from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(311, base_ring=CyclotomicField(310))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(17,311))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(311\) | |
| Conductor: | \(311\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(310\) | 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: | $\Q(\zeta_{155})$ |
| Fixed field: | Number field defined by a degree 310 polynomial (not computed) |
First 31 of 120 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{311}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{155}\right)\) | \(e\left(\frac{82}{155}\right)\) | \(e\left(\frac{22}{155}\right)\) | \(e\left(\frac{98}{155}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{14}{31}\right)\) | \(e\left(\frac{33}{155}\right)\) | \(e\left(\frac{9}{155}\right)\) | \(e\left(\frac{109}{155}\right)\) | \(e\left(\frac{27}{62}\right)\) |
| \(\chi_{311}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{155}\right)\) | \(e\left(\frac{79}{155}\right)\) | \(e\left(\frac{59}{155}\right)\) | \(e\left(\frac{136}{155}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{15}{31}\right)\) | \(e\left(\frac{11}{155}\right)\) | \(e\left(\frac{3}{155}\right)\) | \(e\left(\frac{88}{155}\right)\) | \(e\left(\frac{9}{62}\right)\) |
| \(\chi_{311}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{155}\right)\) | \(e\left(\frac{9}{155}\right)\) | \(e\left(\frac{44}{155}\right)\) | \(e\left(\frac{41}{155}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{28}{31}\right)\) | \(e\left(\frac{66}{155}\right)\) | \(e\left(\frac{18}{155}\right)\) | \(e\left(\frac{63}{155}\right)\) | \(e\left(\frac{23}{62}\right)\) |
| \(\chi_{311}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{155}\right)\) | \(e\left(\frac{111}{155}\right)\) | \(e\left(\frac{26}{155}\right)\) | \(e\left(\frac{144}{155}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{25}{31}\right)\) | \(e\left(\frac{39}{155}\right)\) | \(e\left(\frac{67}{155}\right)\) | \(e\left(\frac{2}{155}\right)\) | \(e\left(\frac{15}{62}\right)\) |
| \(\chi_{311}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{155}\right)\) | \(e\left(\frac{77}{155}\right)\) | \(e\left(\frac{32}{155}\right)\) | \(e\left(\frac{58}{155}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{26}{31}\right)\) | \(e\left(\frac{48}{155}\right)\) | \(e\left(\frac{154}{155}\right)\) | \(e\left(\frac{74}{155}\right)\) | \(e\left(\frac{59}{62}\right)\) |
| \(\chi_{311}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{82}{155}\right)\) | \(e\left(\frac{104}{155}\right)\) | \(e\left(\frac{9}{155}\right)\) | \(e\left(\frac{26}{155}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{17}{31}\right)\) | \(e\left(\frac{91}{155}\right)\) | \(e\left(\frac{53}{155}\right)\) | \(e\left(\frac{108}{155}\right)\) | \(e\left(\frac{35}{62}\right)\) |
| \(\chi_{311}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{34}{155}\right)\) | \(e\left(\frac{28}{155}\right)\) | \(e\left(\frac{68}{155}\right)\) | \(e\left(\frac{7}{155}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{31}\right)\) | \(e\left(\frac{102}{155}\right)\) | \(e\left(\frac{56}{155}\right)\) | \(e\left(\frac{41}{155}\right)\) | \(e\left(\frac{13}{62}\right)\) |
| \(\chi_{311}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{98}{155}\right)\) | \(e\left(\frac{26}{155}\right)\) | \(e\left(\frac{41}{155}\right)\) | \(e\left(\frac{84}{155}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{12}{31}\right)\) | \(e\left(\frac{139}{155}\right)\) | \(e\left(\frac{52}{155}\right)\) | \(e\left(\frac{27}{155}\right)\) | \(e\left(\frac{1}{62}\right)\) |
| \(\chi_{311}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{134}{155}\right)\) | \(e\left(\frac{83}{155}\right)\) | \(e\left(\frac{113}{155}\right)\) | \(e\left(\frac{137}{155}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{24}{31}\right)\) | \(e\left(\frac{92}{155}\right)\) | \(e\left(\frac{11}{155}\right)\) | \(e\left(\frac{116}{155}\right)\) | \(e\left(\frac{33}{62}\right)\) |
