from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6378, base_ring=CyclotomicField(354))
M = H._module
chi = DirichletCharacter(H, M([0,130]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,6378))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6378\) | |
| Conductor: | \(1063\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(177\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1063.i | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{177})$ |
| Fixed field: | Number field defined by a degree 177 polynomial (not computed) |
First 31 of 116 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6378}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{59}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{119}{177}\right)\) | \(e\left(\frac{13}{59}\right)\) | \(e\left(\frac{30}{59}\right)\) | \(e\left(\frac{16}{177}\right)\) | \(e\left(\frac{86}{177}\right)\) | \(e\left(\frac{39}{59}\right)\) | \(e\left(\frac{113}{177}\right)\) | \(e\left(\frac{65}{177}\right)\) |
| \(\chi_{6378}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{59}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{121}{177}\right)\) | \(e\left(\frac{39}{59}\right)\) | \(e\left(\frac{31}{59}\right)\) | \(e\left(\frac{107}{177}\right)\) | \(e\left(\frac{22}{177}\right)\) | \(e\left(\frac{58}{59}\right)\) | \(e\left(\frac{103}{177}\right)\) | \(e\left(\frac{136}{177}\right)\) |
| \(\chi_{6378}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{59}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{32}{177}\right)\) | \(e\left(\frac{3}{59}\right)\) | \(e\left(\frac{16}{59}\right)\) | \(e\left(\frac{40}{177}\right)\) | \(e\left(\frac{38}{177}\right)\) | \(e\left(\frac{9}{59}\right)\) | \(e\left(\frac{17}{177}\right)\) | \(e\left(\frac{74}{177}\right)\) |
| \(\chi_{6378}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{59}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{70}{177}\right)\) | \(e\left(\frac{25}{59}\right)\) | \(e\left(\frac{35}{59}\right)\) | \(e\left(\frac{176}{177}\right)\) | \(e\left(\frac{61}{177}\right)\) | \(e\left(\frac{16}{59}\right)\) | \(e\left(\frac{4}{177}\right)\) | \(e\left(\frac{7}{177}\right)\) |
| \(\chi_{6378}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{59}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{61}{177}\right)\) | \(e\left(\frac{26}{59}\right)\) | \(e\left(\frac{1}{59}\right)\) | \(e\left(\frac{32}{177}\right)\) | \(e\left(\frac{172}{177}\right)\) | \(e\left(\frac{19}{59}\right)\) | \(e\left(\frac{49}{177}\right)\) | \(e\left(\frac{130}{177}\right)\) |
| \(\chi_{6378}(193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{59}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{44}{177}\right)\) | \(e\left(\frac{41}{59}\right)\) | \(e\left(\frac{22}{59}\right)\) | \(e\left(\frac{55}{177}\right)\) | \(e\left(\frac{8}{177}\right)\) | \(e\left(\frac{5}{59}\right)\) | \(e\left(\frac{134}{177}\right)\) | \(e\left(\frac{146}{177}\right)\) |
| \(\chi_{6378}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{59}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{55}{177}\right)\) | \(e\left(\frac{7}{59}\right)\) | \(e\left(\frac{57}{59}\right)\) | \(e\left(\frac{113}{177}\right)\) | \(e\left(\frac{10}{177}\right)\) | \(e\left(\frac{21}{59}\right)\) | \(e\left(\frac{79}{177}\right)\) | \(e\left(\frac{94}{177}\right)\) |
| \(\chi_{6378}(295,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{59}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{169}{177}\right)\) | \(e\left(\frac{14}{59}\right)\) | \(e\left(\frac{55}{59}\right)\) | \(e\left(\frac{167}{177}\right)\) | \(e\left(\frac{79}{177}\right)\) | \(e\left(\frac{42}{59}\right)\) | \(e\left(\frac{40}{177}\right)\) | \(e\left(\frac{70}{177}\right)\) |
| \(\chi_{6378}(307,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{59}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{155}{177}\right)\) | \(e\left(\frac{9}{59}\right)\) | \(e\left(\frac{48}{59}\right)\) | \(e\left(\frac{61}{177}\right)\) | \(e\left(\frac{173}{177}\right)\) | \(e\left(\frac{27}{59}\right)\) | \(e\left(\frac{110}{177}\right)\) | \(e\left(\frac{104}{177}\right)\) |
