from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(83205, base_ring=CyclotomicField(3612))
M = H._module
chi = DirichletCharacter(H, M([602,903,3078]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,83205))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(83205\) | |
| Conductor: | \(83205\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(3612\) | 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: | $\Q(\zeta_{3612})$ |
| Fixed field: | Number field defined by a degree 3612 polynomial (not computed) |
First 20 of 1008 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{83205}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3215}{3612}\right)\) | \(e\left(\frac{1409}{1806}\right)\) | \(e\left(\frac{359}{516}\right)\) | \(e\left(\frac{807}{1204}\right)\) | \(e\left(\frac{1111}{1806}\right)\) | \(e\left(\frac{1189}{3612}\right)\) | \(e\left(\frac{529}{903}\right)\) | \(e\left(\frac{506}{903}\right)\) | \(e\left(\frac{1195}{1204}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{83205}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1627}{3612}\right)\) | \(e\left(\frac{1627}{1806}\right)\) | \(e\left(\frac{247}{516}\right)\) | \(e\left(\frac{423}{1204}\right)\) | \(e\left(\frac{137}{1806}\right)\) | \(e\left(\frac{2333}{3612}\right)\) | \(e\left(\frac{839}{903}\right)\) | \(e\left(\frac{724}{903}\right)\) | \(e\left(\frac{1159}{1204}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{83205}(113,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1681}{3612}\right)\) | \(e\left(\frac{1681}{1806}\right)\) | \(e\left(\frac{37}{516}\right)\) | \(e\left(\frac{477}{1204}\right)\) | \(e\left(\frac{923}{1806}\right)\) | \(e\left(\frac{479}{3612}\right)\) | \(e\left(\frac{485}{903}\right)\) | \(e\left(\frac{778}{903}\right)\) | \(e\left(\frac{769}{1204}\right)\) | \(e\left(\frac{6}{7}\right)\) |
| \(\chi_{83205}(137,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3107}{3612}\right)\) | \(e\left(\frac{1301}{1806}\right)\) | \(e\left(\frac{263}{516}\right)\) | \(e\left(\frac{699}{1204}\right)\) | \(e\left(\frac{1345}{1806}\right)\) | \(e\left(\frac{1285}{3612}\right)\) | \(e\left(\frac{334}{903}\right)\) | \(e\left(\frac{398}{903}\right)\) | \(e\left(\frac{771}{1204}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{83205}(383,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3217}{3612}\right)\) | \(e\left(\frac{1411}{1806}\right)\) | \(e\left(\frac{313}{516}\right)\) | \(e\left(\frac{809}{1204}\right)\) | \(e\left(\frac{605}{1806}\right)\) | \(e\left(\frac{719}{3612}\right)\) | \(e\left(\frac{449}{903}\right)\) | \(e\left(\frac{508}{903}\right)\) | \(e\left(\frac{913}{1204}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{83205}(452,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{887}{3612}\right)\) | \(e\left(\frac{887}{1806}\right)\) | \(e\left(\frac{239}{516}\right)\) | \(e\left(\frac{887}{1204}\right)\) | \(e\left(\frac{1339}{1806}\right)\) | \(e\left(\frac{2857}{3612}\right)\) | \(e\left(\frac{640}{903}\right)\) | \(e\left(\frac{887}{903}\right)\) | \(e\left(\frac{751}{1204}\right)\) | \(e\left(\frac{2}{7}\right)\) |
| \(\chi_{83205}(518,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1549}{3612}\right)\) | \(e\left(\frac{1549}{1806}\right)\) | \(e\left(\frac{493}{516}\right)\) | \(e\left(\frac{345}{1204}\right)\) | \(e\left(\frac{5}{1806}\right)\) | \(e\left(\frac{2603}{3612}\right)\) | \(e\left(\frac{347}{903}\right)\) | \(e\left(\frac{646}{903}\right)\) | \(e\left(\frac{117}{1204}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{83205}(653,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{433}{3612}\right)\) | \(e\left(\frac{433}{1806}\right)\) | \(e\left(\frac{361}{516}\right)\) | \(e\left(\frac{433}{1204}\right)\) | \(e\left(\frac{617}{1806}\right)\) | \(e\left(\frac{1187}{3612}\right)\) | \(e\left(\frac{740}{903}\right)\) | \(e\left(\frac{433}{903}\right)\) | \(e\left(\frac{953}{1204}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{83205}(677,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{3612}\right)\) | \(e\left(\frac{227}{1806}\right)\) | \(e\left(\frac{455}{516}\right)\) | \(e\left(\frac{227}{1204}\right)\) | \(e\left(\frac{361}{1806}\right)\) | \(e\left(\frac{2641}{3612}\right)\) | \(e\left(\frac{853}{903}\right)\) | \(e\left(\frac{227}{903}\right)\) | \(e\left(\frac{1103}{1204}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{83205}(758,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{869}{3612}\right)\) | \(e\left(\frac{869}{1806}\right)\) | \(e\left(\frac{137}{516}\right)\) | \(e\left(\frac{869}{1204}\right)\) | \(e\left(\frac{475}{1806}\right)\) | \(e\left(\frac{3475}{3612}\right)\) | \(e\left(\frac{457}{903}\right)\) | \(e\left(\frac{869}{903}\right)\) | \(e\left(\frac{881}{1204}\right)\) | \(e\left(\frac{6}{7}\right)\) |
