frac{167}{600}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{193}{200}\right)\) | \(e\left(\frac{259}{300}\right)\) | \(e\left(\frac{211}{600}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{179}{200}\right)\) | \(e\left(\frac{241}{300}\right)\) |
| \(\chi_{36000}(5189,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{11}{600}\right)\) | \(e\left(\frac{529}{600}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{191}{200}\right)\) | \(e\left(\frac{233}{300}\right)\) | \(e\left(\frac{557}{600}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{173}{200}\right)\) | \(e\left(\frac{167}{300}\right)\) |
| \(\chi_{36000}(5429,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{600}\right)\) | \(e\left(\frac{173}{600}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{67}{200}\right)\) | \(e\left(\frac{121}{300}\right)\) | \(e\left(\frac{409}{600}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{1}{200}\right)\) | \(e\left(\frac{79}{300}\right)\) |
| \(\chi_{36000}(5789,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{600}\right)\) | \(e\left(\frac{539}{600}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{181}{200}\right)\) | \(e\left(\frac{103}{300}\right)\) | \(e\left(\frac{487}{600}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{143}{200}\right)\) | \(e\left(\frac{97}{300}\right)\) |
| \(\chi_{36000}(5909,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{119}{600}\right)\) | \(e\left(\frac{541}{600}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{139}{200}\right)\) | \(e\left(\frac{257}{300}\right)\) | \(e\left(\frac{353}{600}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{17}{200}\right)\) | \(e\left(\frac{143}{300}\right)\) |
| \(\chi_{36000}(6269,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{233}{600}\right)\) | \(e\left(\frac{187}{600}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{173}{200}\right)\) | \(e\left(\frac{299}{300}\right)\) | \(e\left(\frac{71}{600}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{119}{200}\right)\) | \(e\left(\frac{101}{300}\right)\) |
| \(\chi_{36000}(6509,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{109}{600}\right)\) | \(e\left(\frac{551}{600}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{129}{200}\right)\) | \(e\left(\frac{127}{300}\right)\) | \(e\left(\frac{283}{600}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{187}{200}\right)\) | \(e\left(\frac{73}{300}\right)\) |
| \(\chi_{36000}(6629,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{587}{600}\right)\) | \(e\left(\frac{193}{600}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{47}{200}\right)\) | \(e\left(\frac{161}{300}\right)\) | \(e\left(\frac{269}{600}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{141}{200}\right)\) | \(e\left(\frac{239}{300}\right)\) |
| \(\chi_{36000}(6869,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{223}{600}\right)\) | \(e\left(\frac{197}{600}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{163}{200}\right)\) | \(e\left(\frac{169}{300}\right)\) | \(e\left(\frac{1}{600}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{89}{200}\right)\) | \(e\left(\frac{31}{300}\right)\) |