from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(227, base_ring=CyclotomicField(226))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,227))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(227\) | |
| Conductor: | \(227\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(226\) | 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: | $\Q(\zeta_{113})$ |
| Fixed field: | Number field defined by a degree 226 polynomial (not computed) |
First 31 of 112 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{227}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{226}\right)\) | \(e\left(\frac{23}{113}\right)\) | \(e\left(\frac{1}{113}\right)\) | \(e\left(\frac{11}{226}\right)\) | \(e\left(\frac{47}{226}\right)\) | \(e\left(\frac{77}{113}\right)\) | \(e\left(\frac{3}{226}\right)\) | \(e\left(\frac{46}{113}\right)\) | \(e\left(\frac{6}{113}\right)\) | \(e\left(\frac{14}{113}\right)\) |
| \(\chi_{227}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{226}\right)\) | \(e\left(\frac{27}{113}\right)\) | \(e\left(\frac{11}{113}\right)\) | \(e\left(\frac{121}{226}\right)\) | \(e\left(\frac{65}{226}\right)\) | \(e\left(\frac{56}{113}\right)\) | \(e\left(\frac{33}{226}\right)\) | \(e\left(\frac{54}{113}\right)\) | \(e\left(\frac{66}{113}\right)\) | \(e\left(\frac{41}{113}\right)\) |
| \(\chi_{227}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{226}\right)\) | \(e\left(\frac{64}{113}\right)\) | \(e\left(\frac{47}{113}\right)\) | \(e\left(\frac{65}{226}\right)\) | \(e\left(\frac{175}{226}\right)\) | \(e\left(\frac{3}{113}\right)\) | \(e\left(\frac{141}{226}\right)\) | \(e\left(\frac{15}{113}\right)\) | \(e\left(\frac{56}{113}\right)\) | \(e\left(\frac{93}{113}\right)\) |
| \(\chi_{227}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{226}\right)\) | \(e\left(\frac{69}{113}\right)\) | \(e\left(\frac{3}{113}\right)\) | \(e\left(\frac{33}{226}\right)\) | \(e\left(\frac{141}{226}\right)\) | \(e\left(\frac{5}{113}\right)\) | \(e\left(\frac{9}{226}\right)\) | \(e\left(\frac{25}{113}\right)\) | \(e\left(\frac{18}{113}\right)\) | \(e\left(\frac{42}{113}\right)\) |
| \(\chi_{227}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{226}\right)\) | \(e\left(\frac{47}{113}\right)\) | \(e\left(\frac{61}{113}\right)\) | \(e\left(\frac{219}{226}\right)\) | \(e\left(\frac{155}{226}\right)\) | \(e\left(\frac{64}{113}\right)\) | \(e\left(\frac{183}{226}\right)\) | \(e\left(\frac{94}{113}\right)\) | \(e\left(\frac{27}{113}\right)\) | \(e\left(\frac{63}{113}\right)\) |
| \(\chi_{227}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{226}\right)\) | \(e\left(\frac{62}{113}\right)\) | \(e\left(\frac{42}{113}\right)\) | \(e\left(\frac{123}{226}\right)\) | \(e\left(\frac{53}{226}\right)\) | \(e\left(\frac{70}{113}\right)\) | \(e\left(\frac{13}{226}\right)\) | \(e\left(\frac{11}{113}\right)\) | \(e\left(\frac{26}{113}\right)\) | \(e\left(\frac{23}{113}\right)\) |
| \(\chi_{227}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{226}\right)\) | \(e\left(\frac{68}{113}\right)\) | \(e\left(\frac{57}{113}\right)\) | \(e\left(\frac{175}{226}\right)\) | \(e\left(\frac{193}{226}\right)\) | \(e\left(\frac{95}{113}\right)\) | \(e\left(\frac{171}{226}\right)\) | \(e\left(\frac{23}{113}\right)\) | \(e\left(\frac{3}{113}\right)\) | \(e\left(\frac{7}{113}\right)\) |
