from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(61731, base_ring=CyclotomicField(2166))
M = H._module
chi = DirichletCharacter(H, M([1444,164]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,61731))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(61731\) | |
| Conductor: | \(61731\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1083\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{1083})$ |
| Fixed field: | Number field defined by a degree 1083 polynomial (not computed) |
First 31 of 684 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{61731}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{268}{361}\right)\) | \(e\left(\frac{175}{361}\right)\) | \(e\left(\frac{461}{1083}\right)\) | \(e\left(\frac{995}{1083}\right)\) | \(e\left(\frac{82}{361}\right)\) | \(e\left(\frac{182}{1083}\right)\) | \(e\left(\frac{650}{1083}\right)\) | \(e\left(\frac{240}{361}\right)\) | \(e\left(\frac{716}{1083}\right)\) | \(e\left(\frac{350}{361}\right)\) |
| \(\chi_{61731}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{175}{361}\right)\) | \(e\left(\frac{350}{361}\right)\) | \(e\left(\frac{922}{1083}\right)\) | \(e\left(\frac{907}{1083}\right)\) | \(e\left(\frac{164}{361}\right)\) | \(e\left(\frac{364}{1083}\right)\) | \(e\left(\frac{217}{1083}\right)\) | \(e\left(\frac{119}{361}\right)\) | \(e\left(\frac{349}{1083}\right)\) | \(e\left(\frac{339}{361}\right)\) |
| \(\chi_{61731}(178,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{361}\right)\) | \(e\left(\frac{146}{361}\right)\) | \(e\left(\frac{857}{1083}\right)\) | \(e\left(\frac{170}{1083}\right)\) | \(e\left(\frac{219}{361}\right)\) | \(e\left(\frac{1076}{1083}\right)\) | \(e\left(\frac{1058}{1083}\right)\) | \(e\left(\frac{324}{361}\right)\) | \(e\left(\frac{389}{1083}\right)\) | \(e\left(\frac{292}{361}\right)\) |
| \(\chi_{61731}(220,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{361}\right)\) | \(e\left(\frac{12}{361}\right)\) | \(e\left(\frac{832}{1083}\right)\) | \(e\left(\frac{553}{1083}\right)\) | \(e\left(\frac{18}{361}\right)\) | \(e\left(\frac{850}{1083}\right)\) | \(e\left(\frac{715}{1083}\right)\) | \(e\left(\frac{264}{361}\right)\) | \(e\left(\frac{571}{1083}\right)\) | \(e\left(\frac{24}{361}\right)\) |
| \(\chi_{61731}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{258}{361}\right)\) | \(e\left(\frac{155}{361}\right)\) | \(e\left(\frac{398}{1083}\right)\) | \(e\left(\frac{314}{1083}\right)\) | \(e\left(\frac{52}{361}\right)\) | \(e\left(\frac{89}{1083}\right)\) | \(e\left(\frac{782}{1083}\right)\) | \(e\left(\frac{161}{361}\right)\) | \(e\left(\frac{5}{1083}\right)\) | \(e\left(\frac{310}{361}\right)\) |
| \(\chi_{61731}(391,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{331}{361}\right)\) | \(e\left(\frac{301}{361}\right)\) | \(e\left(\frac{172}{1083}\right)\) | \(e\left(\frac{484}{1083}\right)\) | \(e\left(\frac{271}{361}\right)\) | \(e\left(\frac{82}{1083}\right)\) | \(e\left(\frac{757}{1083}\right)\) | \(e\left(\frac{124}{361}\right)\) | \(e\left(\frac{394}{1083}\right)\) | \(e\left(\frac{241}{361}\right)\) |
| \(\chi_{61731}(520,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{361}\right)\) | \(e\left(\frac{202}{361}\right)\) | \(e\left(\frac{167}{1083}\right)\) | \(e\left(\frac{344}{1083}\right)\) | \(e\left(\frac{303}{361}\right)\) | \(e\left(\frac{470}{1083}\right)\) | \(e\left(\frac{905}{1083}\right)\) | \(e\left(\frac{112}{361}\right)\) | \(e\left(\frac{647}{1083}\right)\) | \(e\left(\frac{43}{361}\right)\) |
