from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(607, base_ring=CyclotomicField(606))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,607))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(607\) | |
| Conductor: | \(607\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(606\) | 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: | $\Q(\zeta_{303})$ |
| Fixed field: | Number field defined by a degree 606 polynomial (not computed) |
First 31 of 200 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{607}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{292}{303}\right)\) | \(e\left(\frac{1}{606}\right)\) | \(e\left(\frac{281}{303}\right)\) | \(e\left(\frac{601}{606}\right)\) | \(e\left(\frac{195}{202}\right)\) | \(e\left(\frac{38}{101}\right)\) | \(e\left(\frac{90}{101}\right)\) | \(e\left(\frac{1}{303}\right)\) | \(e\left(\frac{193}{202}\right)\) | \(e\left(\frac{223}{303}\right)\) |
| \(\chi_{607}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{303}\right)\) | \(e\left(\frac{601}{606}\right)\) | \(e\left(\frac{110}{303}\right)\) | \(e\left(\frac{25}{606}\right)\) | \(e\left(\frac{35}{202}\right)\) | \(e\left(\frac{12}{101}\right)\) | \(e\left(\frac{55}{101}\right)\) | \(e\left(\frac{298}{303}\right)\) | \(e\left(\frac{45}{202}\right)\) | \(e\left(\frac{97}{303}\right)\) |
| \(\chi_{607}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{170}{303}\right)\) | \(e\left(\frac{563}{606}\right)\) | \(e\left(\frac{37}{303}\right)\) | \(e\left(\frac{215}{606}\right)\) | \(e\left(\frac{99}{202}\right)\) | \(e\left(\frac{83}{101}\right)\) | \(e\left(\frac{69}{101}\right)\) | \(e\left(\frac{260}{303}\right)\) | \(e\left(\frac{185}{202}\right)\) | \(e\left(\frac{107}{303}\right)\) |
| \(\chi_{607}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{303}\right)\) | \(e\left(\frac{37}{606}\right)\) | \(e\left(\frac{95}{303}\right)\) | \(e\left(\frac{421}{606}\right)\) | \(e\left(\frac{145}{202}\right)\) | \(e\left(\frac{93}{101}\right)\) | \(e\left(\frac{98}{101}\right)\) | \(e\left(\frac{37}{303}\right)\) | \(e\left(\frac{71}{202}\right)\) | \(e\left(\frac{70}{303}\right)\) |
| \(\chi_{607}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{236}{303}\right)\) | \(e\left(\frac{557}{606}\right)\) | \(e\left(\frac{169}{303}\right)\) | \(e\left(\frac{245}{606}\right)\) | \(e\left(\frac{141}{202}\right)\) | \(e\left(\frac{57}{101}\right)\) | \(e\left(\frac{34}{101}\right)\) | \(e\left(\frac{254}{303}\right)\) | \(e\left(\frac{37}{202}\right)\) | \(e\left(\frac{284}{303}\right)\) |
| \(\chi_{607}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{208}{303}\right)\) | \(e\left(\frac{229}{606}\right)\) | \(e\left(\frac{113}{303}\right)\) | \(e\left(\frac{67}{606}\right)\) | \(e\left(\frac{13}{202}\right)\) | \(e\left(\frac{16}{101}\right)\) | \(e\left(\frac{6}{101}\right)\) | \(e\left(\frac{229}{303}\right)\) | \(e\left(\frac{161}{202}\right)\) | \(e\left(\frac{163}{303}\right)\) |
| \(\chi_{607}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{303}\right)\) | \(e\left(\frac{203}{606}\right)\) | \(e\left(\frac{79}{303}\right)\) | \(e\left(\frac{197}{606}\right)\) | \(e\left(\frac{195}{202}\right)\) | \(e\left(\frac{38}{101}\right)\) | \(e\left(\frac{90}{101}\right)\) | \(e\left(\frac{203}{303}\right)\) | \(e\left(\frac{193}{202}\right)\) | \(e\left(\frac{122}{303}\right)\) |
