from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4011, base_ring=CyclotomicField(570))
M = H._module
chi = DirichletCharacter(H, M([285,190,519]))
chi.galois_orbit()
[g,chi] = znchar(Mod(44,4011))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4011\) | |
| Conductor: | \(4011\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(570\) | 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: | $\Q(\zeta_{285})$ |
| Fixed field: | Number field defined by a degree 570 polynomial (not computed) |
First 31 of 144 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4011}(44,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{570}\right)\) | \(e\left(\frac{131}{285}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{131}{190}\right)\) | \(e\left(\frac{263}{285}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{93}{95}\right)\) | \(e\left(\frac{262}{285}\right)\) | \(e\left(\frac{37}{570}\right)\) | \(e\left(\frac{329}{570}\right)\) |
| \(\chi_{4011}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{523}{570}\right)\) | \(e\left(\frac{238}{285}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{143}{190}\right)\) | \(e\left(\frac{34}{285}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{29}{95}\right)\) | \(e\left(\frac{191}{285}\right)\) | \(e\left(\frac{461}{570}\right)\) | \(e\left(\frac{217}{570}\right)\) |
| \(\chi_{4011}(74,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{283}{570}\right)\) | \(e\left(\frac{283}{285}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{93}{190}\right)\) | \(e\left(\frac{244}{285}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{74}{95}\right)\) | \(e\left(\frac{281}{285}\right)\) | \(e\left(\frac{341}{570}\right)\) | \(e\left(\frac{367}{570}\right)\) |
| \(\chi_{4011}(95,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{367}{570}\right)\) | \(e\left(\frac{82}{285}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{177}{190}\right)\) | \(e\left(\frac{256}{285}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{6}{95}\right)\) | \(e\left(\frac{164}{285}\right)\) | \(e\left(\frac{269}{570}\right)\) | \(e\left(\frac{343}{570}\right)\) |
| \(\chi_{4011}(116,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{487}{570}\right)\) | \(e\left(\frac{202}{285}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{107}{190}\right)\) | \(e\left(\frac{151}{285}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{31}{95}\right)\) | \(e\left(\frac{119}{285}\right)\) | \(e\left(\frac{329}{570}\right)\) | \(e\left(\frac{553}{570}\right)\) |
| \(\chi_{4011}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{349}{570}\right)\) | \(e\left(\frac{64}{285}\right)\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{159}{190}\right)\) | \(e\left(\frac{172}{285}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{7}{95}\right)\) | \(e\left(\frac{128}{285}\right)\) | \(e\left(\frac{203}{570}\right)\) | \(e\left(\frac{511}{570}\right)\) |
| \(\chi_{4011}(179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{570}\right)\) | \(e\left(\frac{43}{285}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{43}{190}\right)\) | \(e\left(\frac{169}{285}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{24}{95}\right)\) | \(e\left(\frac{86}{285}\right)\) | \(e\left(\frac{221}{570}\right)\) | \(e\left(\frac{517}{570}\right)\) |
| \(\chi_{4011}(212,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{359}{570}\right)\) | \(e\left(\frac{74}{285}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{169}{190}\right)\) | \(e\left(\frac{92}{285}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{17}{95}\right)\) | \(e\left(\frac{148}{285}\right)\) | \(e\left(\frac{493}{570}\right)\) | \(e\left(\frac{101}{570}\right)\) |
| \(\chi_{4011}(233,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{467}{570}\right)\) | \(e\left(\frac{182}{285}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{87}{190}\right)\) | \(e\left(\frac{26}{285}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{11}{95}\right)\) | \(e\left(\frac{79}{285}\right)\) | \(e\left(\frac{319}{570}\right)\) | \(e\left(\frac{233}{570}\right)\) |
