from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1061, base_ring=CyclotomicField(1060))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1061))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1061\) | |
| Conductor: | \(1061\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1060\) | 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: | $\Q(\zeta_{1060})$ |
| Fixed field: | Number field defined by a degree 1060 polynomial (not computed) |
First 31 of 416 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1061}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{1060}\right)\) | \(e\left(\frac{167}{1060}\right)\) | \(e\left(\frac{1}{530}\right)\) | \(e\left(\frac{61}{265}\right)\) | \(e\left(\frac{42}{265}\right)\) | \(e\left(\frac{119}{265}\right)\) | \(e\left(\frac{3}{1060}\right)\) | \(e\left(\frac{167}{530}\right)\) | \(e\left(\frac{49}{212}\right)\) | \(e\left(\frac{301}{530}\right)\) |
| \(\chi_{1061}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{1060}\right)\) | \(e\left(\frac{329}{1060}\right)\) | \(e\left(\frac{167}{530}\right)\) | \(e\left(\frac{117}{265}\right)\) | \(e\left(\frac{124}{265}\right)\) | \(e\left(\frac{263}{265}\right)\) | \(e\left(\frac{501}{1060}\right)\) | \(e\left(\frac{329}{530}\right)\) | \(e\left(\frac{127}{212}\right)\) | \(e\left(\frac{447}{530}\right)\) |
| \(\chi_{1061}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{1060}\right)\) | \(e\left(\frac{501}{1060}\right)\) | \(e\left(\frac{3}{530}\right)\) | \(e\left(\frac{183}{265}\right)\) | \(e\left(\frac{126}{265}\right)\) | \(e\left(\frac{92}{265}\right)\) | \(e\left(\frac{9}{1060}\right)\) | \(e\left(\frac{501}{530}\right)\) | \(e\left(\frac{147}{212}\right)\) | \(e\left(\frac{373}{530}\right)\) |
| \(\chi_{1061}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{1060}\right)\) | \(e\left(\frac{663}{1060}\right)\) | \(e\left(\frac{169}{530}\right)\) | \(e\left(\frac{239}{265}\right)\) | \(e\left(\frac{208}{265}\right)\) | \(e\left(\frac{236}{265}\right)\) | \(e\left(\frac{507}{1060}\right)\) | \(e\left(\frac{133}{530}\right)\) | \(e\left(\frac{13}{212}\right)\) | \(e\left(\frac{519}{530}\right)\) |
| \(\chi_{1061}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{411}{1060}\right)\) | \(e\left(\frac{797}{1060}\right)\) | \(e\left(\frac{411}{530}\right)\) | \(e\left(\frac{161}{265}\right)\) | \(e\left(\frac{37}{265}\right)\) | \(e\left(\frac{149}{265}\right)\) | \(e\left(\frac{173}{1060}\right)\) | \(e\left(\frac{267}{530}\right)\) | \(e\left(\frac{211}{212}\right)\) | \(e\left(\frac{221}{530}\right)\) |
| \(\chi_{1061}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{643}{1060}\right)\) | \(e\left(\frac{321}{1060}\right)\) | \(e\left(\frac{113}{530}\right)\) | \(e\left(\frac{3}{265}\right)\) | \(e\left(\frac{241}{265}\right)\) | \(e\left(\frac{197}{265}\right)\) | \(e\left(\frac{869}{1060}\right)\) | \(e\left(\frac{321}{530}\right)\) | \(e\left(\frac{131}{212}\right)\) | \(e\left(\frac{93}{530}\right)\) |
| \(\chi_{1061}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{603}{1060}\right)\) | \(e\left(\frac{1}{1060}\right)\) | \(e\left(\frac{73}{530}\right)\) | \(e\left(\frac{213}{265}\right)\) | \(e\left(\frac{151}{265}\right)\) | \(e\left(\frac{207}{265}\right)\) | \(e\left(\frac{749}{1060}\right)\) | \(e\left(\frac{1}{530}\right)\) | \(e\left(\frac{79}{212}\right)\) | \(e\left(\frac{243}{530}\right)\) |
