from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(251, base_ring=CyclotomicField(250))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(6,251))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(251\) | |
| Conductor: | \(251\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(250\) | 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: | $\Q(\zeta_{125})$ |
| Fixed field: | Number field defined by a degree 250 polynomial (not computed) |
First 31 of 100 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{251}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{8}{125}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{1}{250}\right)\) | \(e\left(\frac{124}{125}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{16}{125}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{211}{250}\right)\) |
| \(\chi_{251}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{63}{125}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{211}{250}\right)\) | \(e\left(\frac{39}{125}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{1}{125}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{21}{250}\right)\) |
| \(\chi_{251}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{114}{125}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{233}{250}\right)\) | \(e\left(\frac{17}{125}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{163}{250}\right)\) |
| \(\chi_{251}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{11}{125}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{17}{250}\right)\) | \(e\left(\frac{108}{125}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{22}{125}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{87}{250}\right)\) |
| \(\chi_{251}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{54}{125}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{163}{250}\right)\) | \(e\left(\frac{87}{125}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{108}{125}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{143}{250}\right)\) |
| \(\chi_{251}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{18}{125}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{221}{250}\right)\) | \(e\left(\frac{29}{125}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{36}{125}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{131}{250}\right)\) |
| \(\chi_{251}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{86}{125}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{167}{250}\right)\) | \(e\left(\frac{83}{125}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{47}{125}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{237}{250}\right)\) |
| \(\chi_{251}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{39}{125}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{83}{250}\right)\) | \(e\left(\frac{42}{125}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{78}{125}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{13}{250}\right)\) |
| \(\chi_{251}(30,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{48}{125}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{131}{250}\right)\) | \(e\left(\frac{119}{125}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{96}{125}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{141}{250}\right)\) |
| \(\chi_{251}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{66}{125}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{227}{250}\right)\) | \(e\left(\frac{23}{125}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{7}{125}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{147}{250}\right)\) |
| \(\chi_{251}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{97}{125}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{59}{250}\right)\) | \(e\left(\frac{66}{125}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{69}{125}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{199}{250}\right)\) |
| \(\chi_{251}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{31}{125}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{207}{250}\right)\) | \(e\left(\frac{43}{125}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{62}{125}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{177}{250}\right)\) |
| \(\chi_{251}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{117}{125}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{249}{250}\right)\) | \(e\left(\frac{1}{125}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{109}{125}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{39}{250}\right)\) |
| \(\chi_{251}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{101}{125}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{247}{250}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{77}{125}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{117}{250}\right)\) |
| \(\chi_{251}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{181}{250}\right)\) | \(e\left(\frac{69}{125}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{21}{125}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{191}{250}\right)\) |
| \(\chi_{251}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{72}{125}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{9}{250}\right)\) | \(e\left(\frac{116}{125}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{19}{125}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{149}{250}\right)\) |
| \(\chi_{251}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{74}{125}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{103}{250}\right)\) | \(e\left(\frac{22}{125}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{23}{125}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{233}{250}\right)\) |
| \(\chi_{251}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{14}{125}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{33}{250}\right)\) | \(e\left(\frac{92}{125}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{28}{125}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{213}{250}\right)\) |
| \(\chi_{251}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{103}{125}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{91}{250}\right)\) | \(e\left(\frac{34}{125}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{81}{125}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{201}{250}\right)\) |
| \(\chi_{251}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{124}{125}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{203}{250}\right)\) | \(e\left(\frac{47}{125}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{123}{125}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{83}{250}\right)\) |
| \(\chi_{251}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{57}{125}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{179}{250}\right)\) | \(e\left(\frac{71}{125}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{114}{125}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{19}{250}\right)\) |
| \(\chi_{251}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{33}{125}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{51}{250}\right)\) | \(e\left(\frac{74}{125}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{66}{125}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{11}{250}\right)\) |
| \(\chi_{251}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{99}{125}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{153}{250}\right)\) | \(e\left(\frac{97}{125}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{73}{125}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{33}{250}\right)\) |
| \(\chi_{251}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{118}{125}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{171}{250}\right)\) | \(e\left(\frac{79}{125}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{111}{125}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{81}{250}\right)\) |
| \(\chi_{251}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{29}{125}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{113}{250}\right)\) | \(e\left(\frac{12}{125}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{58}{125}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{93}{250}\right)\) |
| \(\chi_{251}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{56}{125}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{7}{250}\right)\) | \(e\left(\frac{118}{125}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{112}{125}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{227}{250}\right)\) |
| \(\chi_{251}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{21}{125}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{237}{250}\right)\) | \(e\left(\frac{13}{125}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{42}{125}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{7}{250}\right)\) |
| \(\chi_{251}(76,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{64}{125}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{133}{250}\right)\) | \(e\left(\frac{117}{125}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{3}{125}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{63}{250}\right)\) |
| \(\chi_{251}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{47}{125}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{209}{250}\right)\) | \(e\left(\frac{41}{125}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{94}{125}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{99}{250}\right)\) |
| \(\chi_{251}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{89}{125}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{183}{250}\right)\) | \(e\left(\frac{67}{125}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{53}{125}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{113}{250}\right)\) |
| \(\chi_{251}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{37}{125}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{239}{250}\right)\) | \(e\left(\frac{11}{125}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{74}{125}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{179}{250}\right)\) |