from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6275, base_ring=CyclotomicField(500))
M = H._module
chi = DirichletCharacter(H, M([225,414]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,6275))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6275\) | |
| Conductor: | \(6275\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(500\) | 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: | $\Q(\zeta_{500})$ |
| Fixed field: | Number field defined by a degree 500 polynomial (not computed) |
First 31 of 200 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6275}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{100}\right)\) | \(e\left(\frac{199}{500}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{107}{250}\right)\) | \(e\left(\frac{297}{500}\right)\) | \(e\left(\frac{9}{100}\right)\) | \(e\left(\frac{199}{250}\right)\) | \(e\left(\frac{227}{250}\right)\) | \(e\left(\frac{229}{500}\right)\) | \(e\left(\frac{123}{500}\right)\) |
| \(\chi_{6275}(42,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{100}\right)\) | \(e\left(\frac{243}{500}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{49}{250}\right)\) | \(e\left(\frac{129}{500}\right)\) | \(e\left(\frac{13}{100}\right)\) | \(e\left(\frac{243}{250}\right)\) | \(e\left(\frac{139}{250}\right)\) | \(e\left(\frac{453}{500}\right)\) | \(e\left(\frac{311}{500}\right)\) |
| \(\chi_{6275}(167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{100}\right)\) | \(e\left(\frac{263}{500}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{159}{250}\right)\) | \(e\left(\frac{189}{500}\right)\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{13}{250}\right)\) | \(e\left(\frac{99}{250}\right)\) | \(e\left(\frac{373}{500}\right)\) | \(e\left(\frac{351}{500}\right)\) |
| \(\chi_{6275}(183,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{100}\right)\) | \(e\left(\frac{433}{500}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{219}{250}\right)\) | \(e\left(\frac{199}{500}\right)\) | \(e\left(\frac{3}{100}\right)\) | \(e\left(\frac{183}{250}\right)\) | \(e\left(\frac{9}{250}\right)\) | \(e\left(\frac{443}{500}\right)\) | \(e\left(\frac{441}{500}\right)\) |
| \(\chi_{6275}(203,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{317}{500}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{31}{250}\right)\) | \(e\left(\frac{51}{500}\right)\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{67}{250}\right)\) | \(e\left(\frac{241}{250}\right)\) | \(e\left(\frac{307}{500}\right)\) | \(e\left(\frac{309}{500}\right)\) |
| \(\chi_{6275}(223,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{100}\right)\) | \(e\left(\frac{401}{500}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{193}{250}\right)\) | \(e\left(\frac{3}{500}\right)\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{151}{250}\right)\) | \(e\left(\frac{73}{250}\right)\) | \(e\left(\frac{371}{500}\right)\) | \(e\left(\frac{77}{500}\right)\) |
| \(\chi_{6275}(238,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{100}\right)\) | \(e\left(\frac{149}{500}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{207}{250}\right)\) | \(e\left(\frac{147}{500}\right)\) | \(e\left(\frac{59}{100}\right)\) | \(e\left(\frac{149}{250}\right)\) | \(e\left(\frac{77}{250}\right)\) | \(e\left(\frac{179}{500}\right)\) | \(e\left(\frac{273}{500}\right)\) |
| \(\chi_{6275}(288,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{100}\right)\) | \(e\left(\frac{449}{500}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{107}{250}\right)\) | \(e\left(\frac{47}{500}\right)\) | \(e\left(\frac{59}{100}\right)\) | \(e\left(\frac{199}{250}\right)\) | \(e\left(\frac{227}{250}\right)\) | \(e\left(\frac{479}{500}\right)\) | \(e\left(\frac{373}{500}\right)\) |
| \(\chi_{6275}(308,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{100}\right)\) | \(e\left(\frac{253}{500}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{229}{250}\right)\) | \(e\left(\frac{159}{500}\right)\) | \(e\left(\frac{23}{100}\right)\) | \(e\left(\frac{3}{250}\right)\) | \(e\left(\frac{119}{250}\right)\) | \(e\left(\frac{163}{500}\right)\) | \(e\left(\frac{81}{500}\right)\) |