| \(\chi_{607}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{303}\right)\) | \(e\left(\frac{541}{606}\right)\) | \(e\left(\frac{218}{303}\right)\) | \(e\left(\frac{325}{606}\right)\) | \(e\left(\frac{51}{202}\right)\) | \(e\left(\frac{55}{101}\right)\) | \(e\left(\frac{8}{101}\right)\) | \(e\left(\frac{238}{303}\right)\) | \(e\left(\frac{181}{202}\right)\) | \(e\left(\frac{49}{303}\right)\) |
| \(\chi_{607}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{34}{303}\right)\) | \(e\left(\frac{355}{606}\right)\) | \(e\left(\frac{68}{303}\right)\) | \(e\left(\frac{43}{606}\right)\) | \(e\left(\frac{141}{202}\right)\) | \(e\left(\frac{57}{101}\right)\) | \(e\left(\frac{34}{101}\right)\) | \(e\left(\frac{52}{303}\right)\) | \(e\left(\frac{37}{202}\right)\) | \(e\left(\frac{82}{303}\right)\) |
| \(\chi_{607}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{218}{303}\right)\) | \(e\left(\frac{173}{606}\right)\) | \(e\left(\frac{133}{303}\right)\) | \(e\left(\frac{347}{606}\right)\) | \(e\left(\frac{1}{202}\right)\) | \(e\left(\frac{9}{101}\right)\) | \(e\left(\frac{16}{101}\right)\) | \(e\left(\frac{173}{303}\right)\) | \(e\left(\frac{59}{202}\right)\) | \(e\left(\frac{98}{303}\right)\) |
| \(\chi_{607}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{274}{303}\right)\) | \(e\left(\frac{223}{606}\right)\) | \(e\left(\frac{245}{303}\right)\) | \(e\left(\frac{97}{606}\right)\) | \(e\left(\frac{55}{202}\right)\) | \(e\left(\frac{91}{101}\right)\) | \(e\left(\frac{72}{101}\right)\) | \(e\left(\frac{223}{303}\right)\) | \(e\left(\frac{13}{202}\right)\) | \(e\left(\frac{37}{303}\right)\) |
| \(\chi_{607}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{140}{303}\right)\) | \(e\left(\frac{125}{606}\right)\) | \(e\left(\frac{280}{303}\right)\) | \(e\left(\frac{587}{606}\right)\) | \(e\left(\frac{135}{202}\right)\) | \(e\left(\frac{3}{101}\right)\) | \(e\left(\frac{39}{101}\right)\) | \(e\left(\frac{125}{303}\right)\) | \(e\left(\frac{87}{202}\right)\) | \(e\left(\frac{302}{303}\right)\) |
| \(\chi_{607}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{175}{303}\right)\) | \(e\left(\frac{535}{606}\right)\) | \(e\left(\frac{47}{303}\right)\) | \(e\left(\frac{355}{606}\right)\) | \(e\left(\frac{93}{202}\right)\) | \(e\left(\frac{29}{101}\right)\) | \(e\left(\frac{74}{101}\right)\) | \(e\left(\frac{232}{303}\right)\) | \(e\left(\frac{33}{202}\right)\) | \(e\left(\frac{226}{303}\right)\) |
| \(\chi_{607}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{130}{303}\right)\) | \(e\left(\frac{181}{606}\right)\) | \(e\left(\frac{260}{303}\right)\) | \(e\left(\frac{307}{606}\right)\) | \(e\left(\frac{147}{202}\right)\) | \(e\left(\frac{10}{101}\right)\) | \(e\left(\frac{29}{101}\right)\) | \(e\left(\frac{181}{303}\right)\) | \(e\left(\frac{189}{202}\right)\) | \(e\left(\frac{64}{303}\right)\) |
| \(\chi_{607}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{209}{303}\right)\) | \(e\left(\frac{587}{606}\right)\) | \(e\left(\frac{115}{303}\right)\) | \(e\left(\frac{95}{606}\right)\) | \(e\left(\frac{133}{202}\right)\) | \(e\left(\frac{86}{101}\right)\) | \(e\left(\frac{7}{101}\right)\) | \(e\left(\frac{284}{303}\right)\) | \(e\left(\frac{171}{202}\right)\) | \(e\left(\frac{5}{303}\right)\) |
