from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9747, base_ring=CyclotomicField(342))
M = H._module
chi = DirichletCharacter(H, M([114,136]))
chi.galois_orbit()
[g,chi] = znchar(Mod(226,9747))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(9747\) | |
| Conductor: | \(3249\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(171\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 3249.cb | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{171})$ |
| Fixed field: | Number field defined by a degree 171 polynomial (not computed) |
First 31 of 108 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{9747}(226,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{171}\right)\) | \(e\left(\frac{79}{171}\right)\) | \(e\left(\frac{158}{171}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{112}{171}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{85}{171}\right)\) | \(e\left(\frac{122}{171}\right)\) | \(e\left(\frac{158}{171}\right)\) |
| \(\chi_{9747}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{169}{171}\right)\) | \(e\left(\frac{167}{171}\right)\) | \(e\left(\frac{163}{171}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{161}{171}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{26}{171}\right)\) | \(e\left(\frac{154}{171}\right)\) | \(e\left(\frac{163}{171}\right)\) |
| \(\chi_{9747}(370,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{82}{171}\right)\) | \(e\left(\frac{164}{171}\right)\) | \(e\left(\frac{157}{171}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{68}{171}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{131}{171}\right)\) | \(e\left(\frac{13}{171}\right)\) | \(e\left(\frac{157}{171}\right)\) |
| \(\chi_{9747}(424,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{171}\right)\) | \(e\left(\frac{143}{171}\right)\) | \(e\left(\frac{115}{171}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{101}{171}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{11}{171}\right)\) | \(e\left(\frac{52}{171}\right)\) | \(e\left(\frac{115}{171}\right)\) |
| \(\chi_{9747}(442,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{164}{171}\right)\) | \(e\left(\frac{157}{171}\right)\) | \(e\left(\frac{143}{171}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{136}{171}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{91}{171}\right)\) | \(e\left(\frac{26}{171}\right)\) | \(e\left(\frac{143}{171}\right)\) |
| \(\chi_{9747}(739,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{170}{171}\right)\) | \(e\left(\frac{169}{171}\right)\) | \(e\left(\frac{167}{171}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{166}{171}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{13}{171}\right)\) | \(e\left(\frac{77}{171}\right)\) | \(e\left(\frac{167}{171}\right)\) |
| \(\chi_{9747}(802,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{171}\right)\) | \(e\left(\frac{122}{171}\right)\) | \(e\left(\frac{73}{171}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{134}{171}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{62}{171}\right)\) | \(e\left(\frac{91}{171}\right)\) | \(e\left(\frac{73}{171}\right)\) |
| \(\chi_{9747}(883,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{171}\right)\) | \(e\left(\frac{11}{171}\right)\) | \(e\left(\frac{22}{171}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{113}{171}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{14}{171}\right)\) | \(e\left(\frac{4}{171}\right)\) | \(e\left(\frac{22}{171}\right)\) |
| \(\chi_{9747}(928,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{171}\right)\) | \(e\left(\frac{136}{171}\right)\) | \(e\left(\frac{101}{171}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{169}{171}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{142}{171}\right)\) | \(e\left(\frac{65}{171}\right)\) | \(e\left(\frac{101}{171}\right)\) |
| \(\chi_{9747}(937,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{171}\right)\) | \(e\left(\frac{170}{171}\right)\) | \(e\left(\frac{169}{171}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{83}{171}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{92}{171}\right)\) | \(e\left(\frac{124}{171}\right)\) | \(e\left(\frac{169}{171}\right)\) |
| \(\chi_{9747}(955,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{146}{171}\right)\) | \(e\left(\frac{121}{171}\right)\) | \(e\left(\frac{71}{171}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{154}{171}\right)\) | \(e\left(\frac{44}{171}\right)\) | \(e\left(\frac{71}{171}\right)\) |
| \(\chi_{9747}(1252,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{171}\right)\) | \(e\left(\frac{88}{171}\right)\) | \(e\left(\frac{5}{171}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{49}{171}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{112}{171}\right)\) | \(e\left(\frac{32}{171}\right)\) | \(e\left(\frac{5}{171}\right)\) |
| \(\chi_{9747}(1315,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{124}{171}\right)\) | \(e\left(\frac{77}{171}\right)\) | \(e\left(\frac{154}{171}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{107}{171}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{98}{171}\right)\) | \(e\left(\frac{28}{171}\right)\) | \(e\left(\frac{154}{171}\right)\) |
| \(\chi_{9747}(1396,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{100}{171}\right)\) | \(e\left(\frac{29}{171}\right)\) | \(e\left(\frac{58}{171}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{158}{171}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{68}{171}\right)\) | \(e\left(\frac{166}{171}\right)\) | \(e\left(\frac{58}{171}\right)\) |
| \(\chi_{9747}(1441,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{171}\right)\) | \(e\left(\frac{82}{171}\right)\) | \(e\left(\frac{164}{171}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{34}{171}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{151}{171}\right)\) | \(e\left(\frac{92}{171}\right)\) | \(e\left(\frac{164}{171}\right)\) |
