from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(53361, base_ring=CyclotomicField(2310))
M = H._module
chi = DirichletCharacter(H, M([1540,55,2163]))
chi.galois_orbit()
[g,chi] = znchar(Mod(52,53361))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(53361\) | |
| Conductor: | \(53361\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2310\) | 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: | $\Q(\zeta_{1155})$ |
| Fixed field: | Number field defined by a degree 2310 polynomial (not computed) |
First 14 of 480 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{53361}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{171}{770}\right)\) | \(e\left(\frac{171}{385}\right)\) | \(e\left(\frac{727}{2310}\right)\) | \(e\left(\frac{513}{770}\right)\) | \(e\left(\frac{124}{231}\right)\) | \(e\left(\frac{799}{1155}\right)\) | \(e\left(\frac{342}{385}\right)\) | \(e\left(\frac{551}{1155}\right)\) | \(e\left(\frac{91}{165}\right)\) | \(e\left(\frac{1753}{2310}\right)\) |
| \(\chi_{53361}(292,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{603}{770}\right)\) | \(e\left(\frac{218}{385}\right)\) | \(e\left(\frac{821}{2310}\right)\) | \(e\left(\frac{269}{770}\right)\) | \(e\left(\frac{32}{231}\right)\) | \(e\left(\frac{467}{1155}\right)\) | \(e\left(\frac{51}{385}\right)\) | \(e\left(\frac{403}{1155}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{2129}{2310}\right)\) |
| \(\chi_{53361}(304,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{389}{770}\right)\) | \(e\left(\frac{4}{385}\right)\) | \(e\left(\frac{1933}{2310}\right)\) | \(e\left(\frac{397}{770}\right)\) | \(e\left(\frac{79}{231}\right)\) | \(e\left(\frac{496}{1155}\right)\) | \(e\left(\frac{8}{385}\right)\) | \(e\left(\frac{569}{1155}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{1957}{2310}\right)\) |
| \(\chi_{53361}(556,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{593}{770}\right)\) | \(e\left(\frac{208}{385}\right)\) | \(e\left(\frac{31}{2310}\right)\) | \(e\left(\frac{239}{770}\right)\) | \(e\left(\frac{181}{231}\right)\) | \(e\left(\frac{382}{1155}\right)\) | \(e\left(\frac{31}{385}\right)\) | \(e\left(\frac{713}{1155}\right)\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{1279}{2310}\right)\) |
| \(\chi_{53361}(733,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{379}{770}\right)\) | \(e\left(\frac{379}{385}\right)\) | \(e\left(\frac{1913}{2310}\right)\) | \(e\left(\frac{367}{770}\right)\) | \(e\left(\frac{74}{231}\right)\) | \(e\left(\frac{26}{1155}\right)\) | \(e\left(\frac{373}{385}\right)\) | \(e\left(\frac{109}{1155}\right)\) | \(e\left(\frac{74}{165}\right)\) | \(e\left(\frac{1877}{2310}\right)\) |
| \(\chi_{53361}(745,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{361}{770}\right)\) | \(e\left(\frac{361}{385}\right)\) | \(e\left(\frac{337}{2310}\right)\) | \(e\left(\frac{313}{770}\right)\) | \(e\left(\frac{142}{231}\right)\) | \(e\left(\frac{874}{1155}\right)\) | \(e\left(\frac{337}{385}\right)\) | \(e\left(\frac{821}{1155}\right)\) | \(e\left(\frac{76}{165}\right)\) | \(e\left(\frac{193}{2310}\right)\) |