| \(\chi_{6378}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{59}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{82}{177}\right)\) | \(e\left(\frac{4}{59}\right)\) | \(e\left(\frac{41}{59}\right)\) | \(e\left(\frac{14}{177}\right)\) | \(e\left(\frac{31}{177}\right)\) | \(e\left(\frac{12}{59}\right)\) | \(e\left(\frac{121}{177}\right)\) | \(e\left(\frac{79}{177}\right)\) |
| \(\chi_{6378}(481,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{59}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{151}{177}\right)\) | \(e\left(\frac{16}{59}\right)\) | \(e\left(\frac{46}{59}\right)\) | \(e\left(\frac{56}{177}\right)\) | \(e\left(\frac{124}{177}\right)\) | \(e\left(\frac{48}{59}\right)\) | \(e\left(\frac{130}{177}\right)\) | \(e\left(\frac{139}{177}\right)\) |
| \(\chi_{6378}(535,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{42}{59}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{43}{177}\right)\) | \(e\left(\frac{28}{59}\right)\) | \(e\left(\frac{51}{59}\right)\) | \(e\left(\frac{98}{177}\right)\) | \(e\left(\frac{40}{177}\right)\) | \(e\left(\frac{25}{59}\right)\) | \(e\left(\frac{139}{177}\right)\) | \(e\left(\frac{22}{177}\right)\) |
| \(\chi_{6378}(619,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{59}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{164}{177}\right)\) | \(e\left(\frac{8}{59}\right)\) | \(e\left(\frac{23}{59}\right)\) | \(e\left(\frac{28}{177}\right)\) | \(e\left(\frac{62}{177}\right)\) | \(e\left(\frac{24}{59}\right)\) | \(e\left(\frac{65}{177}\right)\) | \(e\left(\frac{158}{177}\right)\) |
| \(\chi_{6378}(625,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{59}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{65}{177}\right)\) | \(e\left(\frac{19}{59}\right)\) | \(e\left(\frac{3}{59}\right)\) | \(e\left(\frac{37}{177}\right)\) | \(e\left(\frac{44}{177}\right)\) | \(e\left(\frac{57}{59}\right)\) | \(e\left(\frac{29}{177}\right)\) | \(e\left(\frac{95}{177}\right)\) |
| \(\chi_{6378}(763,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{59}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{76}{177}\right)\) | \(e\left(\frac{44}{59}\right)\) | \(e\left(\frac{38}{59}\right)\) | \(e\left(\frac{95}{177}\right)\) | \(e\left(\frac{46}{177}\right)\) | \(e\left(\frac{14}{59}\right)\) | \(e\left(\frac{151}{177}\right)\) | \(e\left(\frac{43}{177}\right)\) |
| \(\chi_{6378}(847,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{59}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{112}{177}\right)\) | \(e\left(\frac{40}{59}\right)\) | \(e\left(\frac{56}{59}\right)\) | \(e\left(\frac{140}{177}\right)\) | \(e\left(\frac{133}{177}\right)\) | \(e\left(\frac{2}{59}\right)\) | \(e\left(\frac{148}{177}\right)\) | \(e\left(\frac{82}{177}\right)\) |
| \(\chi_{6378}(907,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{59}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{74}{177}\right)\) | \(e\left(\frac{18}{59}\right)\) | \(e\left(\frac{37}{59}\right)\) | \(e\left(\frac{4}{177}\right)\) | \(e\left(\frac{110}{177}\right)\) | \(e\left(\frac{54}{59}\right)\) | \(e\left(\frac{161}{177}\right)\) | \(e\left(\frac{149}{177}\right)\) |
| \(\chi_{6378}(967,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{59}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{62}{177}\right)\) | \(e\left(\frac{39}{59}\right)\) | \(e\left(\frac{31}{59}\right)\) | \(e\left(\frac{166}{177}\right)\) | \(e\left(\frac{140}{177}\right)\) | \(e\left(\frac{58}{59}\right)\) | \(e\left(\frac{44}{177}\right)\) | \(e\left(\frac{77}{177}\right)\) |
| \(\chi_{6378}(997,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{59}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{113}{177}\right)\) | \(e\left(\frac{53}{59}\right)\) | \(e\left(\frac{27}{59}\right)\) | \(e\left(\frac{97}{177}\right)\) | \(e\left(\frac{101}{177}\right)\) | \(e\left(\frac{41}{59}\right)\) | \(e\left(\frac{143}{177}\right)\) | \(e\left(\frac{29}{177}\right)\) |
| \(\chi_{6378}(1021,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{59}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{175}{177}\right)\) | \(e\left(\frac{33}{59}\right)\) | \(e\left(\frac{58}{59}\right)\) | \(e\left(\frac{86}{177}\right)\) | \(e\left(\frac{64}{177}\right)\) | \(e\left(\frac{40}{59}\right)\) | \(e\left(\frac{10}{177}\right)\) | \(e\left(\frac{106}{177}\right)\) |
