from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1776, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,18,31]))
pari: [g,chi] = znchar(Mod(59,1776))
Basic properties
Modulus: | \(1776\) | |
Conductor: | \(1776\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | 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)
|
Galois orbit 1776.ec
\(\chi_{1776}(59,\cdot)\) \(\chi_{1776}(131,\cdot)\) \(\chi_{1776}(683,\cdot)\) \(\chi_{1776}(755,\cdot)\) \(\chi_{1776}(827,\cdot)\) \(\chi_{1776}(923,\cdot)\) \(\chi_{1776}(1091,\cdot)\) \(\chi_{1776}(1115,\cdot)\) \(\chi_{1776}(1475,\cdot)\) \(\chi_{1776}(1499,\cdot)\) \(\chi_{1776}(1667,\cdot)\) \(\chi_{1776}(1763,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((223,1333,593,1297)\) → \((-1,i,-1,e\left(\frac{31}{36}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 1776 }(59, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)