| \(\chi_{61731}(562,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{361}\right)\) | \(e\left(\frac{134}{361}\right)\) | \(e\left(\frac{25}{1083}\right)\) | \(e\left(\frac{700}{1083}\right)\) | \(e\left(\frac{201}{361}\right)\) | \(e\left(\frac{226}{1083}\right)\) | \(e\left(\frac{343}{1083}\right)\) | \(e\left(\frac{60}{361}\right)\) | \(e\left(\frac{901}{1083}\right)\) | \(e\left(\frac{268}{361}\right)\) |
| \(\chi_{61731}(691,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{324}{361}\right)\) | \(e\left(\frac{287}{361}\right)\) | \(e\left(\frac{164}{1083}\right)\) | \(e\left(\frac{260}{1083}\right)\) | \(e\left(\frac{250}{361}\right)\) | \(e\left(\frac{53}{1083}\right)\) | \(e\left(\frac{344}{1083}\right)\) | \(e\left(\frac{177}{361}\right)\) | \(e\left(\frac{149}{1083}\right)\) | \(e\left(\frac{213}{361}\right)\) |
| \(\chi_{61731}(733,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{297}{361}\right)\) | \(e\left(\frac{233}{361}\right)\) | \(e\left(\frac{391}{1083}\right)\) | \(e\left(\frac{118}{1083}\right)\) | \(e\left(\frac{169}{361}\right)\) | \(e\left(\frac{199}{1083}\right)\) | \(e\left(\frac{556}{1083}\right)\) | \(e\left(\frac{72}{361}\right)\) | \(e\left(\frac{1009}{1083}\right)\) | \(e\left(\frac{105}{361}\right)\) |
| \(\chi_{61731}(862,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{205}{361}\right)\) | \(e\left(\frac{49}{361}\right)\) | \(e\left(\frac{389}{1083}\right)\) | \(e\left(\frac{62}{1083}\right)\) | \(e\left(\frac{254}{361}\right)\) | \(e\left(\frac{1004}{1083}\right)\) | \(e\left(\frac{182}{1083}\right)\) | \(e\left(\frac{356}{361}\right)\) | \(e\left(\frac{677}{1083}\right)\) | \(e\left(\frac{98}{361}\right)\) |
| \(\chi_{61731}(904,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{299}{361}\right)\) | \(e\left(\frac{237}{361}\right)\) | \(e\left(\frac{187}{1083}\right)\) | \(e\left(\frac{904}{1083}\right)\) | \(e\left(\frac{175}{361}\right)\) | \(e\left(\frac{1}{1083}\right)\) | \(e\left(\frac{313}{1083}\right)\) | \(e\left(\frac{160}{361}\right)\) | \(e\left(\frac{718}{1083}\right)\) | \(e\left(\frac{113}{361}\right)\) |
| \(\chi_{61731}(1033,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{105}{361}\right)\) | \(e\left(\frac{210}{361}\right)\) | \(e\left(\frac{842}{1083}\right)\) | \(e\left(\frac{833}{1083}\right)\) | \(e\left(\frac{315}{361}\right)\) | \(e\left(\frac{74}{1083}\right)\) | \(e\left(\frac{419}{1083}\right)\) | \(e\left(\frac{288}{361}\right)\) | \(e\left(\frac{65}{1083}\right)\) | \(e\left(\frac{59}{361}\right)\) |
| \(\chi_{61731}(1075,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{361}\right)\) | \(e\left(\frac{146}{361}\right)\) | \(e\left(\frac{496}{1083}\right)\) | \(e\left(\frac{892}{1083}\right)\) | \(e\left(\frac{219}{361}\right)\) | \(e\left(\frac{715}{1083}\right)\) | \(e\left(\frac{697}{1083}\right)\) | \(e\left(\frac{324}{361}\right)\) | \(e\left(\frac{28}{1083}\right)\) | \(e\left(\frac{292}{361}\right)\) |
| \(\chi_{61731}(1204,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{361}\right)\) | \(e\left(\frac{48}{361}\right)\) | \(e\left(\frac{440}{1083}\right)\) | \(e\left(\frac{407}{1083}\right)\) | \(e\left(\frac{72}{361}\right)\) | \(e\left(\frac{512}{1083}\right)\) | \(e\left(\frac{1055}{1083}\right)\) | \(e\left(\frac{334}{361}\right)\) | \(e\left(\frac{479}{1083}\right)\) | \(e\left(\frac{96}{361}\right)\) |