| \(\chi_{6275}(327,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{100}\right)\) | \(e\left(\frac{431}{500}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{233}{250}\right)\) | \(e\left(\frac{93}{500}\right)\) | \(e\left(\frac{21}{100}\right)\) | \(e\left(\frac{181}{250}\right)\) | \(e\left(\frac{13}{250}\right)\) | \(e\left(\frac{1}{500}\right)\) | \(e\left(\frac{387}{500}\right)\) |
| \(\chi_{6275}(333,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{100}\right)\) | \(e\left(\frac{173}{500}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{39}{250}\right)\) | \(e\left(\frac{419}{500}\right)\) | \(e\left(\frac{43}{100}\right)\) | \(e\left(\frac{173}{250}\right)\) | \(e\left(\frac{29}{250}\right)\) | \(e\left(\frac{483}{500}\right)\) | \(e\left(\frac{421}{500}\right)\) |
| \(\chi_{6275}(347,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{100}\right)\) | \(e\left(\frac{87}{500}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{141}{250}\right)\) | \(e\left(\frac{361}{500}\right)\) | \(e\left(\frac{17}{100}\right)\) | \(e\left(\frac{87}{250}\right)\) | \(e\left(\frac{201}{250}\right)\) | \(e\left(\frac{477}{500}\right)\) | \(e\left(\frac{99}{500}\right)\) |
| \(\chi_{6275}(378,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{217}{500}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{231}{250}\right)\) | \(e\left(\frac{251}{500}\right)\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{217}{250}\right)\) | \(e\left(\frac{191}{250}\right)\) | \(e\left(\frac{207}{500}\right)\) | \(e\left(\frac{109}{500}\right)\) |
| \(\chi_{6275}(417,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{303}{500}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{129}{250}\right)\) | \(e\left(\frac{309}{500}\right)\) | \(e\left(\frac{73}{100}\right)\) | \(e\left(\frac{53}{250}\right)\) | \(e\left(\frac{19}{250}\right)\) | \(e\left(\frac{213}{500}\right)\) | \(e\left(\frac{431}{500}\right)\) |
| \(\chi_{6275}(437,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{100}\right)\) | \(e\left(\frac{59}{500}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{87}{250}\right)\) | \(e\left(\frac{377}{500}\right)\) | \(e\left(\frac{69}{100}\right)\) | \(e\left(\frac{59}{250}\right)\) | \(e\left(\frac{7}{250}\right)\) | \(e\left(\frac{289}{500}\right)\) | \(e\left(\frac{343}{500}\right)\) |
| \(\chi_{6275}(453,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{100}\right)\) | \(e\left(\frac{97}{500}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{71}{250}\right)\) | \(e\left(\frac{391}{500}\right)\) | \(e\left(\frac{27}{100}\right)\) | \(e\left(\frac{97}{250}\right)\) | \(e\left(\frac{181}{250}\right)\) | \(e\left(\frac{187}{500}\right)\) | \(e\left(\frac{369}{500}\right)\) |
| \(\chi_{6275}(622,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{100}\right)\) | \(e\left(\frac{207}{500}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{51}{250}\right)\) | \(e\left(\frac{221}{500}\right)\) | \(e\left(\frac{37}{100}\right)\) | \(e\left(\frac{207}{250}\right)\) | \(e\left(\frac{211}{250}\right)\) | \(e\left(\frac{497}{500}\right)\) | \(e\left(\frac{339}{500}\right)\) |
| \(\chi_{6275}(678,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{100}\right)\) | \(e\left(\frac{57}{500}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{101}{250}\right)\) | \(e\left(\frac{271}{500}\right)\) | \(e\left(\frac{87}{100}\right)\) | \(e\left(\frac{57}{250}\right)\) | \(e\left(\frac{11}{250}\right)\) | \(e\left(\frac{347}{500}\right)\) | \(e\left(\frac{289}{500}\right)\) |
| \(\chi_{6275}(687,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{100}\right)\) | \(e\left(\frac{359}{500}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{237}{250}\right)\) | \(e\left(\frac{277}{500}\right)\) | \(e\left(\frac{69}{100}\right)\) | \(e\left(\frac{109}{250}\right)\) | \(e\left(\frac{157}{250}\right)\) | \(e\left(\frac{89}{500}\right)\) | \(e\left(\frac{443}{500}\right)\) |
| \(\chi_{6275}(688,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{100}\right)\) | \(e\left(\frac{309}{500}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{87}{250}\right)\) | \(e\left(\frac{127}{500}\right)\) | \(e\left(\frac{19}{100}\right)\) | \(e\left(\frac{59}{250}\right)\) | \(e\left(\frac{7}{250}\right)\) | \(e\left(\frac{39}{500}\right)\) | \(e\left(\frac{93}{500}\right)\) |