| \(\chi_{9747}(1450,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{171}\right)\) | \(e\left(\frac{26}{171}\right)\) | \(e\left(\frac{52}{171}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{65}{171}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{2}{171}\right)\) | \(e\left(\frac{25}{171}\right)\) | \(e\left(\frac{52}{171}\right)\) |
| \(\chi_{9747}(1468,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{128}{171}\right)\) | \(e\left(\frac{85}{171}\right)\) | \(e\left(\frac{170}{171}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{127}{171}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{62}{171}\right)\) | \(e\left(\frac{170}{171}\right)\) |
| \(\chi_{9747}(1765,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{171}\right)\) | \(e\left(\frac{7}{171}\right)\) | \(e\left(\frac{14}{171}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{103}{171}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{40}{171}\right)\) | \(e\left(\frac{158}{171}\right)\) | \(e\left(\frac{14}{171}\right)\) |
| \(\chi_{9747}(1828,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{171}\right)\) | \(e\left(\frac{32}{171}\right)\) | \(e\left(\frac{64}{171}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{80}{171}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{134}{171}\right)\) | \(e\left(\frac{136}{171}\right)\) | \(e\left(\frac{64}{171}\right)\) |
| \(\chi_{9747}(1909,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{171}\right)\) | \(e\left(\frac{47}{171}\right)\) | \(e\left(\frac{94}{171}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{32}{171}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{122}{171}\right)\) | \(e\left(\frac{157}{171}\right)\) | \(e\left(\frac{94}{171}\right)\) |
| \(\chi_{9747}(1954,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{171}\right)\) | \(e\left(\frac{28}{171}\right)\) | \(e\left(\frac{56}{171}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{70}{171}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{160}{171}\right)\) | \(e\left(\frac{119}{171}\right)\) | \(e\left(\frac{56}{171}\right)\) |
| \(\chi_{9747}(1963,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{112}{171}\right)\) | \(e\left(\frac{53}{171}\right)\) | \(e\left(\frac{106}{171}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{47}{171}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{83}{171}\right)\) | \(e\left(\frac{97}{171}\right)\) | \(e\left(\frac{106}{171}\right)\) |
| \(\chi_{9747}(1981,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{110}{171}\right)\) | \(e\left(\frac{49}{171}\right)\) | \(e\left(\frac{98}{171}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{37}{171}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{109}{171}\right)\) | \(e\left(\frac{80}{171}\right)\) | \(e\left(\frac{98}{171}\right)\) |
| \(\chi_{9747}(2278,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{134}{171}\right)\) | \(e\left(\frac{97}{171}\right)\) | \(e\left(\frac{23}{171}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{157}{171}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{139}{171}\right)\) | \(e\left(\frac{113}{171}\right)\) | \(e\left(\frac{23}{171}\right)\) |
| \(\chi_{9747}(2341,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{171}\right)\) | \(e\left(\frac{158}{171}\right)\) | \(e\left(\frac{145}{171}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{53}{171}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{170}{171}\right)\) | \(e\left(\frac{73}{171}\right)\) | \(e\left(\frac{145}{171}\right)\) |
| \(\chi_{9747}(2422,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{118}{171}\right)\) | \(e\left(\frac{65}{171}\right)\) | \(e\left(\frac{130}{171}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{77}{171}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{5}{171}\right)\) | \(e\left(\frac{148}{171}\right)\) | \(e\left(\frac{130}{171}\right)\) |
| \(\chi_{9747}(2467,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{158}{171}\right)\) | \(e\left(\frac{145}{171}\right)\) | \(e\left(\frac{119}{171}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{106}{171}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{169}{171}\right)\) | \(e\left(\frac{146}{171}\right)\) | \(e\left(\frac{119}{171}\right)\) |
| \(\chi_{9747}(2476,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{171}\right)\) | \(e\left(\frac{80}{171}\right)\) | \(e\left(\frac{160}{171}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{29}{171}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{164}{171}\right)\) | \(e\left(\frac{169}{171}\right)\) | \(e\left(\frac{160}{171}\right)\) |
| \(\chi_{9747}(2494,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{92}{171}\right)\) | \(e\left(\frac{13}{171}\right)\) | \(e\left(\frac{26}{171}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{118}{171}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{1}{171}\right)\) | \(e\left(\frac{98}{171}\right)\) | \(e\left(\frac{26}{171}\right)\) |
| \(\chi_{9747}(2791,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{171}\right)\) | \(e\left(\frac{16}{171}\right)\) | \(e\left(\frac{32}{171}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{40}{171}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{67}{171}\right)\) | \(e\left(\frac{68}{171}\right)\) | \(e\left(\frac{32}{171}\right)\) |
| \(\chi_{9747}(2854,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{142}{171}\right)\) | \(e\left(\frac{113}{171}\right)\) | \(e\left(\frac{55}{171}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{26}{171}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{35}{171}\right)\) | \(e\left(\frac{10}{171}\right)\) | \(e\left(\frac{55}{171}\right)\) |