| \(\chi_{61731}(1246,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{341}{361}\right)\) | \(e\left(\frac{321}{361}\right)\) | \(e\left(\frac{235}{1083}\right)\) | \(e\left(\frac{82}{1083}\right)\) | \(e\left(\frac{301}{361}\right)\) | \(e\left(\frac{175}{1083}\right)\) | \(e\left(\frac{625}{1083}\right)\) | \(e\left(\frac{203}{361}\right)\) | \(e\left(\frac{22}{1083}\right)\) | \(e\left(\frac{281}{361}\right)\) |
| \(\chi_{61731}(1417,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{361}\right)\) | \(e\left(\frac{40}{361}\right)\) | \(e\left(\frac{487}{1083}\right)\) | \(e\left(\frac{640}{1083}\right)\) | \(e\left(\frac{60}{361}\right)\) | \(e\left(\frac{547}{1083}\right)\) | \(e\left(\frac{97}{1083}\right)\) | \(e\left(\frac{158}{361}\right)\) | \(e\left(\frac{700}{1083}\right)\) | \(e\left(\frac{80}{361}\right)\) |
| \(\chi_{61731}(1546,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{280}{361}\right)\) | \(e\left(\frac{199}{361}\right)\) | \(e\left(\frac{320}{1083}\right)\) | \(e\left(\frac{296}{1083}\right)\) | \(e\left(\frac{118}{361}\right)\) | \(e\left(\frac{77}{1083}\right)\) | \(e\left(\frac{275}{1083}\right)\) | \(e\left(\frac{46}{361}\right)\) | \(e\left(\frac{53}{1083}\right)\) | \(e\left(\frac{37}{361}\right)\) |
| \(\chi_{61731}(1588,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{361}\right)\) | \(e\left(\frac{25}{361}\right)\) | \(e\left(\frac{169}{1083}\right)\) | \(e\left(\frac{400}{1083}\right)\) | \(e\left(\frac{218}{361}\right)\) | \(e\left(\frac{748}{1083}\right)\) | \(e\left(\frac{196}{1083}\right)\) | \(e\left(\frac{189}{361}\right)\) | \(e\left(\frac{979}{1083}\right)\) | \(e\left(\frac{50}{361}\right)\) |
| \(\chi_{61731}(1717,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{256}{361}\right)\) | \(e\left(\frac{151}{361}\right)\) | \(e\left(\frac{602}{1083}\right)\) | \(e\left(\frac{611}{1083}\right)\) | \(e\left(\frac{46}{361}\right)\) | \(e\left(\frac{287}{1083}\right)\) | \(e\left(\frac{1025}{1083}\right)\) | \(e\left(\frac{73}{361}\right)\) | \(e\left(\frac{296}{1083}\right)\) | \(e\left(\frac{302}{361}\right)\) |
| \(\chi_{61731}(1759,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{138}{361}\right)\) | \(e\left(\frac{276}{361}\right)\) | \(e\left(\frac{364}{1083}\right)\) | \(e\left(\frac{445}{1083}\right)\) | \(e\left(\frac{53}{361}\right)\) | \(e\left(\frac{778}{1083}\right)\) | \(e\left(\frac{922}{1083}\right)\) | \(e\left(\frac{296}{361}\right)\) | \(e\left(\frac{859}{1083}\right)\) | \(e\left(\frac{191}{361}\right)\) |
| \(\chi_{61731}(1888,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{361}\right)\) | \(e\left(\frac{141}{361}\right)\) | \(e\left(\frac{29}{1083}\right)\) | \(e\left(\frac{812}{1083}\right)\) | \(e\left(\frac{31}{361}\right)\) | \(e\left(\frac{782}{1083}\right)\) | \(e\left(\frac{8}{1083}\right)\) | \(e\left(\frac{214}{361}\right)\) | \(e\left(\frac{482}{1083}\right)\) | \(e\left(\frac{282}{361}\right)\) |
| \(\chi_{61731}(1930,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{216}{361}\right)\) | \(e\left(\frac{71}{361}\right)\) | \(e\left(\frac{1072}{1083}\right)\) | \(e\left(\frac{775}{1083}\right)\) | \(e\left(\frac{287}{361}\right)\) | \(e\left(\frac{637}{1083}\right)\) | \(e\left(\frac{109}{1083}\right)\) | \(e\left(\frac{118}{361}\right)\) | \(e\left(\frac{340}{1083}\right)\) | \(e\left(\frac{142}{361}\right)\) |