| \(\chi_{607}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{205}{303}\right)\) | \(e\left(\frac{367}{606}\right)\) | \(e\left(\frac{107}{303}\right)\) | \(e\left(\frac{589}{606}\right)\) | \(e\left(\frac{57}{202}\right)\) | \(e\left(\frac{8}{101}\right)\) | \(e\left(\frac{3}{101}\right)\) | \(e\left(\frac{64}{303}\right)\) | \(e\left(\frac{131}{202}\right)\) | \(e\left(\frac{31}{303}\right)\) |
| \(\chi_{607}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{303}\right)\) | \(e\left(\frac{151}{606}\right)\) | \(e\left(\frac{11}{303}\right)\) | \(e\left(\frac{457}{606}\right)\) | \(e\left(\frac{155}{202}\right)\) | \(e\left(\frac{82}{101}\right)\) | \(e\left(\frac{56}{101}\right)\) | \(e\left(\frac{151}{303}\right)\) | \(e\left(\frac{55}{202}\right)\) | \(e\left(\frac{40}{303}\right)\) |
| \(\chi_{607}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{206}{303}\right)\) | \(e\left(\frac{119}{606}\right)\) | \(e\left(\frac{109}{303}\right)\) | \(e\left(\frac{11}{606}\right)\) | \(e\left(\frac{177}{202}\right)\) | \(e\left(\frac{78}{101}\right)\) | \(e\left(\frac{4}{101}\right)\) | \(e\left(\frac{119}{303}\right)\) | \(e\left(\frac{141}{202}\right)\) | \(e\left(\frac{176}{303}\right)\) |
| \(\chi_{607}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{303}\right)\) | \(e\left(\frac{425}{606}\right)\) | \(e\left(\frac{43}{303}\right)\) | \(e\left(\frac{299}{606}\right)\) | \(e\left(\frac{55}{202}\right)\) | \(e\left(\frac{91}{101}\right)\) | \(e\left(\frac{72}{101}\right)\) | \(e\left(\frac{122}{303}\right)\) | \(e\left(\frac{13}{202}\right)\) | \(e\left(\frac{239}{303}\right)\) |
| \(\chi_{607}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{203}{303}\right)\) | \(e\left(\frac{257}{606}\right)\) | \(e\left(\frac{103}{303}\right)\) | \(e\left(\frac{533}{606}\right)\) | \(e\left(\frac{19}{202}\right)\) | \(e\left(\frac{70}{101}\right)\) | \(e\left(\frac{1}{101}\right)\) | \(e\left(\frac{257}{303}\right)\) | \(e\left(\frac{111}{202}\right)\) | \(e\left(\frac{44}{303}\right)\) |
| \(\chi_{607}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{303}\right)\) | \(e\left(\frac{599}{606}\right)\) | \(e\left(\frac{154}{303}\right)\) | \(e\left(\frac{35}{606}\right)\) | \(e\left(\frac{49}{202}\right)\) | \(e\left(\frac{37}{101}\right)\) | \(e\left(\frac{77}{101}\right)\) | \(e\left(\frac{296}{303}\right)\) | \(e\left(\frac{63}{202}\right)\) | \(e\left(\frac{257}{303}\right)\) |
| \(\chi_{607}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{303}\right)\) | \(e\left(\frac{365}{606}\right)\) | \(e\left(\frac{151}{303}\right)\) | \(e\left(\frac{599}{606}\right)\) | \(e\left(\frac{71}{202}\right)\) | \(e\left(\frac{33}{101}\right)\) | \(e\left(\frac{25}{101}\right)\) | \(e\left(\frac{62}{303}\right)\) | \(e\left(\frac{149}{202}\right)\) | \(e\left(\frac{191}{303}\right)\) |
| \(\chi_{607}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{303}\right)\) | \(e\left(\frac{103}{606}\right)\) | \(e\left(\frac{158}{303}\right)\) | \(e\left(\frac{91}{606}\right)\) | \(e\left(\frac{87}{202}\right)\) | \(e\left(\frac{76}{101}\right)\) | \(e\left(\frac{79}{101}\right)\) | \(e\left(\frac{103}{303}\right)\) | \(e\left(\frac{83}{202}\right)\) | \(e\left(\frac{244}{303}\right)\) |