| \(\chi_{6275}(722,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{100}\right)\) | \(e\left(\frac{427}{500}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{11}{250}\right)\) | \(e\left(\frac{381}{500}\right)\) | \(e\left(\frac{57}{100}\right)\) | \(e\left(\frac{177}{250}\right)\) | \(e\left(\frac{21}{250}\right)\) | \(e\left(\frac{117}{500}\right)\) | \(e\left(\frac{279}{500}\right)\) |
| \(\chi_{6275}(797,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{100}\right)\) | \(e\left(\frac{267}{500}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{131}{250}\right)\) | \(e\left(\frac{401}{500}\right)\) | \(e\left(\frac{97}{100}\right)\) | \(e\left(\frac{17}{250}\right)\) | \(e\left(\frac{91}{250}\right)\) | \(e\left(\frac{257}{500}\right)\) | \(e\left(\frac{459}{500}\right)\) |
| \(\chi_{6275}(823,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{100}\right)\) | \(e\left(\frac{41}{500}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{213}{250}\right)\) | \(e\left(\frac{423}{500}\right)\) | \(e\left(\frac{31}{100}\right)\) | \(e\left(\frac{41}{250}\right)\) | \(e\left(\frac{43}{250}\right)\) | \(e\left(\frac{311}{500}\right)\) | \(e\left(\frac{357}{500}\right)\) |
| \(\chi_{6275}(848,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{100}\right)\) | \(e\left(\frac{301}{500}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{143}{250}\right)\) | \(e\left(\frac{203}{500}\right)\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{51}{250}\right)\) | \(e\left(\frac{23}{250}\right)\) | \(e\left(\frac{271}{500}\right)\) | \(e\left(\frac{377}{500}\right)\) |
| \(\chi_{6275}(852,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{451}{500}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{93}{250}\right)\) | \(e\left(\frac{153}{500}\right)\) | \(e\left(\frac{41}{100}\right)\) | \(e\left(\frac{201}{250}\right)\) | \(e\left(\frac{223}{250}\right)\) | \(e\left(\frac{421}{500}\right)\) | \(e\left(\frac{427}{500}\right)\) |
| \(\chi_{6275}(862,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{100}\right)\) | \(e\left(\frac{319}{500}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{17}{250}\right)\) | \(e\left(\frac{157}{500}\right)\) | \(e\left(\frac{29}{100}\right)\) | \(e\left(\frac{69}{250}\right)\) | \(e\left(\frac{237}{250}\right)\) | \(e\left(\frac{249}{500}\right)\) | \(e\left(\frac{363}{500}\right)\) |
| \(\chi_{6275}(898,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{100}\right)\) | \(e\left(\frac{241}{500}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{63}{250}\right)\) | \(e\left(\frac{23}{500}\right)\) | \(e\left(\frac{31}{100}\right)\) | \(e\left(\frac{241}{250}\right)\) | \(e\left(\frac{143}{250}\right)\) | \(e\left(\frac{11}{500}\right)\) | \(e\left(\frac{257}{500}\right)\) |
| \(\chi_{6275}(903,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{100}\right)\) | \(e\left(\frac{77}{500}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{211}{250}\right)\) | \(e\left(\frac{331}{500}\right)\) | \(e\left(\frac{7}{100}\right)\) | \(e\left(\frac{77}{250}\right)\) | \(e\left(\frac{221}{250}\right)\) | \(e\left(\frac{267}{500}\right)\) | \(e\left(\frac{329}{500}\right)\) |
| \(\chi_{6275}(938,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{100}\right)\) | \(e\left(\frac{109}{500}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{237}{250}\right)\) | \(e\left(\frac{27}{500}\right)\) | \(e\left(\frac{19}{100}\right)\) | \(e\left(\frac{109}{250}\right)\) | \(e\left(\frac{157}{250}\right)\) | \(e\left(\frac{339}{500}\right)\) | \(e\left(\frac{193}{500}\right)\) |
| \(\chi_{6275}(977,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{100}\right)\) | \(e\left(\frac{211}{500}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{23}{250}\right)\) | \(e\left(\frac{433}{500}\right)\) | \(e\left(\frac{1}{100}\right)\) | \(e\left(\frac{211}{250}\right)\) | \(e\left(\frac{203}{250}\right)\) | \(e\left(\frac{381}{500}\right)\) | \(e\left(\frac{447}{500}\right)\) |
| \(\chi_{6275}(983,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{100}\right)\) | \(e\left(\frac{473}{500}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{189}{250}\right)\) | \(e\left(\frac{319}{500}\right)\) | \(e\left(\frac{43}{100}\right)\) | \(e\left(\frac{223}{250}\right)\) | \(e\left(\frac{179}{250}\right)\) | \(e\left(\frac{283}{500}\right)\) | \(e\left(\frac{21}{500}\right)\) |