from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1337, base_ring=CyclotomicField(190))
M = H._module
chi = DirichletCharacter(H, M([95,112]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,1337))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1337\) | |
| Conductor: | \(1337\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(190\) | 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: | $\Q(\zeta_{95})$ |
| Fixed field: | Number field defined by a degree 190 polynomial (not computed) |
First 31 of 72 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1337}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{95}\right)\) | \(e\left(\frac{167}{190}\right)\) | \(e\left(\frac{83}{95}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{77}{95}\right)\) | \(e\left(\frac{72}{95}\right)\) | \(e\left(\frac{173}{190}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{143}{190}\right)\) |
| \(\chi_{1337}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{95}\right)\) | \(e\left(\frac{143}{190}\right)\) | \(e\left(\frac{87}{95}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{83}{95}\right)\) | \(e\left(\frac{48}{95}\right)\) | \(e\left(\frac{147}{190}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{127}{190}\right)\) |
| \(\chi_{1337}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{56}{95}\right)\) | \(e\left(\frac{183}{190}\right)\) | \(e\left(\frac{17}{95}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{73}{95}\right)\) | \(e\left(\frac{88}{95}\right)\) | \(e\left(\frac{127}{190}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{27}{190}\right)\) |
| \(\chi_{1337}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{84}{95}\right)\) | \(e\left(\frac{37}{190}\right)\) | \(e\left(\frac{73}{95}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{62}{95}\right)\) | \(e\left(\frac{37}{95}\right)\) | \(e\left(\frac{143}{190}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{183}{190}\right)\) |
| \(\chi_{1337}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{95}\right)\) | \(e\left(\frac{147}{190}\right)\) | \(e\left(\frac{23}{95}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{82}{95}\right)\) | \(e\left(\frac{52}{95}\right)\) | \(e\left(\frac{183}{190}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{3}{190}\right)\) |
| \(\chi_{1337}(90,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{95}\right)\) | \(e\left(\frac{101}{190}\right)\) | \(e\left(\frac{94}{95}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{46}{95}\right)\) | \(e\left(\frac{6}{95}\right)\) | \(e\left(\frac{149}{190}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{99}{190}\right)\) |
| \(\chi_{1337}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{64}{95}\right)\) | \(e\left(\frac{87}{190}\right)\) | \(e\left(\frac{33}{95}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{2}{95}\right)\) | \(e\left(\frac{87}{95}\right)\) | \(e\left(\frac{23}{190}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{153}{190}\right)\) |
| \(\chi_{1337}(104,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{48}{95}\right)\) | \(e\left(\frac{89}{190}\right)\) | \(e\left(\frac{1}{95}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{49}{95}\right)\) | \(e\left(\frac{89}{95}\right)\) | \(e\left(\frac{41}{190}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{91}{190}\right)\) |
| \(\chi_{1337}(118,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{95}\right)\) | \(e\left(\frac{153}{190}\right)\) | \(e\left(\frac{22}{95}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{33}{95}\right)\) | \(e\left(\frac{58}{95}\right)\) | \(e\left(\frac{47}{190}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{7}{190}\right)\) |
| \(\chi_{1337}(195,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{36}{95}\right)\) | \(e\left(\frac{43}{190}\right)\) | \(e\left(\frac{72}{95}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{13}{95}\right)\) | \(e\left(\frac{43}{95}\right)\) | \(e\left(\frac{7}{190}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{187}{190}\right)\) |
| \(\chi_{1337}(209,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{87}{95}\right)\) | \(e\left(\frac{1}{190}\right)\) | \(e\left(\frac{79}{95}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{71}{95}\right)\) | \(e\left(\frac{1}{95}\right)\) | \(e\left(\frac{9}{190}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{159}{190}\right)\) |
| \(\chi_{1337}(237,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{95}\right)\) | \(e\left(\frac{33}{190}\right)\) | \(e\left(\frac{42}{95}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{63}{95}\right)\) | \(e\left(\frac{33}{95}\right)\) | \(e\left(\frac{107}{190}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{117}{190}\right)\) |
| \(\chi_{1337}(251,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{78}{95}\right)\) | \(e\left(\frac{109}{190}\right)\) | \(e\left(\frac{61}{95}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{44}{95}\right)\) | \(e\left(\frac{14}{95}\right)\) | \(e\left(\frac{31}{190}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{41}{190}\right)\) |
| \(\chi_{1337}(258,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{86}{95}\right)\) | \(e\left(\frac{13}{190}\right)\) | \(e\left(\frac{77}{95}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{68}{95}\right)\) | \(e\left(\frac{13}{95}\right)\) | \(e\left(\frac{117}{190}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{167}{190}\right)\) |
| \(\chi_{1337}(272,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{95}\right)\) | \(e\left(\frac{149}{190}\right)\) | \(e\left(\frac{86}{95}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{34}{95}\right)\) | \(e\left(\frac{54}{95}\right)\) | \(e\left(\frac{11}{190}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{131}{190}\right)\) |