| \(\chi_{227}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{99}{226}\right)\) | \(e\left(\frac{17}{113}\right)\) | \(e\left(\frac{99}{113}\right)\) | \(e\left(\frac{185}{226}\right)\) | \(e\left(\frac{133}{226}\right)\) | \(e\left(\frac{52}{113}\right)\) | \(e\left(\frac{71}{226}\right)\) | \(e\left(\frac{34}{113}\right)\) | \(e\left(\frac{29}{113}\right)\) | \(e\left(\frac{30}{113}\right)\) |
| \(\chi_{227}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{93}{226}\right)\) | \(e\left(\frac{105}{113}\right)\) | \(e\left(\frac{93}{113}\right)\) | \(e\left(\frac{119}{226}\right)\) | \(e\left(\frac{77}{226}\right)\) | \(e\left(\frac{42}{113}\right)\) | \(e\left(\frac{53}{226}\right)\) | \(e\left(\frac{97}{113}\right)\) | \(e\left(\frac{106}{113}\right)\) | \(e\left(\frac{59}{113}\right)\) |
| \(\chi_{227}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{226}\right)\) | \(e\left(\frac{73}{113}\right)\) | \(e\left(\frac{13}{113}\right)\) | \(e\left(\frac{143}{226}\right)\) | \(e\left(\frac{159}{226}\right)\) | \(e\left(\frac{97}{113}\right)\) | \(e\left(\frac{39}{226}\right)\) | \(e\left(\frac{33}{113}\right)\) | \(e\left(\frac{78}{113}\right)\) | \(e\left(\frac{69}{113}\right)\) |
| \(\chi_{227}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{226}\right)\) | \(e\left(\frac{102}{113}\right)\) | \(e\left(\frac{29}{113}\right)\) | \(e\left(\frac{93}{226}\right)\) | \(e\left(\frac{7}{226}\right)\) | \(e\left(\frac{86}{113}\right)\) | \(e\left(\frac{87}{226}\right)\) | \(e\left(\frac{91}{113}\right)\) | \(e\left(\frac{61}{113}\right)\) | \(e\left(\frac{67}{113}\right)\) |
| \(\chi_{227}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{226}\right)\) | \(e\left(\frac{110}{113}\right)\) | \(e\left(\frac{49}{113}\right)\) | \(e\left(\frac{87}{226}\right)\) | \(e\left(\frac{43}{226}\right)\) | \(e\left(\frac{44}{113}\right)\) | \(e\left(\frac{147}{226}\right)\) | \(e\left(\frac{107}{113}\right)\) | \(e\left(\frac{68}{113}\right)\) | \(e\left(\frac{8}{113}\right)\) |
| \(\chi_{227}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{197}{226}\right)\) | \(e\left(\frac{11}{113}\right)\) | \(e\left(\frac{84}{113}\right)\) | \(e\left(\frac{133}{226}\right)\) | \(e\left(\frac{219}{226}\right)\) | \(e\left(\frac{27}{113}\right)\) | \(e\left(\frac{139}{226}\right)\) | \(e\left(\frac{22}{113}\right)\) | \(e\left(\frac{52}{113}\right)\) | \(e\left(\frac{46}{113}\right)\) |
| \(\chi_{227}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{226}\right)\) | \(e\left(\frac{2}{113}\right)\) | \(e\left(\frac{5}{113}\right)\) | \(e\left(\frac{55}{226}\right)\) | \(e\left(\frac{9}{226}\right)\) | \(e\left(\frac{46}{113}\right)\) | \(e\left(\frac{15}{226}\right)\) | \(e\left(\frac{4}{113}\right)\) | \(e\left(\frac{30}{113}\right)\) | \(e\left(\frac{70}{113}\right)\) |
| \(\chi_{227}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{165}{226}\right)\) | \(e\left(\frac{66}{113}\right)\) | \(e\left(\frac{52}{113}\right)\) | \(e\left(\frac{7}{226}\right)\) | \(e\left(\frac{71}{226}\right)\) | \(e\left(\frac{49}{113}\right)\) | \(e\left(\frac{43}{226}\right)\) | \(e\left(\frac{19}{113}\right)\) | \(e\left(\frac{86}{113}\right)\) | \(e\left(\frac{50}{113}\right)\) |