| \(\chi_{4011}(254,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{281}{570}\right)\) | \(e\left(\frac{281}{285}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{91}{190}\right)\) | \(e\left(\frac{203}{285}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{53}{95}\right)\) | \(e\left(\frac{277}{285}\right)\) | \(e\left(\frac{397}{570}\right)\) | \(e\left(\frac{449}{570}\right)\) |
| \(\chi_{4011}(284,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{570}\right)\) | \(e\left(\frac{127}{285}\right)\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{127}{190}\right)\) | \(e\left(\frac{181}{285}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{51}{95}\right)\) | \(e\left(\frac{254}{285}\right)\) | \(e\left(\frac{149}{570}\right)\) | \(e\left(\frac{493}{570}\right)\) |
| \(\chi_{4011}(296,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{570}\right)\) | \(e\left(\frac{119}{285}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{119}{190}\right)\) | \(e\left(\frac{17}{285}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{62}{95}\right)\) | \(e\left(\frac{238}{285}\right)\) | \(e\left(\frac{373}{570}\right)\) | \(e\left(\frac{251}{570}\right)\) |
| \(\chi_{4011}(305,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{570}\right)\) | \(e\left(\frac{67}{285}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{67}{190}\right)\) | \(e\left(\frac{91}{285}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{86}{95}\right)\) | \(e\left(\frac{134}{285}\right)\) | \(e\left(\frac{119}{570}\right)\) | \(e\left(\frac{103}{570}\right)\) |
| \(\chi_{4011}(317,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{389}{570}\right)\) | \(e\left(\frac{104}{285}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{9}{190}\right)\) | \(e\left(\frac{137}{285}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{47}{95}\right)\) | \(e\left(\frac{208}{285}\right)\) | \(e\left(\frac{223}{570}\right)\) | \(e\left(\frac{11}{570}\right)\) |
| \(\chi_{4011}(359,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{570}\right)\) | \(e\left(\frac{113}{285}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{113}{190}\right)\) | \(e\left(\frac{179}{285}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{94}{95}\right)\) | \(e\left(\frac{226}{285}\right)\) | \(e\left(\frac{541}{570}\right)\) | \(e\left(\frac{497}{570}\right)\) |
| \(\chi_{4011}(380,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{203}{570}\right)\) | \(e\left(\frac{203}{285}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{13}{190}\right)\) | \(e\left(\frac{29}{285}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{89}{95}\right)\) | \(e\left(\frac{121}{285}\right)\) | \(e\left(\frac{301}{570}\right)\) | \(e\left(\frac{227}{570}\right)\) |
| \(\chi_{4011}(401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{570}\right)\) | \(e\left(\frac{227}{285}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{37}{190}\right)\) | \(e\left(\frac{236}{285}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{56}{95}\right)\) | \(e\left(\frac{169}{285}\right)\) | \(e\left(\frac{199}{570}\right)\) | \(e\left(\frac{383}{570}\right)\) |
| \(\chi_{4011}(410,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{463}{570}\right)\) | \(e\left(\frac{178}{285}\right)\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{83}{190}\right)\) | \(e\left(\frac{229}{285}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{64}{95}\right)\) | \(e\left(\frac{71}{285}\right)\) | \(e\left(\frac{431}{570}\right)\) | \(e\left(\frac{397}{570}\right)\) |
| \(\chi_{4011}(443,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{497}{570}\right)\) | \(e\left(\frac{212}{285}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{117}{190}\right)\) | \(e\left(\frac{71}{285}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{41}{95}\right)\) | \(e\left(\frac{139}{285}\right)\) | \(e\left(\frac{49}{570}\right)\) | \(e\left(\frac{143}{570}\right)\) |
| \(\chi_{4011}(473,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{570}\right)\) | \(e\left(\frac{211}{285}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{21}{190}\right)\) | \(e\left(\frac{193}{285}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{78}{95}\right)\) | \(e\left(\frac{137}{285}\right)\) | \(e\left(\frac{77}{570}\right)\) | \(e\left(\frac{469}{570}\right)\) |
