from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1171, base_ring=CyclotomicField(1170))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1171))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1171\) | |
| Conductor: | \(1171\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1170\) | 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: | $\Q(\zeta_{585})$ |
| Fixed field: | Number field defined by a degree 1170 polynomial (not computed) |
First 31 of 288 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1171}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{1170}\right)\) | \(e\left(\frac{31}{234}\right)\) | \(e\left(\frac{1}{585}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{161}{234}\right)\) | \(e\left(\frac{1}{390}\right)\) | \(e\left(\frac{31}{117}\right)\) | \(e\left(\frac{599}{1170}\right)\) | \(e\left(\frac{67}{390}\right)\) |
| \(\chi_{1171}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{599}{1170}\right)\) | \(e\left(\frac{83}{234}\right)\) | \(e\left(\frac{14}{585}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{31}{234}\right)\) | \(e\left(\frac{209}{390}\right)\) | \(e\left(\frac{83}{117}\right)\) | \(e\left(\frac{781}{1170}\right)\) | \(e\left(\frac{353}{390}\right)\) |
| \(\chi_{1171}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{1170}\right)\) | \(e\left(\frac{187}{234}\right)\) | \(e\left(\frac{157}{585}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{5}{234}\right)\) | \(e\left(\frac{157}{390}\right)\) | \(e\left(\frac{70}{117}\right)\) | \(e\left(\frac{443}{1170}\right)\) | \(e\left(\frac{379}{390}\right)\) |
| \(\chi_{1171}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{311}{1170}\right)\) | \(e\left(\frac{47}{234}\right)\) | \(e\left(\frac{311}{585}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{229}{234}\right)\) | \(e\left(\frac{311}{390}\right)\) | \(e\left(\frac{47}{117}\right)\) | \(e\left(\frac{259}{1170}\right)\) | \(e\left(\frac{167}{390}\right)\) |
| \(\chi_{1171}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{847}{1170}\right)\) | \(e\left(\frac{49}{234}\right)\) | \(e\left(\frac{262}{585}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{179}{234}\right)\) | \(e\left(\frac{67}{390}\right)\) | \(e\left(\frac{49}{117}\right)\) | \(e\left(\frac{743}{1170}\right)\) | \(e\left(\frac{199}{390}\right)\) |
| \(\chi_{1171}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{1170}\right)\) | \(e\left(\frac{203}{234}\right)\) | \(e\left(\frac{233}{585}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{73}{234}\right)\) | \(e\left(\frac{233}{390}\right)\) | \(e\left(\frac{86}{117}\right)\) | \(e\left(\frac{337}{1170}\right)\) | \(e\left(\frac{11}{390}\right)\) |
| \(\chi_{1171}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{841}{1170}\right)\) | \(e\left(\frac{97}{234}\right)\) | \(e\left(\frac{256}{585}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{149}{234}\right)\) | \(e\left(\frac{61}{390}\right)\) | \(e\left(\frac{97}{117}\right)\) | \(e\left(\frac{659}{1170}\right)\) | \(e\left(\frac{187}{390}\right)\) |
| \(\chi_{1171}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{601}{1170}\right)\) | \(e\left(\frac{145}{234}\right)\) | \(e\left(\frac{16}{585}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{119}{234}\right)\) | \(e\left(\frac{211}{390}\right)\) | \(e\left(\frac{28}{117}\right)\) | \(e\left(\frac{809}{1170}\right)\) | \(e\left(\frac{97}{390}\right)\) |
| \(\chi_{1171}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{961}{1170}\right)\) | \(e\left(\frac{73}{234}\right)\) | \(e\left(\frac{376}{585}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{47}{234}\right)\) | \(e\left(\frac{181}{390}\right)\) | \(e\left(\frac{73}{117}\right)\) | \(e\left(\frac{1169}{1170}\right)\) | \(e\left(\frac{37}{390}\right)\) |