| \(\chi_{227}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{226}\right)\) | \(e\left(\frac{76}{113}\right)\) | \(e\left(\frac{77}{113}\right)\) | \(e\left(\frac{169}{226}\right)\) | \(e\left(\frac{3}{226}\right)\) | \(e\left(\frac{53}{113}\right)\) | \(e\left(\frac{5}{226}\right)\) | \(e\left(\frac{39}{113}\right)\) | \(e\left(\frac{10}{113}\right)\) | \(e\left(\frac{61}{113}\right)\) |
| \(\chi_{227}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{226}\right)\) | \(e\left(\frac{49}{113}\right)\) | \(e\left(\frac{66}{113}\right)\) | \(e\left(\frac{161}{226}\right)\) | \(e\left(\frac{51}{226}\right)\) | \(e\left(\frac{110}{113}\right)\) | \(e\left(\frac{85}{226}\right)\) | \(e\left(\frac{98}{113}\right)\) | \(e\left(\frac{57}{113}\right)\) | \(e\left(\frac{20}{113}\right)\) |
| \(\chi_{227}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{226}\right)\) | \(e\left(\frac{88}{113}\right)\) | \(e\left(\frac{107}{113}\right)\) | \(e\left(\frac{47}{226}\right)\) | \(e\left(\frac{57}{226}\right)\) | \(e\left(\frac{103}{113}\right)\) | \(e\left(\frac{95}{226}\right)\) | \(e\left(\frac{63}{113}\right)\) | \(e\left(\frac{77}{113}\right)\) | \(e\left(\frac{29}{113}\right)\) |
| \(\chi_{227}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{226}\right)\) | \(e\left(\frac{75}{113}\right)\) | \(e\left(\frac{18}{113}\right)\) | \(e\left(\frac{85}{226}\right)\) | \(e\left(\frac{55}{226}\right)\) | \(e\left(\frac{30}{113}\right)\) | \(e\left(\frac{167}{226}\right)\) | \(e\left(\frac{37}{113}\right)\) | \(e\left(\frac{108}{113}\right)\) | \(e\left(\frac{26}{113}\right)\) |
| \(\chi_{227}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{201}{226}\right)\) | \(e\left(\frac{103}{113}\right)\) | \(e\left(\frac{88}{113}\right)\) | \(e\left(\frac{177}{226}\right)\) | \(e\left(\frac{181}{226}\right)\) | \(e\left(\frac{109}{113}\right)\) | \(e\left(\frac{151}{226}\right)\) | \(e\left(\frac{93}{113}\right)\) | \(e\left(\frac{76}{113}\right)\) | \(e\left(\frac{102}{113}\right)\) |
| \(\chi_{227}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{226}\right)\) | \(e\left(\frac{109}{113}\right)\) | \(e\left(\frac{103}{113}\right)\) | \(e\left(\frac{3}{226}\right)\) | \(e\left(\frac{95}{226}\right)\) | \(e\left(\frac{21}{113}\right)\) | \(e\left(\frac{83}{226}\right)\) | \(e\left(\frac{105}{113}\right)\) | \(e\left(\frac{53}{113}\right)\) | \(e\left(\frac{86}{113}\right)\) |
| \(\chi_{227}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{226}\right)\) | \(e\left(\frac{14}{113}\right)\) | \(e\left(\frac{35}{113}\right)\) | \(e\left(\frac{159}{226}\right)\) | \(e\left(\frac{63}{226}\right)\) | \(e\left(\frac{96}{113}\right)\) | \(e\left(\frac{105}{226}\right)\) | \(e\left(\frac{28}{113}\right)\) | \(e\left(\frac{97}{113}\right)\) | \(e\left(\frac{38}{113}\right)\) |
| \(\chi_{227}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{226}\right)\) | \(e\left(\frac{77}{113}\right)\) | \(e\left(\frac{23}{113}\right)\) | \(e\left(\frac{27}{226}\right)\) | \(e\left(\frac{177}{226}\right)\) | \(e\left(\frac{76}{113}\right)\) | \(e\left(\frac{69}{226}\right)\) | \(e\left(\frac{41}{113}\right)\) | \(e\left(\frac{25}{113}\right)\) | \(e\left(\frac{96}{113}\right)\) |