| \(\chi_{83205}(887,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3235}{3612}\right)\) | \(e\left(\frac{1429}{1806}\right)\) | \(e\left(\frac{415}{516}\right)\) | \(e\left(\frac{827}{1204}\right)\) | \(e\left(\frac{1469}{1806}\right)\) | \(e\left(\frac{101}{3612}\right)\) | \(e\left(\frac{632}{903}\right)\) | \(e\left(\frac{526}{903}\right)\) | \(e\left(\frac{783}{1204}\right)\) | \(e\left(\frac{6}{7}\right)\) |
| \(\chi_{83205}(968,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3253}{3612}\right)\) | \(e\left(\frac{1447}{1806}\right)\) | \(e\left(\frac{1}{516}\right)\) | \(e\left(\frac{845}{1204}\right)\) | \(e\left(\frac{527}{1806}\right)\) | \(e\left(\frac{3095}{3612}\right)\) | \(e\left(\frac{815}{903}\right)\) | \(e\left(\frac{544}{903}\right)\) | \(e\left(\frac{653}{1204}\right)\) | \(e\left(\frac{2}{7}\right)\) |
| \(\chi_{83205}(1028,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3581}{3612}\right)\) | \(e\left(\frac{1775}{1806}\right)\) | \(e\left(\frac{197}{516}\right)\) | \(e\left(\frac{1173}{1204}\right)\) | \(e\left(\frac{619}{1806}\right)\) | \(e\left(\frac{1867}{3612}\right)\) | \(e\left(\frac{337}{903}\right)\) | \(e\left(\frac{872}{903}\right)\) | \(e\left(\frac{157}{1204}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{83205}(1157,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{403}{3612}\right)\) | \(e\left(\frac{403}{1806}\right)\) | \(e\left(\frac{19}{516}\right)\) | \(e\left(\frac{403}{1204}\right)\) | \(e\left(\frac{983}{1806}\right)\) | \(e\left(\frac{1013}{3612}\right)\) | \(e\left(\frac{134}{903}\right)\) | \(e\left(\frac{403}{903}\right)\) | \(e\left(\frac{367}{1204}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{83205}(1163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{821}{3612}\right)\) | \(e\left(\frac{821}{1806}\right)\) | \(e\left(\frac{209}{516}\right)\) | \(e\left(\frac{821}{1204}\right)\) | \(e\left(\frac{1783}{1806}\right)\) | \(e\left(\frac{307}{3612}\right)\) | \(e\left(\frac{571}{903}\right)\) | \(e\left(\frac{821}{903}\right)\) | \(e\left(\frac{425}{1204}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{83205}(1193,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{913}{3612}\right)\) | \(e\left(\frac{913}{1806}\right)\) | \(e\left(\frac{157}{516}\right)\) | \(e\left(\frac{913}{1204}\right)\) | \(e\left(\frac{179}{1806}\right)\) | \(e\left(\frac{359}{3612}\right)\) | \(e\left(\frac{503}{903}\right)\) | \(e\left(\frac{10}{903}\right)\) | \(e\left(\frac{697}{1204}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{83205}(1292,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1759}{3612}\right)\) | \(e\left(\frac{1759}{1806}\right)\) | \(e\left(\frac{307}{516}\right)\) | \(e\left(\frac{555}{1204}\right)\) | \(e\left(\frac{1055}{1806}\right)\) | \(e\left(\frac{209}{3612}\right)\) | \(e\left(\frac{74}{903}\right)\) | \(e\left(\frac{856}{903}\right)\) | \(e\left(\frac{607}{1204}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{83205}(1298,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2057}{3612}\right)\) | \(e\left(\frac{251}{1806}\right)\) | \(e\left(\frac{161}{516}\right)\) | \(e\left(\frac{853}{1204}\right)\) | \(e\left(\frac{1513}{1806}\right)\) | \(e\left(\frac{2419}{3612}\right)\) | \(e\left(\frac{796}{903}\right)\) | \(e\left(\frac{251}{903}\right)\) | \(e\left(\frac{729}{1204}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{83205}(1427,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2743}{3612}\right)\) | \(e\left(\frac{937}{1806}\right)\) | \(e\left(\frac{379}{516}\right)\) | \(e\left(\frac{335}{1204}\right)\) | \(e\left(\frac{1331}{1806}\right)\) | \(e\left(\frac{137}{3612}\right)\) | \(e\left(\frac{446}{903}\right)\) | \(e\left(\frac{34}{903}\right)\) | \(e\left(\frac{323}{1204}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{83205}(1532,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2423}{3612}\right)\) | \(e\left(\frac{617}{1806}\right)\) | \(e\left(\frac{515}{516}\right)\) | \(e\left(\frac{15}{1204}\right)\) | \(e\left(\frac{1021}{1806}\right)\) | \(e\left(\frac{3097}{3612}\right)\) | \(e\left(\frac{604}{903}\right)\) | \(e\left(\frac{617}{903}\right)\) | \(e\left(\frac{895}{1204}\right)\) | \(e\left(\frac{6}{7}\right)\) |