| \(\chi_{61731}(2059,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{265}{361}\right)\) | \(e\left(\frac{169}{361}\right)\) | \(e\left(\frac{767}{1083}\right)\) | \(e\left(\frac{899}{1083}\right)\) | \(e\left(\frac{73}{361}\right)\) | \(e\left(\frac{479}{1083}\right)\) | \(e\left(\frac{473}{1083}\right)\) | \(e\left(\frac{108}{361}\right)\) | \(e\left(\frac{611}{1083}\right)\) | \(e\left(\frac{338}{361}\right)\) |
| \(\chi_{61731}(2101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{66}{361}\right)\) | \(e\left(\frac{132}{361}\right)\) | \(e\left(\frac{127}{1083}\right)\) | \(e\left(\frac{307}{1083}\right)\) | \(e\left(\frac{198}{361}\right)\) | \(e\left(\frac{325}{1083}\right)\) | \(e\left(\frac{1006}{1083}\right)\) | \(e\left(\frac{16}{361}\right)\) | \(e\left(\frac{505}{1083}\right)\) | \(e\left(\frac{264}{361}\right)\) |
| \(\chi_{61731}(2230,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{298}{361}\right)\) | \(e\left(\frac{235}{361}\right)\) | \(e\left(\frac{650}{1083}\right)\) | \(e\left(\frac{872}{1083}\right)\) | \(e\left(\frac{172}{361}\right)\) | \(e\left(\frac{461}{1083}\right)\) | \(e\left(\frac{254}{1083}\right)\) | \(e\left(\frac{116}{361}\right)\) | \(e\left(\frac{683}{1083}\right)\) | \(e\left(\frac{109}{361}\right)\) |
| \(\chi_{61731}(2272,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{361}\right)\) | \(e\left(\frac{98}{361}\right)\) | \(e\left(\frac{778}{1083}\right)\) | \(e\left(\frac{124}{1083}\right)\) | \(e\left(\frac{147}{361}\right)\) | \(e\left(\frac{925}{1083}\right)\) | \(e\left(\frac{364}{1083}\right)\) | \(e\left(\frac{351}{361}\right)\) | \(e\left(\frac{271}{1083}\right)\) | \(e\left(\frac{196}{361}\right)\) |
| \(\chi_{61731}(2401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{350}{361}\right)\) | \(e\left(\frac{339}{361}\right)\) | \(e\left(\frac{761}{1083}\right)\) | \(e\left(\frac{731}{1083}\right)\) | \(e\left(\frac{328}{361}\right)\) | \(e\left(\frac{728}{1083}\right)\) | \(e\left(\frac{434}{1083}\right)\) | \(e\left(\frac{238}{361}\right)\) | \(e\left(\frac{698}{1083}\right)\) | \(e\left(\frac{317}{361}\right)\) |
| \(\chi_{61731}(2443,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{165}{361}\right)\) | \(e\left(\frac{330}{361}\right)\) | \(e\left(\frac{859}{1083}\right)\) | \(e\left(\frac{226}{1083}\right)\) | \(e\left(\frac{134}{361}\right)\) | \(e\left(\frac{271}{1083}\right)\) | \(e\left(\frac{349}{1083}\right)\) | \(e\left(\frac{40}{361}\right)\) | \(e\left(\frac{721}{1083}\right)\) | \(e\left(\frac{299}{361}\right)\) |
| \(\chi_{61731}(2572,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{60}{361}\right)\) | \(e\left(\frac{120}{361}\right)\) | \(e\left(\frac{17}{1083}\right)\) | \(e\left(\frac{476}{1083}\right)\) | \(e\left(\frac{180}{361}\right)\) | \(e\left(\frac{197}{1083}\right)\) | \(e\left(\frac{1013}{1083}\right)\) | \(e\left(\frac{113}{361}\right)\) | \(e\left(\frac{656}{1083}\right)\) | \(e\left(\frac{240}{361}\right)\) |
| \(\chi_{61731}(2614,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{361}\right)\) | \(e\left(\frac{106}{361}\right)\) | \(e\left(\frac{370}{1083}\right)\) | \(e\left(\frac{613}{1083}\right)\) | \(e\left(\frac{159}{361}\right)\) | \(e\left(\frac{529}{1083}\right)\) | \(e\left(\frac{961}{1083}\right)\) | \(e\left(\frac{166}{361}\right)\) | \(e\left(\frac{772}{1083}\right)\) | \(e\left(\frac{212}{361}\right)\) |