| \(\chi_{1337}(293,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{95}\right)\) | \(e\left(\frac{3}{190}\right)\) | \(e\left(\frac{47}{95}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{23}{95}\right)\) | \(e\left(\frac{3}{95}\right)\) | \(e\left(\frac{27}{190}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{97}{190}\right)\) |
| \(\chi_{1337}(321,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{95}\right)\) | \(e\left(\frac{51}{190}\right)\) | \(e\left(\frac{39}{95}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{11}{95}\right)\) | \(e\left(\frac{51}{95}\right)\) | \(e\left(\frac{79}{190}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{129}{190}\right)\) |
| \(\chi_{1337}(335,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{46}{95}\right)\) | \(e\left(\frac{113}{190}\right)\) | \(e\left(\frac{92}{95}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{43}{95}\right)\) | \(e\left(\frac{18}{95}\right)\) | \(e\left(\frac{67}{190}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{107}{190}\right)\) |
| \(\chi_{1337}(349,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{52}{95}\right)\) | \(e\left(\frac{41}{190}\right)\) | \(e\left(\frac{9}{95}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{61}{95}\right)\) | \(e\left(\frac{41}{95}\right)\) | \(e\left(\frac{179}{190}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{59}{190}\right)\) |
| \(\chi_{1337}(363,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{95}\right)\) | \(e\left(\frac{21}{190}\right)\) | \(e\left(\frac{44}{95}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{66}{95}\right)\) | \(e\left(\frac{21}{95}\right)\) | \(e\left(\frac{189}{190}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{109}{190}\right)\) |
| \(\chi_{1337}(384,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{18}{95}\right)\) | \(e\left(\frac{69}{190}\right)\) | \(e\left(\frac{36}{95}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{54}{95}\right)\) | \(e\left(\frac{69}{95}\right)\) | \(e\left(\frac{51}{190}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{141}{190}\right)\) |
| \(\chi_{1337}(391,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{95}\right)\) | \(e\left(\frac{27}{190}\right)\) | \(e\left(\frac{43}{95}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{17}{95}\right)\) | \(e\left(\frac{27}{95}\right)\) | \(e\left(\frac{53}{190}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{113}{190}\right)\) |
| \(\chi_{1337}(398,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{72}{95}\right)\) | \(e\left(\frac{181}{190}\right)\) | \(e\left(\frac{49}{95}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{26}{95}\right)\) | \(e\left(\frac{86}{95}\right)\) | \(e\left(\frac{109}{190}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{89}{190}\right)\) |
| \(\chi_{1337}(405,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{95}\right)\) | \(e\left(\frac{59}{190}\right)\) | \(e\left(\frac{6}{95}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{9}{95}\right)\) | \(e\left(\frac{59}{95}\right)\) | \(e\left(\frac{151}{190}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{71}{190}\right)\) |
| \(\chi_{1337}(433,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{95}\right)\) | \(e\left(\frac{29}{190}\right)\) | \(e\left(\frac{11}{95}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{64}{95}\right)\) | \(e\left(\frac{29}{95}\right)\) | \(e\left(\frac{71}{190}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{51}{190}\right)\) |
| \(\chi_{1337}(447,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{95}\right)\) | \(e\left(\frac{77}{190}\right)\) | \(e\left(\frac{3}{95}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{52}{95}\right)\) | \(e\left(\frac{77}{95}\right)\) | \(e\left(\frac{123}{190}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{83}{190}\right)\) |
| \(\chi_{1337}(454,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{28}{95}\right)\) | \(e\left(\frac{139}{190}\right)\) | \(e\left(\frac{56}{95}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{84}{95}\right)\) | \(e\left(\frac{44}{95}\right)\) | \(e\left(\frac{111}{190}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{61}{190}\right)\) |
| \(\chi_{1337}(461,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{34}{95}\right)\) | \(e\left(\frac{67}{190}\right)\) | \(e\left(\frac{68}{95}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{7}{95}\right)\) | \(e\left(\frac{67}{95}\right)\) | \(e\left(\frac{33}{190}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{13}{190}\right)\) |
| \(\chi_{1337}(468,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{95}\right)\) | \(e\left(\frac{47}{190}\right)\) | \(e\left(\frac{8}{95}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{12}{95}\right)\) | \(e\left(\frac{47}{95}\right)\) | \(e\left(\frac{43}{190}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{63}{190}\right)\) |
| \(\chi_{1337}(482,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{95}\right)\) | \(e\left(\frac{53}{190}\right)\) | \(e\left(\frac{7}{95}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{58}{95}\right)\) | \(e\left(\frac{53}{95}\right)\) | \(e\left(\frac{97}{190}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{67}{190}\right)\) |
| \(\chi_{1337}(510,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{95}\right)\) | \(e\left(\frac{103}{190}\right)\) | \(e\left(\frac{62}{95}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{93}{95}\right)\) | \(e\left(\frac{8}{95}\right)\) | \(e\left(\frac{167}{190}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{37}{190}\right)\) |