| \(\chi_{311}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{155}\right)\) | \(e\left(\frac{23}{155}\right)\) | \(e\left(\frac{78}{155}\right)\) | \(e\left(\frac{122}{155}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{31}\right)\) | \(e\left(\frac{117}{155}\right)\) | \(e\left(\frac{46}{155}\right)\) | \(e\left(\frac{6}{155}\right)\) | \(e\left(\frac{45}{62}\right)\) |
| \(\chi_{311}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{155}\right)\) | \(e\left(\frac{41}{155}\right)\) | \(e\left(\frac{11}{155}\right)\) | \(e\left(\frac{49}{155}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{31}\right)\) | \(e\left(\frac{94}{155}\right)\) | \(e\left(\frac{82}{155}\right)\) | \(e\left(\frac{132}{155}\right)\) | \(e\left(\frac{29}{62}\right)\) |
| \(\chi_{311}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{155}\right)\) | \(e\left(\frac{108}{155}\right)\) | \(e\left(\frac{63}{155}\right)\) | \(e\left(\frac{27}{155}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{26}{31}\right)\) | \(e\left(\frac{17}{155}\right)\) | \(e\left(\frac{61}{155}\right)\) | \(e\left(\frac{136}{155}\right)\) | \(e\left(\frac{59}{62}\right)\) |
| \(\chi_{311}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{76}{155}\right)\) | \(e\left(\frac{17}{155}\right)\) | \(e\left(\frac{152}{155}\right)\) | \(e\left(\frac{43}{155}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{15}{31}\right)\) | \(e\left(\frac{73}{155}\right)\) | \(e\left(\frac{34}{155}\right)\) | \(e\left(\frac{119}{155}\right)\) | \(e\left(\frac{9}{62}\right)\) |
| \(\chi_{311}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{155}\right)\) | \(e\left(\frac{42}{155}\right)\) | \(e\left(\frac{102}{155}\right)\) | \(e\left(\frac{88}{155}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{31}\right)\) | \(e\left(\frac{153}{155}\right)\) | \(e\left(\frac{84}{155}\right)\) | \(e\left(\frac{139}{155}\right)\) | \(e\left(\frac{35}{62}\right)\) |
| \(\chi_{311}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{155}\right)\) | \(e\left(\frac{21}{155}\right)\) | \(e\left(\frac{51}{155}\right)\) | \(e\left(\frac{44}{155}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{24}{31}\right)\) | \(e\left(\frac{154}{155}\right)\) | \(e\left(\frac{42}{155}\right)\) | \(e\left(\frac{147}{155}\right)\) | \(e\left(\frac{33}{62}\right)\) |
| \(\chi_{311}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{52}{155}\right)\) | \(e\left(\frac{134}{155}\right)\) | \(e\left(\frac{104}{155}\right)\) | \(e\left(\frac{111}{155}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{31}\right)\) | \(e\left(\frac{1}{155}\right)\) | \(e\left(\frac{113}{155}\right)\) | \(e\left(\frac{8}{155}\right)\) | \(e\left(\frac{29}{62}\right)\) |
| \(\chi_{311}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{155}\right)\) | \(e\left(\frac{48}{155}\right)\) | \(e\left(\frac{28}{155}\right)\) | \(e\left(\frac{12}{155}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{15}{31}\right)\) | \(e\left(\frac{42}{155}\right)\) | \(e\left(\frac{96}{155}\right)\) | \(e\left(\frac{26}{155}\right)\) | \(e\left(\frac{9}{62}\right)\) |
| \(\chi_{311}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{155}\right)\) | \(e\left(\frac{127}{155}\right)\) | \(e\left(\frac{87}{155}\right)\) | \(e\left(\frac{148}{155}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{30}{31}\right)\) | \(e\left(\frac{53}{155}\right)\) | \(e\left(\frac{99}{155}\right)\) | \(e\left(\frac{114}{155}\right)\) | \(e\left(\frac{49}{62}\right)\) |
| \(\chi_{311}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{112}{155}\right)\) | \(e\left(\frac{74}{155}\right)\) | \(e\left(\frac{69}{155}\right)\) | \(e\left(\frac{96}{155}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{27}{31}\right)\) | \(e\left(\frac{26}{155}\right)\) | \(e\left(\frac{148}{155}\right)\) | \(e\left(\frac{53}{155}\right)\) | \(e\left(\frac{41}{62}\right)\) |
| \(\chi_{311}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{123}{155}\right)\) | \(e\left(\frac{1}{155}\right)\) | \(e\left(\frac{91}{155}\right)\) | \(e\left(\frac{39}{155}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{10}{31}\right)\) | \(e\left(\frac{59}{155}\right)\) | \(e\left(\frac{2}{155}\right)\) | \(e\left(\frac{7}{155}\right)\) | \(e\left(\frac{37}{62}\right)\) |