| \(\chi_{1061}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{501}{1060}\right)\) | \(e\left(\frac{987}{1060}\right)\) | \(e\left(\frac{501}{530}\right)\) | \(e\left(\frac{86}{265}\right)\) | \(e\left(\frac{107}{265}\right)\) | \(e\left(\frac{259}{265}\right)\) | \(e\left(\frac{443}{1060}\right)\) | \(e\left(\frac{457}{530}\right)\) | \(e\left(\frac{169}{212}\right)\) | \(e\left(\frac{281}{530}\right)\) |
| \(\chi_{1061}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{769}{1060}\right)\) | \(e\left(\frac{163}{1060}\right)\) | \(e\left(\frac{239}{530}\right)\) | \(e\left(\frac{4}{265}\right)\) | \(e\left(\frac{233}{265}\right)\) | \(e\left(\frac{86}{265}\right)\) | \(e\left(\frac{187}{1060}\right)\) | \(e\left(\frac{163}{530}\right)\) | \(e\left(\frac{157}{212}\right)\) | \(e\left(\frac{389}{530}\right)\) |
| \(\chi_{1061}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{749}{1060}\right)\) | \(e\left(\frac{3}{1060}\right)\) | \(e\left(\frac{219}{530}\right)\) | \(e\left(\frac{109}{265}\right)\) | \(e\left(\frac{188}{265}\right)\) | \(e\left(\frac{91}{265}\right)\) | \(e\left(\frac{127}{1060}\right)\) | \(e\left(\frac{3}{530}\right)\) | \(e\left(\frac{25}{212}\right)\) | \(e\left(\frac{199}{530}\right)\) |
| \(\chi_{1061}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{247}{1060}\right)\) | \(e\left(\frac{969}{1060}\right)\) | \(e\left(\frac{247}{530}\right)\) | \(e\left(\frac{227}{265}\right)\) | \(e\left(\frac{39}{265}\right)\) | \(e\left(\frac{243}{265}\right)\) | \(e\left(\frac{741}{1060}\right)\) | \(e\left(\frac{439}{530}\right)\) | \(e\left(\frac{19}{212}\right)\) | \(e\left(\frac{147}{530}\right)\) |
| \(\chi_{1061}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1049}{1060}\right)\) | \(e\left(\frac{283}{1060}\right)\) | \(e\left(\frac{519}{530}\right)\) | \(e\left(\frac{124}{265}\right)\) | \(e\left(\frac{68}{265}\right)\) | \(e\left(\frac{16}{265}\right)\) | \(e\left(\frac{1027}{1060}\right)\) | \(e\left(\frac{283}{530}\right)\) | \(e\left(\frac{97}{212}\right)\) | \(e\left(\frac{399}{530}\right)\) |
| \(\chi_{1061}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{879}{1060}\right)\) | \(e\left(\frac{513}{1060}\right)\) | \(e\left(\frac{349}{530}\right)\) | \(e\left(\frac{89}{265}\right)\) | \(e\left(\frac{83}{265}\right)\) | \(e\left(\frac{191}{265}\right)\) | \(e\left(\frac{517}{1060}\right)\) | \(e\left(\frac{513}{530}\right)\) | \(e\left(\frac{35}{212}\right)\) | \(e\left(\frac{109}{530}\right)\) |
| \(\chi_{1061}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{171}{1060}\right)\) | \(e\left(\frac{997}{1060}\right)\) | \(e\left(\frac{171}{530}\right)\) | \(e\left(\frac{96}{265}\right)\) | \(e\left(\frac{27}{265}\right)\) | \(e\left(\frac{209}{265}\right)\) | \(e\left(\frac{513}{1060}\right)\) | \(e\left(\frac{467}{530}\right)\) | \(e\left(\frac{111}{212}\right)\) | \(e\left(\frac{61}{530}\right)\) |
| \(\chi_{1061}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{489}{1060}\right)\) | \(e\left(\frac{43}{1060}\right)\) | \(e\left(\frac{489}{530}\right)\) | \(e\left(\frac{149}{265}\right)\) | \(e\left(\frac{133}{265}\right)\) | \(e\left(\frac{156}{265}\right)\) | \(e\left(\frac{407}{1060}\right)\) | \(e\left(\frac{43}{530}\right)\) | \(e\left(\frac{5}{212}\right)\) | \(e\left(\frac{379}{530}\right)\) |