| \(\chi_{227}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{226}\right)\) | \(e\left(\frac{58}{113}\right)\) | \(e\left(\frac{32}{113}\right)\) | \(e\left(\frac{13}{226}\right)\) | \(e\left(\frac{35}{226}\right)\) | \(e\left(\frac{91}{113}\right)\) | \(e\left(\frac{209}{226}\right)\) | \(e\left(\frac{3}{113}\right)\) | \(e\left(\frac{79}{113}\right)\) | \(e\left(\frac{109}{113}\right)\) |
| \(\chi_{227}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{226}\right)\) | \(e\left(\frac{93}{113}\right)\) | \(e\left(\frac{63}{113}\right)\) | \(e\left(\frac{15}{226}\right)\) | \(e\left(\frac{23}{226}\right)\) | \(e\left(\frac{105}{113}\right)\) | \(e\left(\frac{189}{226}\right)\) | \(e\left(\frac{73}{113}\right)\) | \(e\left(\frac{39}{113}\right)\) | \(e\left(\frac{91}{113}\right)\) |
| \(\chi_{227}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{226}\right)\) | \(e\left(\frac{33}{113}\right)\) | \(e\left(\frac{26}{113}\right)\) | \(e\left(\frac{173}{226}\right)\) | \(e\left(\frac{205}{226}\right)\) | \(e\left(\frac{81}{113}\right)\) | \(e\left(\frac{191}{226}\right)\) | \(e\left(\frac{66}{113}\right)\) | \(e\left(\frac{43}{113}\right)\) | \(e\left(\frac{25}{113}\right)\) |
| \(\chi_{227}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{226}\right)\) | \(e\left(\frac{106}{113}\right)\) | \(e\left(\frac{39}{113}\right)\) | \(e\left(\frac{203}{226}\right)\) | \(e\left(\frac{25}{226}\right)\) | \(e\left(\frac{65}{113}\right)\) | \(e\left(\frac{117}{226}\right)\) | \(e\left(\frac{99}{113}\right)\) | \(e\left(\frac{8}{113}\right)\) | \(e\left(\frac{94}{113}\right)\) |
| \(\chi_{227}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{226}\right)\) | \(e\left(\frac{108}{113}\right)\) | \(e\left(\frac{44}{113}\right)\) | \(e\left(\frac{145}{226}\right)\) | \(e\left(\frac{147}{226}\right)\) | \(e\left(\frac{111}{113}\right)\) | \(e\left(\frac{19}{226}\right)\) | \(e\left(\frac{103}{113}\right)\) | \(e\left(\frac{38}{113}\right)\) | \(e\left(\frac{51}{113}\right)\) |
| \(\chi_{227}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{226}\right)\) | \(e\left(\frac{94}{113}\right)\) | \(e\left(\frac{9}{113}\right)\) | \(e\left(\frac{99}{226}\right)\) | \(e\left(\frac{197}{226}\right)\) | \(e\left(\frac{15}{113}\right)\) | \(e\left(\frac{27}{226}\right)\) | \(e\left(\frac{75}{113}\right)\) | \(e\left(\frac{54}{113}\right)\) | \(e\left(\frac{13}{113}\right)\) |
| \(\chi_{227}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{226}\right)\) | \(e\left(\frac{1}{113}\right)\) | \(e\left(\frac{59}{113}\right)\) | \(e\left(\frac{197}{226}\right)\) | \(e\left(\frac{61}{226}\right)\) | \(e\left(\frac{23}{113}\right)\) | \(e\left(\frac{177}{226}\right)\) | \(e\left(\frac{2}{113}\right)\) | \(e\left(\frac{15}{113}\right)\) | \(e\left(\frac{35}{113}\right)\) |
| \(\chi_{227}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{226}\right)\) | \(e\left(\frac{84}{113}\right)\) | \(e\left(\frac{97}{113}\right)\) | \(e\left(\frac{163}{226}\right)\) | \(e\left(\frac{39}{226}\right)\) | \(e\left(\frac{11}{113}\right)\) | \(e\left(\frac{65}{226}\right)\) | \(e\left(\frac{55}{113}\right)\) | \(e\left(\frac{17}{113}\right)\) | \(e\left(\frac{2}{113}\right)\) |