from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(166410, base_ring=CyclotomicField(3612))
M = H._module
chi = DirichletCharacter(H, M([3010,2709,678]))
chi.galois_orbit()
[g,chi] = znchar(Mod(113,166410))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(166410\) | |
| Conductor: | \(83205\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(3612\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 83205.ja | 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: | $\Q(\zeta_{3612})$ |
| Fixed field: | Number field defined by a degree 3612 polynomial (not computed) |
First 10 of 1008 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{166410}(113,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{516}\right)\) | \(e\left(\frac{923}{1806}\right)\) | \(e\left(\frac{479}{3612}\right)\) | \(e\left(\frac{769}{1204}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1937}{3612}\right)\) | \(e\left(\frac{53}{1806}\right)\) | \(e\left(\frac{149}{903}\right)\) | \(e\left(\frac{71}{172}\right)\) | \(e\left(\frac{1495}{1806}\right)\) |
| \(\chi_{166410}(137,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{263}{516}\right)\) | \(e\left(\frac{1345}{1806}\right)\) | \(e\left(\frac{1285}{3612}\right)\) | \(e\left(\frac{771}{1204}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{2263}{3612}\right)\) | \(e\left(\frac{1081}{1806}\right)\) | \(e\left(\frac{313}{903}\right)\) | \(e\left(\frac{77}{172}\right)\) | \(e\left(\frac{983}{1806}\right)\) |
| \(\chi_{166410}(383,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{313}{516}\right)\) | \(e\left(\frac{605}{1806}\right)\) | \(e\left(\frac{719}{3612}\right)\) | \(e\left(\frac{913}{1204}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{125}{3612}\right)\) | \(e\left(\frac{23}{1806}\right)\) | \(e\left(\frac{218}{903}\right)\) | \(e\left(\frac{159}{172}\right)\) | \(e\left(\frac{751}{1806}\right)\) |
| \(\chi_{166410}(653,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{361}{516}\right)\) | \(e\left(\frac{617}{1806}\right)\) | \(e\left(\frac{1187}{3612}\right)\) | \(e\left(\frac{953}{1204}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1829}{3612}\right)\) | \(e\left(\frac{1319}{1806}\right)\) | \(e\left(\frac{488}{903}\right)\) | \(e\left(\frac{107}{172}\right)\) | \(e\left(\frac{745}{1806}\right)\) |
| \(\chi_{166410}(677,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{455}{516}\right)\) | \(e\left(\frac{361}{1806}\right)\) | \(e\left(\frac{2641}{3612}\right)\) | \(e\left(\frac{1103}{1204}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3403}{3612}\right)\) | \(e\left(\frac{1363}{1806}\right)\) | \(e\left(\frac{748}{903}\right)\) | \(e\left(\frac{41}{172}\right)\) | \(e\left(\frac{1475}{1806}\right)\) |
| \(\chi_{166410}(887,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{415}{516}\right)\) | \(e\left(\frac{1469}{1806}\right)\) | \(e\left(\frac{101}{3612}\right)\) | \(e\left(\frac{783}{1204}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1811}{3612}\right)\) | \(e\left(\frac{1229}{1806}\right)\) | \(e\left(\frac{695}{903}\right)\) | \(e\left(\frac{113}{172}\right)\) | \(e\left(\frac{319}{1806}\right)\) |
| \(\chi_{166410}(1157,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{516}\right)\) | \(e\left(\frac{983}{1806}\right)\) | \(e\left(\frac{1013}{3612}\right)\) | \(e\left(\frac{367}{1204}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1427}{3612}\right)\) | \(e\left(\frac{1115}{1806}\right)\) | \(e\left(\frac{596}{903}\right)\) | \(e\left(\frac{69}{172}\right)\) | \(e\left(\frac{1465}{1806}\right)\) |
| \(\chi_{166410}(1163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{209}{516}\right)\) | \(e\left(\frac{1783}{1806}\right)\) | \(e\left(\frac{307}{3612}\right)\) | \(e\left(\frac{425}{1204}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1249}{3612}\right)\) | \(e\left(\frac{1429}{1806}\right)\) | \(e\left(\frac{235}{903}\right)\) | \(e\left(\frac{71}{172}\right)\) | \(e\left(\frac{1667}{1806}\right)\) |
| \(\chi_{166410}(1193,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{516}\right)\) | \(e\left(\frac{179}{1806}\right)\) | \(e\left(\frac{359}{3612}\right)\) | \(e\left(\frac{697}{1204}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1037}{3612}\right)\) | \(e\left(\frac{971}{1806}\right)\) | \(e\left(\frac{566}{903}\right)\) | \(e\left(\frac{27}{172}\right)\) | \(e\left(\frac{61}{1806}\right)\) |
| \(\chi_{166410}(1427,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{379}{516}\right)\) | \(e\left(\frac{1331}{1806}\right)\) | \(e\left(\frac{137}{3612}\right)\) | \(e\left(\frac{323}{1204}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{275}{3612}\right)\) | \(e\left(\frac{773}{1806}\right)\) | \(e\left(\frac{299}{903}\right)\) | \(e\left(\frac{109}{172}\right)\) | \(e\left(\frac{1291}{1806}\right)\) |