| \(\chi_{1061}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{927}{1060}\right)\) | \(e\left(\frac{49}{1060}\right)\) | \(e\left(\frac{397}{530}\right)\) | \(e\left(\frac{102}{265}\right)\) | \(e\left(\frac{244}{265}\right)\) | \(e\left(\frac{73}{265}\right)\) | \(e\left(\frac{661}{1060}\right)\) | \(e\left(\frac{49}{530}\right)\) | \(e\left(\frac{55}{212}\right)\) | \(e\left(\frac{247}{530}\right)\) |
| \(\chi_{1061}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{479}{1060}\right)\) | \(e\left(\frac{493}{1060}\right)\) | \(e\left(\frac{479}{530}\right)\) | \(e\left(\frac{69}{265}\right)\) | \(e\left(\frac{243}{265}\right)\) | \(e\left(\frac{26}{265}\right)\) | \(e\left(\frac{377}{1060}\right)\) | \(e\left(\frac{493}{530}\right)\) | \(e\left(\frac{151}{212}\right)\) | \(e\left(\frac{19}{530}\right)\) |
| \(\chi_{1061}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{413}{1060}\right)\) | \(e\left(\frac{71}{1060}\right)\) | \(e\left(\frac{413}{530}\right)\) | \(e\left(\frac{18}{265}\right)\) | \(e\left(\frac{121}{265}\right)\) | \(e\left(\frac{122}{265}\right)\) | \(e\left(\frac{179}{1060}\right)\) | \(e\left(\frac{71}{530}\right)\) | \(e\left(\frac{97}{212}\right)\) | \(e\left(\frac{293}{530}\right)\) |
| \(\chi_{1061}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{511}{1060}\right)\) | \(e\left(\frac{537}{1060}\right)\) | \(e\left(\frac{511}{530}\right)\) | \(e\left(\frac{166}{265}\right)\) | \(e\left(\frac{262}{265}\right)\) | \(e\left(\frac{124}{265}\right)\) | \(e\left(\frac{473}{1060}\right)\) | \(e\left(\frac{7}{530}\right)\) | \(e\left(\frac{23}{212}\right)\) | \(e\left(\frac{111}{530}\right)\) |
| \(\chi_{1061}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{1060}\right)\) | \(e\left(\frac{183}{1060}\right)\) | \(e\left(\frac{109}{530}\right)\) | \(e\left(\frac{24}{265}\right)\) | \(e\left(\frac{73}{265}\right)\) | \(e\left(\frac{251}{265}\right)\) | \(e\left(\frac{327}{1060}\right)\) | \(e\left(\frac{183}{530}\right)\) | \(e\left(\frac{41}{212}\right)\) | \(e\left(\frac{479}{530}\right)\) |
| \(\chi_{1061}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{497}{1060}\right)\) | \(e\left(\frac{319}{1060}\right)\) | \(e\left(\frac{497}{530}\right)\) | \(e\left(\frac{107}{265}\right)\) | \(e\left(\frac{204}{265}\right)\) | \(e\left(\frac{48}{265}\right)\) | \(e\left(\frac{431}{1060}\right)\) | \(e\left(\frac{319}{530}\right)\) | \(e\left(\frac{185}{212}\right)\) | \(e\left(\frac{137}{530}\right)\) |
| \(\chi_{1061}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{721}{1060}\right)\) | \(e\left(\frac{627}{1060}\right)\) | \(e\left(\frac{191}{530}\right)\) | \(e\left(\frac{256}{265}\right)\) | \(e\left(\frac{72}{265}\right)\) | \(e\left(\frac{204}{265}\right)\) | \(e\left(\frac{43}{1060}\right)\) | \(e\left(\frac{97}{530}\right)\) | \(e\left(\frac{137}{212}\right)\) | \(e\left(\frac{251}{530}\right)\) |
| \(\chi_{1061}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{651}{1060}\right)\) | \(e\left(\frac{597}{1060}\right)\) | \(e\left(\frac{121}{530}\right)\) | \(e\left(\frac{226}{265}\right)\) | \(e\left(\frac{47}{265}\right)\) | \(e\left(\frac{89}{265}\right)\) | \(e\left(\frac{893}{1060}\right)\) | \(e\left(\frac{67}{530}\right)\) | \(e\left(\frac{99}{212}\right)\) | \(e\left(\frac{381}{530}\right)\) |