| \(\chi_{607}(84,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{86}{303}\right)\) | \(e\left(\frac{185}{606}\right)\) | \(e\left(\frac{172}{303}\right)\) | \(e\left(\frac{287}{606}\right)\) | \(e\left(\frac{119}{202}\right)\) | \(e\left(\frac{61}{101}\right)\) | \(e\left(\frac{86}{101}\right)\) | \(e\left(\frac{185}{303}\right)\) | \(e\left(\frac{153}{202}\right)\) | \(e\left(\frac{47}{303}\right)\) |
| \(\chi_{607}(90,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{275}{303}\right)\) | \(e\left(\frac{581}{606}\right)\) | \(e\left(\frac{247}{303}\right)\) | \(e\left(\frac{125}{606}\right)\) | \(e\left(\frac{175}{202}\right)\) | \(e\left(\frac{60}{101}\right)\) | \(e\left(\frac{73}{101}\right)\) | \(e\left(\frac{278}{303}\right)\) | \(e\left(\frac{23}{202}\right)\) | \(e\left(\frac{182}{303}\right)\) |
| \(\chi_{607}(96,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{290}{303}\right)\) | \(e\left(\frac{497}{606}\right)\) | \(e\left(\frac{277}{303}\right)\) | \(e\left(\frac{545}{606}\right)\) | \(e\left(\frac{157}{202}\right)\) | \(e\left(\frac{100}{101}\right)\) | \(e\left(\frac{88}{101}\right)\) | \(e\left(\frac{194}{303}\right)\) | \(e\left(\frac{173}{202}\right)\) | \(e\left(\frac{236}{303}\right)\) |
| \(\chi_{607}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{303}\right)\) | \(e\left(\frac{467}{606}\right)\) | \(e\left(\frac{28}{303}\right)\) | \(e\left(\frac{89}{606}\right)\) | \(e\left(\frac{165}{202}\right)\) | \(e\left(\frac{71}{101}\right)\) | \(e\left(\frac{14}{101}\right)\) | \(e\left(\frac{164}{303}\right)\) | \(e\left(\frac{39}{202}\right)\) | \(e\left(\frac{212}{303}\right)\) |
| \(\chi_{607}(108,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{148}{303}\right)\) | \(e\left(\frac{565}{606}\right)\) | \(e\left(\frac{296}{303}\right)\) | \(e\left(\frac{205}{606}\right)\) | \(e\left(\frac{85}{202}\right)\) | \(e\left(\frac{58}{101}\right)\) | \(e\left(\frac{47}{101}\right)\) | \(e\left(\frac{262}{303}\right)\) | \(e\left(\frac{167}{202}\right)\) | \(e\left(\frac{250}{303}\right)\) |
| \(\chi_{607}(110,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{239}{303}\right)\) | \(e\left(\frac{419}{606}\right)\) | \(e\left(\frac{175}{303}\right)\) | \(e\left(\frac{329}{606}\right)\) | \(e\left(\frac{97}{202}\right)\) | \(e\left(\frac{65}{101}\right)\) | \(e\left(\frac{37}{101}\right)\) | \(e\left(\frac{116}{303}\right)\) | \(e\left(\frac{67}{202}\right)\) | \(e\left(\frac{113}{303}\right)\) |
| \(\chi_{607}(113,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{92}{303}\right)\) | \(e\left(\frac{515}{606}\right)\) | \(e\left(\frac{184}{303}\right)\) | \(e\left(\frac{455}{606}\right)\) | \(e\left(\frac{31}{202}\right)\) | \(e\left(\frac{77}{101}\right)\) | \(e\left(\frac{92}{101}\right)\) | \(e\left(\frac{212}{303}\right)\) | \(e\left(\frac{11}{202}\right)\) | \(e\left(\frac{8}{303}\right)\) |
| \(\chi_{607}(114,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{303}\right)\) | \(e\left(\frac{89}{606}\right)\) | \(e\left(\frac{163}{303}\right)\) | \(e\left(\frac{161}{606}\right)\) | \(e\left(\frac{185}{202}\right)\) | \(e\left(\frac{49}{101}\right)\) | \(e\left(\frac{31}{101}\right)\) | \(e\left(\frac{89}{303}\right)\) | \(e\left(\frac{7}{202}\right)\) | \(e\left(\frac{152}{303}\right)\) |