| \(\chi_{1171}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{719}{1170}\right)\) | \(e\left(\frac{59}{234}\right)\) | \(e\left(\frac{134}{585}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{163}{234}\right)\) | \(e\left(\frac{329}{390}\right)\) | \(e\left(\frac{59}{117}\right)\) | \(e\left(\frac{121}{1170}\right)\) | \(e\left(\frac{203}{390}\right)\) |
| \(\chi_{1171}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{203}{1170}\right)\) | \(e\left(\frac{209}{234}\right)\) | \(e\left(\frac{203}{585}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{157}{234}\right)\) | \(e\left(\frac{203}{390}\right)\) | \(e\left(\frac{92}{117}\right)\) | \(e\left(\frac{1087}{1170}\right)\) | \(e\left(\frac{341}{390}\right)\) |
| \(\chi_{1171}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{349}{1170}\right)\) | \(e\left(\frac{55}{234}\right)\) | \(e\left(\frac{349}{585}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{29}{234}\right)\) | \(e\left(\frac{349}{390}\right)\) | \(e\left(\frac{55}{117}\right)\) | \(e\left(\frac{791}{1170}\right)\) | \(e\left(\frac{373}{390}\right)\) |
| \(\chi_{1171}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{823}{1170}\right)\) | \(e\left(\frac{7}{234}\right)\) | \(e\left(\frac{238}{585}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{59}{234}\right)\) | \(e\left(\frac{43}{390}\right)\) | \(e\left(\frac{7}{117}\right)\) | \(e\left(\frac{407}{1170}\right)\) | \(e\left(\frac{151}{390}\right)\) |
| \(\chi_{1171}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{799}{1170}\right)\) | \(e\left(\frac{199}{234}\right)\) | \(e\left(\frac{214}{585}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{173}{234}\right)\) | \(e\left(\frac{19}{390}\right)\) | \(e\left(\frac{82}{117}\right)\) | \(e\left(\frac{71}{1170}\right)\) | \(e\left(\frac{103}{390}\right)\) |
| \(\chi_{1171}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1057}{1170}\right)\) | \(e\left(\frac{7}{234}\right)\) | \(e\left(\frac{472}{585}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{59}{234}\right)\) | \(e\left(\frac{277}{390}\right)\) | \(e\left(\frac{7}{117}\right)\) | \(e\left(\frac{173}{1170}\right)\) | \(e\left(\frac{229}{390}\right)\) |
| \(\chi_{1171}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1073}{1170}\right)\) | \(e\left(\frac{35}{234}\right)\) | \(e\left(\frac{488}{585}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{61}{234}\right)\) | \(e\left(\frac{293}{390}\right)\) | \(e\left(\frac{35}{117}\right)\) | \(e\left(\frac{397}{1170}\right)\) | \(e\left(\frac{131}{390}\right)\) |
| \(\chi_{1171}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{503}{1170}\right)\) | \(e\left(\frac{149}{234}\right)\) | \(e\left(\frac{503}{585}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{19}{234}\right)\) | \(e\left(\frac{113}{390}\right)\) | \(e\left(\frac{32}{117}\right)\) | \(e\left(\frac{607}{1170}\right)\) | \(e\left(\frac{161}{390}\right)\) |
| \(\chi_{1171}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{313}{1170}\right)\) | \(e\left(\frac{109}{234}\right)\) | \(e\left(\frac{313}{585}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{83}{234}\right)\) | \(e\left(\frac{313}{390}\right)\) | \(e\left(\frac{109}{117}\right)\) | \(e\left(\frac{287}{1170}\right)\) | \(e\left(\frac{301}{390}\right)\) |
| \(\chi_{1171}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{1170}\right)\) | \(e\left(\frac{229}{234}\right)\) | \(e\left(\frac{181}{585}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{125}{234}\right)\) | \(e\left(\frac{181}{390}\right)\) | \(e\left(\frac{112}{117}\right)\) | \(e\left(\frac{779}{1170}\right)\) | \(e\left(\frac{37}{390}\right)\) |
| \(\chi_{1171}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{1170}\right)\) | \(e\left(\frac{5}{234}\right)\) | \(e\left(\frac{53}{585}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{109}{234}\right)\) | \(e\left(\frac{53}{390}\right)\) | \(e\left(\frac{5}{117}\right)\) | \(e\left(\frac{157}{1170}\right)\) | \(e\left(\frac{41}{390}\right)\) |
