from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(125, base_ring=CyclotomicField(100))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,125))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(125\) | |
| Conductor: | \(125\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(100\) | 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: | $\Q(\zeta_{100})$ |
| Fixed field: | Number field defined by a degree 100 polynomial |
First 31 of 40 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{125}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{100}\right)\) | \(e\left(\frac{7}{100}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{100}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{9}{100}\right)\) | \(e\left(\frac{39}{100}\right)\) |
| \(\chi_{125}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{100}\right)\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{21}{100}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{63}{100}\right)\) | \(e\left(\frac{73}{100}\right)\) |
| \(\chi_{125}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{100}\right)\) | \(e\left(\frac{21}{100}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{100}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{27}{100}\right)\) | \(e\left(\frac{17}{100}\right)\) |
| \(\chi_{125}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{100}\right)\) | \(e\left(\frac{63}{100}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{27}{100}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{81}{100}\right)\) | \(e\left(\frac{51}{100}\right)\) |
| \(\chi_{125}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{100}\right)\) | \(e\left(\frac{73}{100}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{100}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{51}{100}\right)\) | \(e\left(\frac{21}{100}\right)\) |
| \(\chi_{125}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{100}\right)\) | \(e\left(\frac{11}{100}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{100}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{57}{100}\right)\) | \(e\left(\frac{47}{100}\right)\) |
| \(\chi_{125}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{100}\right)\) | \(e\left(\frac{39}{100}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{31}{100}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{93}{100}\right)\) | \(e\left(\frac{3}{100}\right)\) |
| \(\chi_{125}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{100}\right)\) | \(e\left(\frac{17}{100}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{93}{100}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{79}{100}\right)\) | \(e\left(\frac{9}{100}\right)\) |
| \(\chi_{125}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{100}\right)\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{63}{100}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{89}{100}\right)\) | \(e\left(\frac{19}{100}\right)\) |
| \(\chi_{125}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{87}{100}\right)\) | \(e\left(\frac{9}{100}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{61}{100}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{83}{100}\right)\) | \(e\left(\frac{93}{100}\right)\) |
| \(\chi_{125}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{100}\right)\) | \(e\left(\frac{81}{100}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{37}{100}\right)\) |
| \(\chi_{125}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{100}\right)\) | \(e\left(\frac{3}{100}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{87}{100}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{61}{100}\right)\) | \(e\left(\frac{31}{100}\right)\) |
| \(\chi_{125}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{100}\right)\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{57}{100}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{71}{100}\right)\) | \(e\left(\frac{41}{100}\right)\) |
| \(\chi_{125}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{93}{100}\right)\) | \(e\left(\frac{51}{100}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{79}{100}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{37}{100}\right)\) | \(e\left(\frac{27}{100}\right)\) |
| \(\chi_{125}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{100}\right)\) | \(e\left(\frac{79}{100}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{73}{100}\right)\) | \(e\left(\frac{83}{100}\right)\) |
| \(\chi_{125}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{100}\right)\) | \(e\left(\frac{77}{100}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{99}{100}\right)\) | \(e\left(\frac{29}{100}\right)\) |
| \(\chi_{125}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{100}\right)\) | \(e\left(\frac{87}{100}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{23}{100}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{69}{100}\right)\) | \(e\left(\frac{99}{100}\right)\) |
| \(\chi_{125}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{100}\right)\) | \(e\left(\frac{69}{100}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{100}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{3}{100}\right)\) | \(e\left(\frac{13}{100}\right)\) |
| \(\chi_{125}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{100}\right)\) | \(e\left(\frac{41}{100}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{89}{100}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{67}{100}\right)\) | \(e\left(\frac{57}{100}\right)\) |
| \(\chi_{125}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{43}{100}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{41}{100}\right)\) | \(e\left(\frac{11}{100}\right)\) |
| \(\chi_{125}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{99}{100}\right)\) | \(e\left(\frac{93}{100}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{97}{100}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{61}{100}\right)\) |
| \(\chi_{125}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{100}\right)\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{39}{100}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{17}{100}\right)\) | \(e\left(\frac{7}{100}\right)\) |
| \(\chi_{125}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{100}\right)\) | \(e\left(\frac{19}{100}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{51}{100}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{53}{100}\right)\) | \(e\left(\frac{63}{100}\right)\) |
| \(\chi_{125}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{37}{100}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{73}{100}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{19}{100}\right)\) | \(e\left(\frac{49}{100}\right)\) |
| \(\chi_{125}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{100}\right)\) | \(e\left(\frac{27}{100}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{83}{100}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{79}{100}\right)\) |
| \(\chi_{125}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{100}\right)\) | \(e\left(\frac{29}{100}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{41}{100}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{23}{100}\right)\) | \(e\left(\frac{33}{100}\right)\) |
| \(\chi_{125}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{100}\right)\) | \(e\left(\frac{1}{100}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{29}{100}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{87}{100}\right)\) | \(e\left(\frac{77}{100}\right)\) |
| \(\chi_{125}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{100}\right)\) | \(e\left(\frac{83}{100}\right)\) | \(e\left(\frac{19}{50}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{100}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{21}{100}\right)\) | \(e\left(\frac{91}{100}\right)\) |
| \(\chi_{125}(88,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{100}\right)\) | \(e\left(\frac{53}{100}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{37}{100}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{11}{100}\right)\) | \(e\left(\frac{81}{100}\right)\) |
| \(\chi_{125}(92,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{31}{100}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{99}{100}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{97}{100}\right)\) | \(e\left(\frac{87}{100}\right)\) |
| \(\chi_{125}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{100}\right)\) | \(e\left(\frac{59}{100}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{100}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{43}{100}\right)\) |