from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4725, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([7,0,9]))
pari: [g,chi] = znchar(Mod(776,4725))
Basic properties
Modulus: | \(4725\) | |
Conductor: | \(189\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{189}(20,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4725.dl
\(\chi_{4725}(776,\cdot)\) \(\chi_{4725}(1301,\cdot)\) \(\chi_{4725}(2351,\cdot)\) \(\chi_{4725}(2876,\cdot)\) \(\chi_{4725}(3926,\cdot)\) \(\chi_{4725}(4451,\cdot)\)
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: | \(\Q(\zeta_{9})\) |
Fixed field: | Number field defined by a degree 18 polynomial |
Values on generators
\((4376,1702,2026)\) → \((e\left(\frac{7}{18}\right),1,-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
\( \chi_{ 4725 }(776, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) |
sage: chi.jacobi_sum(n)