| \(\chi_{1171}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{1170}\right)\) | \(e\left(\frac{193}{234}\right)\) | \(e\left(\frac{127}{585}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{89}{234}\right)\) | \(e\left(\frac{127}{390}\right)\) | \(e\left(\frac{76}{117}\right)\) | \(e\left(\frac{23}{1170}\right)\) | \(e\left(\frac{319}{390}\right)\) |
| \(\chi_{1171}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{1170}\right)\) | \(e\left(\frac{185}{234}\right)\) | \(e\left(\frac{89}{585}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{55}{234}\right)\) | \(e\left(\frac{89}{390}\right)\) | \(e\left(\frac{68}{117}\right)\) | \(e\left(\frac{661}{1170}\right)\) | \(e\left(\frac{113}{390}\right)\) |
| \(\chi_{1171}(94,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1147}{1170}\right)\) | \(e\left(\frac{223}{234}\right)\) | \(e\left(\frac{562}{585}\right)\) | \(e\left(\frac{11}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{41}{234}\right)\) | \(e\left(\frac{367}{390}\right)\) | \(e\left(\frac{106}{117}\right)\) | \(e\left(\frac{263}{1170}\right)\) | \(e\left(\frac{19}{390}\right)\) |
| \(\chi_{1171}(99,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{511}{1170}\right)\) | \(e\left(\frac{163}{234}\right)\) | \(e\left(\frac{511}{585}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{137}{234}\right)\) | \(e\left(\frac{121}{390}\right)\) | \(e\left(\frac{46}{117}\right)\) | \(e\left(\frac{719}{1170}\right)\) | \(e\left(\frac{307}{390}\right)\) |
| \(\chi_{1171}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1139}{1170}\right)\) | \(e\left(\frac{209}{234}\right)\) | \(e\left(\frac{554}{585}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{157}{234}\right)\) | \(e\left(\frac{359}{390}\right)\) | \(e\left(\frac{92}{117}\right)\) | \(e\left(\frac{151}{1170}\right)\) | \(e\left(\frac{263}{390}\right)\) |
| \(\chi_{1171}(108,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{467}{1170}\right)\) | \(e\left(\frac{203}{234}\right)\) | \(e\left(\frac{467}{585}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{73}{234}\right)\) | \(e\left(\frac{77}{390}\right)\) | \(e\left(\frac{86}{117}\right)\) | \(e\left(\frac{103}{1170}\right)\) | \(e\left(\frac{89}{390}\right)\) |
| \(\chi_{1171}(112,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{809}{1170}\right)\) | \(e\left(\frac{41}{234}\right)\) | \(e\left(\frac{224}{585}\right)\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{145}{234}\right)\) | \(e\left(\frac{29}{390}\right)\) | \(e\left(\frac{41}{117}\right)\) | \(e\left(\frac{211}{1170}\right)\) | \(e\left(\frac{383}{390}\right)\) |
| \(\chi_{1171}(116,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{187}{1170}\right)\) | \(e\left(\frac{181}{234}\right)\) | \(e\left(\frac{187}{585}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{155}{234}\right)\) | \(e\left(\frac{187}{390}\right)\) | \(e\left(\frac{64}{117}\right)\) | \(e\left(\frac{863}{1170}\right)\) | \(e\left(\frac{49}{390}\right)\) |
| \(\chi_{1171}(118,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{881}{1170}\right)\) | \(e\left(\frac{167}{234}\right)\) | \(e\left(\frac{296}{585}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{37}{234}\right)\) | \(e\left(\frac{101}{390}\right)\) | \(e\left(\frac{50}{117}\right)\) | \(e\left(\frac{49}{1170}\right)\) | \(e\left(\frac{137}{390}\right)\) |
| \(\chi_{1171}(123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{281}{1170}\right)\) | \(e\left(\frac{53}{234}\right)\) | \(e\left(\frac{281}{585}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{79}{234}\right)\) | \(e\left(\frac{281}{390}\right)\) | \(e\left(\frac{53}{117}\right)\) | \(e\left(\frac{1009}{1170}\right)\) | \(e\left(\frac{107}{390}\right)\) |
| \(\chi_{1171}(128,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{1170}\right)\) | \(e\left(\frac{217}{234}\right)\) | \(e\left(\frac{7}{585}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{191}{234}\right)\) | \(e\left(\frac{7}{390}\right)\) | \(e\left(\frac{100}{117}\right)\) | \(e\left(\frac{683}{1170}\right)\) | \(e\left(\frac{79}{390}\right)\) |