| \(\chi_{1061}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{337}{1060}\right)\) | \(e\left(\frac{99}{1060}\right)\) | \(e\left(\frac{337}{530}\right)\) | \(e\left(\frac{152}{265}\right)\) | \(e\left(\frac{109}{265}\right)\) | \(e\left(\frac{88}{265}\right)\) | \(e\left(\frac{1011}{1060}\right)\) | \(e\left(\frac{99}{530}\right)\) | \(e\left(\frac{189}{212}\right)\) | \(e\left(\frac{207}{530}\right)\) |
| \(\chi_{1061}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{541}{1060}\right)\) | \(e\left(\frac{247}{1060}\right)\) | \(e\left(\frac{11}{530}\right)\) | \(e\left(\frac{141}{265}\right)\) | \(e\left(\frac{197}{265}\right)\) | \(e\left(\frac{249}{265}\right)\) | \(e\left(\frac{563}{1060}\right)\) | \(e\left(\frac{247}{530}\right)\) | \(e\left(\frac{9}{212}\right)\) | \(e\left(\frac{131}{530}\right)\) |
| \(\chi_{1061}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{1060}\right)\) | \(e\left(\frac{211}{1060}\right)\) | \(e\left(\frac{33}{530}\right)\) | \(e\left(\frac{158}{265}\right)\) | \(e\left(\frac{61}{265}\right)\) | \(e\left(\frac{217}{265}\right)\) | \(e\left(\frac{99}{1060}\right)\) | \(e\left(\frac{211}{530}\right)\) | \(e\left(\frac{133}{212}\right)\) | \(e\left(\frac{393}{530}\right)\) |
| \(\chi_{1061}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{1060}\right)\) | \(e\left(\frac{967}{1060}\right)\) | \(e\left(\frac{101}{530}\right)\) | \(e\left(\frac{66}{265}\right)\) | \(e\left(\frac{2}{265}\right)\) | \(e\left(\frac{94}{265}\right)\) | \(e\left(\frac{303}{1060}\right)\) | \(e\left(\frac{437}{530}\right)\) | \(e\left(\frac{73}{212}\right)\) | \(e\left(\frac{191}{530}\right)\) |
| \(\chi_{1061}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{379}{1060}\right)\) | \(e\left(\frac{753}{1060}\right)\) | \(e\left(\frac{379}{530}\right)\) | \(e\left(\frac{64}{265}\right)\) | \(e\left(\frac{18}{265}\right)\) | \(e\left(\frac{51}{265}\right)\) | \(e\left(\frac{77}{1060}\right)\) | \(e\left(\frac{223}{530}\right)\) | \(e\left(\frac{127}{212}\right)\) | \(e\left(\frac{129}{530}\right)\) |
| \(\chi_{1061}(85,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{739}{1060}\right)\) | \(e\left(\frac{453}{1060}\right)\) | \(e\left(\frac{209}{530}\right)\) | \(e\left(\frac{29}{265}\right)\) | \(e\left(\frac{33}{265}\right)\) | \(e\left(\frac{226}{265}\right)\) | \(e\left(\frac{97}{1060}\right)\) | \(e\left(\frac{453}{530}\right)\) | \(e\left(\frac{171}{212}\right)\) | \(e\left(\frac{369}{530}\right)\) |
| \(\chi_{1061}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{307}{1060}\right)\) | \(e\left(\frac{389}{1060}\right)\) | \(e\left(\frac{307}{530}\right)\) | \(e\left(\frac{177}{265}\right)\) | \(e\left(\frac{174}{265}\right)\) | \(e\left(\frac{228}{265}\right)\) | \(e\left(\frac{921}{1060}\right)\) | \(e\left(\frac{389}{530}\right)\) | \(e\left(\frac{203}{212}\right)\) | \(e\left(\frac{187}{530}\right)\) |
| \(\chi_{1061}(90,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{579}{1060}\right)\) | \(e\left(\frac{233}{1060}\right)\) | \(e\left(\frac{49}{530}\right)\) | \(e\left(\frac{74}{265}\right)\) | \(e\left(\frac{203}{265}\right)\) | \(e\left(\frac{1}{265}\right)\) | \(e\left(\frac{677}{1060}\right)\) | \(e\left(\frac{233}{530}\right)\) | \(e\left(\frac{175}{212}\right)\) | \(e\left(\frac{439}{530}\right)\) |