| \(\chi_{6378}(1081,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{59}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{157}{177}\right)\) | \(e\left(\frac{35}{59}\right)\) | \(e\left(\frac{49}{59}\right)\) | \(e\left(\frac{152}{177}\right)\) | \(e\left(\frac{109}{177}\right)\) | \(e\left(\frac{46}{59}\right)\) | \(e\left(\frac{100}{177}\right)\) | \(e\left(\frac{175}{177}\right)\) |
| \(\chi_{6378}(1123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{59}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{104}{177}\right)\) | \(e\left(\frac{54}{59}\right)\) | \(e\left(\frac{52}{59}\right)\) | \(e\left(\frac{130}{177}\right)\) | \(e\left(\frac{35}{177}\right)\) | \(e\left(\frac{44}{59}\right)\) | \(e\left(\frac{11}{177}\right)\) | \(e\left(\frac{152}{177}\right)\) |
| \(\chi_{6378}(1249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{59}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{98}{177}\right)\) | \(e\left(\frac{35}{59}\right)\) | \(e\left(\frac{49}{59}\right)\) | \(e\left(\frac{34}{177}\right)\) | \(e\left(\frac{50}{177}\right)\) | \(e\left(\frac{46}{59}\right)\) | \(e\left(\frac{41}{177}\right)\) | \(e\left(\frac{116}{177}\right)\) |
| \(\chi_{6378}(1369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{59}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{64}{177}\right)\) | \(e\left(\frac{6}{59}\right)\) | \(e\left(\frac{32}{59}\right)\) | \(e\left(\frac{80}{177}\right)\) | \(e\left(\frac{76}{177}\right)\) | \(e\left(\frac{18}{59}\right)\) | \(e\left(\frac{34}{177}\right)\) | \(e\left(\frac{148}{177}\right)\) |
| \(\chi_{6378}(1387,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{59}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{137}{177}\right)\) | \(e\left(\frac{11}{59}\right)\) | \(e\left(\frac{39}{59}\right)\) | \(e\left(\frac{127}{177}\right)\) | \(e\left(\frac{41}{177}\right)\) | \(e\left(\frac{33}{59}\right)\) | \(e\left(\frac{23}{177}\right)\) | \(e\left(\frac{173}{177}\right)\) |
| \(\chi_{6378}(1393,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{59}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{85}{177}\right)\) | \(e\left(\frac{43}{59}\right)\) | \(e\left(\frac{13}{59}\right)\) | \(e\left(\frac{62}{177}\right)\) | \(e\left(\frac{112}{177}\right)\) | \(e\left(\frac{11}{59}\right)\) | \(e\left(\frac{106}{177}\right)\) | \(e\left(\frac{97}{177}\right)\) |
| \(\chi_{6378}(1429,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{59}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{167}{177}\right)\) | \(e\left(\frac{47}{59}\right)\) | \(e\left(\frac{54}{59}\right)\) | \(e\left(\frac{76}{177}\right)\) | \(e\left(\frac{143}{177}\right)\) | \(e\left(\frac{23}{59}\right)\) | \(e\left(\frac{50}{177}\right)\) | \(e\left(\frac{176}{177}\right)\) |
| \(\chi_{6378}(1513,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{59}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{101}{177}\right)\) | \(e\left(\frac{15}{59}\right)\) | \(e\left(\frac{21}{59}\right)\) | \(e\left(\frac{82}{177}\right)\) | \(e\left(\frac{131}{177}\right)\) | \(e\left(\frac{45}{59}\right)\) | \(e\left(\frac{26}{177}\right)\) | \(e\left(\frac{134}{177}\right)\) |
| \(\chi_{6378}(1537,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{59}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{170}{177}\right)\) | \(e\left(\frac{27}{59}\right)\) | \(e\left(\frac{26}{59}\right)\) | \(e\left(\frac{124}{177}\right)\) | \(e\left(\frac{47}{177}\right)\) | \(e\left(\frac{22}{59}\right)\) | \(e\left(\frac{35}{177}\right)\) | \(e\left(\frac{17}{177}\right)\) |
| \(\chi_{6378}(1543,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{59}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{34}{177}\right)\) | \(e\left(\frac{29}{59}\right)\) | \(e\left(\frac{17}{59}\right)\) | \(e\left(\frac{131}{177}\right)\) | \(e\left(\frac{151}{177}\right)\) | \(e\left(\frac{28}{59}\right)\) | \(e\left(\frac{7}{177}\right)\) | \(e\left(\frac{145}{177}\right)\) |
| \(\chi_{6378}(1555,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{59}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{38}{177}\right)\) | \(e\left(\frac{22}{59}\right)\) | \(e\left(\frac{19}{59}\right)\) | \(e\left(\frac{136}{177}\right)\) | \(e\left(\frac{23}{177}\right)\) | \(e\left(\frac{7}{59}\right)\) | \(e\left(\frac{164}{177}\right)\) | \(e\left(\frac{110}{177}\right)\) |