| \(\chi_{311}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{66}{155}\right)\) | \(e\left(\frac{27}{155}\right)\) | \(e\left(\frac{132}{155}\right)\) | \(e\left(\frac{123}{155}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{22}{31}\right)\) | \(e\left(\frac{43}{155}\right)\) | \(e\left(\frac{54}{155}\right)\) | \(e\left(\frac{34}{155}\right)\) | \(e\left(\frac{7}{62}\right)\) |
| \(\chi_{311}(76,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{126}{155}\right)\) | \(e\left(\frac{122}{155}\right)\) | \(e\left(\frac{97}{155}\right)\) | \(e\left(\frac{108}{155}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{11}{31}\right)\) | \(e\left(\frac{68}{155}\right)\) | \(e\left(\frac{89}{155}\right)\) | \(e\left(\frac{79}{155}\right)\) | \(e\left(\frac{19}{62}\right)\) |
| \(\chi_{311}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{147}{155}\right)\) | \(e\left(\frac{39}{155}\right)\) | \(e\left(\frac{139}{155}\right)\) | \(e\left(\frac{126}{155}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{18}{31}\right)\) | \(e\left(\frac{131}{155}\right)\) | \(e\left(\frac{78}{155}\right)\) | \(e\left(\frac{118}{155}\right)\) | \(e\left(\frac{17}{62}\right)\) |
| \(\chi_{311}(85,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{152}{155}\right)\) | \(e\left(\frac{34}{155}\right)\) | \(e\left(\frac{149}{155}\right)\) | \(e\left(\frac{86}{155}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{30}{31}\right)\) | \(e\left(\frac{146}{155}\right)\) | \(e\left(\frac{68}{155}\right)\) | \(e\left(\frac{83}{155}\right)\) | \(e\left(\frac{49}{62}\right)\) |
| \(\chi_{311}(88,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{155}\right)\) | \(e\left(\frac{52}{155}\right)\) | \(e\left(\frac{82}{155}\right)\) | \(e\left(\frac{13}{155}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{24}{31}\right)\) | \(e\left(\frac{123}{155}\right)\) | \(e\left(\frac{104}{155}\right)\) | \(e\left(\frac{54}{155}\right)\) | \(e\left(\frac{33}{62}\right)\) |
| \(\chi_{311}(92,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{32}{155}\right)\) | \(e\left(\frac{154}{155}\right)\) | \(e\left(\frac{64}{155}\right)\) | \(e\left(\frac{116}{155}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{21}{31}\right)\) | \(e\left(\frac{96}{155}\right)\) | \(e\left(\frac{153}{155}\right)\) | \(e\left(\frac{148}{155}\right)\) | \(e\left(\frac{25}{62}\right)\) |
| \(\chi_{311}(93,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{155}\right)\) | \(e\left(\frac{67}{155}\right)\) | \(e\left(\frac{52}{155}\right)\) | \(e\left(\frac{133}{155}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{31}\right)\) | \(e\left(\frac{78}{155}\right)\) | \(e\left(\frac{134}{155}\right)\) | \(e\left(\frac{4}{155}\right)\) | \(e\left(\frac{61}{62}\right)\) |
| \(\chi_{311}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{155}\right)\) | \(e\left(\frac{32}{155}\right)\) | \(e\left(\frac{122}{155}\right)\) | \(e\left(\frac{8}{155}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{10}{31}\right)\) | \(e\left(\frac{28}{155}\right)\) | \(e\left(\frac{64}{155}\right)\) | \(e\left(\frac{69}{155}\right)\) | \(e\left(\frac{37}{62}\right)\) |
| \(\chi_{311}(99,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{155}\right)\) | \(e\left(\frac{146}{155}\right)\) | \(e\left(\frac{111}{155}\right)\) | \(e\left(\frac{114}{155}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{31}\right)\) | \(e\left(\frac{89}{155}\right)\) | \(e\left(\frac{137}{155}\right)\) | \(e\left(\frac{92}{155}\right)\) | \(e\left(\frac{39}{62}\right)\) |
| \(\chi_{311}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{155}\right)\) | \(e\left(\frac{24}{155}\right)\) | \(e\left(\frac{14}{155}\right)\) | \(e\left(\frac{6}{155}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{23}{31}\right)\) | \(e\left(\frac{21}{155}\right)\) | \(e\left(\frac{48}{155}\right)\) | \(e\left(\frac{13}{155}\right)\) | \(e\left(\frac{51}{62}\right)\) |
| \(\chi_{311}(102,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{42}{155}\right)\) | \(e\left(\frac{144}{155}\right)\) | \(e\left(\frac{84}{155}\right)\) | \(e\left(\frac{36}{155}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{14}{31}\right)\) | \(e\left(\frac{126}{155}\right)\) | \(e\left(\frac{133}{155}\right)\) | \(e\left(\frac{78}{155}\right)\) | \(e\left(\frac{27}{62}\right)\) |