| \(\chi_{4011}(494,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{570}\right)\) | \(e\left(\frac{109}{285}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{109}{190}\right)\) | \(e\left(\frac{97}{285}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{52}{95}\right)\) | \(e\left(\frac{218}{285}\right)\) | \(e\left(\frac{83}{570}\right)\) | \(e\left(\frac{91}{570}\right)\) |
| \(\chi_{4011}(506,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{570}\right)\) | \(e\left(\frac{41}{285}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{41}{190}\right)\) | \(e\left(\frac{128}{285}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{3}{95}\right)\) | \(e\left(\frac{82}{285}\right)\) | \(e\left(\frac{277}{570}\right)\) | \(e\left(\frac{29}{570}\right)\) |
| \(\chi_{4011}(527,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{221}{570}\right)\) | \(e\left(\frac{221}{285}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{31}{190}\right)\) | \(e\left(\frac{113}{285}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{88}{95}\right)\) | \(e\left(\frac{157}{285}\right)\) | \(e\left(\frac{367}{570}\right)\) | \(e\left(\frac{59}{570}\right)\) |
| \(\chi_{4011}(557,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{337}{570}\right)\) | \(e\left(\frac{52}{285}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{147}{190}\right)\) | \(e\left(\frac{211}{285}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{71}{95}\right)\) | \(e\left(\frac{104}{285}\right)\) | \(e\left(\frac{539}{570}\right)\) | \(e\left(\frac{433}{570}\right)\) |
| \(\chi_{4011}(569,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{311}{570}\right)\) | \(e\left(\frac{26}{285}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{121}{190}\right)\) | \(e\left(\frac{248}{285}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{83}{95}\right)\) | \(e\left(\frac{52}{285}\right)\) | \(e\left(\frac{127}{570}\right)\) | \(e\left(\frac{359}{570}\right)\) |
| \(\chi_{4011}(620,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{181}{570}\right)\) | \(e\left(\frac{181}{285}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{181}{190}\right)\) | \(e\left(\frac{148}{285}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{48}{95}\right)\) | \(e\left(\frac{77}{285}\right)\) | \(e\left(\frac{347}{570}\right)\) | \(e\left(\frac{559}{570}\right)\) |
| \(\chi_{4011}(662,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{331}{570}\right)\) | \(e\left(\frac{46}{285}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{141}{190}\right)\) | \(e\left(\frac{88}{285}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{8}{95}\right)\) | \(e\left(\frac{92}{285}\right)\) | \(e\left(\frac{137}{570}\right)\) | \(e\left(\frac{109}{570}\right)\) |
| \(\chi_{4011}(674,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{377}{570}\right)\) | \(e\left(\frac{92}{285}\right)\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{187}{190}\right)\) | \(e\left(\frac{176}{285}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{16}{95}\right)\) | \(e\left(\frac{184}{285}\right)\) | \(e\left(\frac{559}{570}\right)\) | \(e\left(\frac{503}{570}\right)\) |
| \(\chi_{4011}(683,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{570}\right)\) | \(e\left(\frac{163}{285}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{163}{190}\right)\) | \(e\left(\frac{64}{285}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{49}{95}\right)\) | \(e\left(\frac{41}{285}\right)\) | \(e\left(\frac{281}{570}\right)\) | \(e\left(\frac{157}{570}\right)\) |
| \(\chi_{4011}(704,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{373}{570}\right)\) | \(e\left(\frac{88}{285}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{183}{190}\right)\) | \(e\left(\frac{94}{285}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{69}{95}\right)\) | \(e\left(\frac{176}{285}\right)\) | \(e\left(\frac{101}{570}\right)\) | \(e\left(\frac{97}{570}\right)\) |
| \(\chi_{4011}(716,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{449}{570}\right)\) | \(e\left(\frac{164}{285}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{69}{190}\right)\) | \(e\left(\frac{227}{285}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{12}{95}\right)\) | \(e\left(\frac{43}{285}\right)\) | \(e\left(\frac{253}{570}\right)\) | \(e\left(\frac{401}{570}\right)\) |