from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(166410, base_ring=CyclotomicField(3612))
M = H._module
chi = DirichletCharacter(H, M([3010,903,3114]))
pari: [g,chi] = znchar(Mod(887,166410))
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 \(\chi_{83205}(887,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 166410.iy
\(\chi_{166410}(113,\cdot)\) \(\chi_{166410}(137,\cdot)\) \(\chi_{166410}(383,\cdot)\) \(\chi_{166410}(653,\cdot)\) \(\chi_{166410}(677,\cdot)\) \(\chi_{166410}(887,\cdot)\) \(\chi_{166410}(1157,\cdot)\) \(\chi_{166410}(1163,\cdot)\) \(\chi_{166410}(1193,\cdot)\) \(\chi_{166410}(1427,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | $\Q(\zeta_{3612})$ |
| Fixed field: | Number field defined by a degree 3612 polynomial (not computed) |
Values on generators
\((129431,99847,40681)\) → \((e\left(\frac{5}{6}\right),i,e\left(\frac{519}{602}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 166410 }(887, a) \) | \(-1\) | \(1\) | \(e\left(\frac{415}{516}\right)\) | \(e\left(\frac{1469}{1806}\right)\) | \(e\left(\frac{101}{3612}\right)\) | \(e\left(\frac{783}{1204}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1811}{3612}\right)\) | \(e\left(\frac{1229}{1806}\right)\) | \(e\left(\frac{695}{903}\right)\) | \(e\left(\frac{113}{172}\right)\) | \(e\left(\frac{319}{1806}\right)\) |
sage: chi.jacobi_sum(n)