| \(\chi_{53361}(871,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{770}\right)\) | \(e\left(\frac{197}{385}\right)\) | \(e\left(\frac{1549}{2310}\right)\) | \(e\left(\frac{591}{770}\right)\) | \(e\left(\frac{214}{231}\right)\) | \(e\left(\frac{943}{1155}\right)\) | \(e\left(\frac{9}{385}\right)\) | \(e\left(\frac{977}{1155}\right)\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{421}{2310}\right)\) |
| \(\chi_{53361}(985,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{563}{770}\right)\) | \(e\left(\frac{178}{385}\right)\) | \(e\left(\frac{1511}{2310}\right)\) | \(e\left(\frac{149}{770}\right)\) | \(e\left(\frac{89}{231}\right)\) | \(e\left(\frac{512}{1155}\right)\) | \(e\left(\frac{356}{385}\right)\) | \(e\left(\frac{103}{1155}\right)\) | \(e\left(\frac{23}{165}\right)\) | \(e\left(\frac{269}{2310}\right)\) |
| \(\chi_{53361}(997,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{770}\right)\) | \(e\left(\frac{229}{385}\right)\) | \(e\left(\frac{73}{2310}\right)\) | \(e\left(\frac{687}{770}\right)\) | \(e\left(\frac{76}{231}\right)\) | \(e\left(\frac{676}{1155}\right)\) | \(e\left(\frac{73}{385}\right)\) | \(e\left(\frac{524}{1155}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{1447}{2310}\right)\) |
| \(\chi_{53361}(1174,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{421}{770}\right)\) | \(e\left(\frac{36}{385}\right)\) | \(e\left(\frac{1997}{2310}\right)\) | \(e\left(\frac{493}{770}\right)\) | \(e\left(\frac{95}{231}\right)\) | \(e\left(\frac{614}{1155}\right)\) | \(e\left(\frac{72}{385}\right)\) | \(e\left(\frac{886}{1155}\right)\) | \(e\left(\frac{161}{165}\right)\) | \(e\left(\frac{2213}{2310}\right)\) |
| \(\chi_{53361}(1249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{770}\right)\) | \(e\left(\frac{223}{385}\right)\) | \(e\left(\frac{61}{2310}\right)\) | \(e\left(\frac{669}{770}\right)\) | \(e\left(\frac{73}{231}\right)\) | \(e\left(\frac{1087}{1155}\right)\) | \(e\left(\frac{61}{385}\right)\) | \(e\left(\frac{248}{1155}\right)\) | \(e\left(\frac{73}{165}\right)\) | \(e\left(\frac{1399}{2310}\right)\) |
| \(\chi_{53361}(1300,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{770}\right)\) | \(e\left(\frac{107}{385}\right)\) | \(e\left(\frac{599}{2310}\right)\) | \(e\left(\frac{321}{770}\right)\) | \(e\left(\frac{92}{231}\right)\) | \(e\left(\frac{563}{1155}\right)\) | \(e\left(\frac{214}{385}\right)\) | \(e\left(\frac{1072}{1155}\right)\) | \(e\left(\frac{92}{165}\right)\) | \(e\left(\frac{1241}{2310}\right)\) |
| \(\chi_{53361}(1426,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{549}{770}\right)\) | \(e\left(\frac{164}{385}\right)\) | \(e\left(\frac{713}{2310}\right)\) | \(e\left(\frac{107}{770}\right)\) | \(e\left(\frac{5}{231}\right)\) | \(e\left(\frac{701}{1155}\right)\) | \(e\left(\frac{328}{385}\right)\) | \(e\left(\frac{229}{1155}\right)\) | \(e\left(\frac{104}{165}\right)\) | \(e\left(\frac{1697}{2310}\right)\) |
| \(\chi_{53361}(1438,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{551}{770}\right)\) | \(e\left(\frac{166}{385}\right)\) | \(e\left(\frac{2257}{2310}\right)\) | \(e\left(\frac{113}{770}\right)\) | \(e\left(\frac{160}{231}\right)\) | \(e\left(\frac{949}{1155}\right)\) | \(e\left(\frac{332}{385}\right)\) | \(e\left(\frac{1091}{1155}\right)\) | \(e\left(\frac{61}{165}\right)\) | \(e\left(\frac{943}{2310}\right)\) |