from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(360000, base_ring=CyclotomicField(6000))
M = H._module
chi = DirichletCharacter(H, M([3000,4125,1000,3204]))
pari: [g,chi] = znchar(Mod(803,360000))
Basic properties
Modulus: | \(360000\) | |
Conductor: | \(360000\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(6000\) | 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 360000.vy
\(\chi_{360000}(83,\cdot)\) \(\chi_{360000}(347,\cdot)\) \(\chi_{360000}(563,\cdot)\) \(\chi_{360000}(587,\cdot)\) \(\chi_{360000}(803,\cdot)\) \(\chi_{360000}(1067,\cdot)\) \(\chi_{360000}(1283,\cdot)\) \(\chi_{360000}(1523,\cdot)\)
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: | $\Q(\zeta_{6000})$ |
Fixed field: | Number field defined by a degree 6000 polynomial (not computed) |
Values on generators
\((258751,202501,280001,29377)\) → \((-1,e\left(\frac{11}{16}\right),e\left(\frac{1}{6}\right),e\left(\frac{267}{500}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 360000 }(803, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{600}\right)\) | \(e\left(\frac{1729}{6000}\right)\) | \(e\left(\frac{5231}{6000}\right)\) | \(e\left(\frac{33}{250}\right)\) | \(e\left(\frac{1049}{2000}\right)\) | \(e\left(\frac{337}{3000}\right)\) | \(e\left(\frac{1423}{6000}\right)\) | \(e\left(\frac{362}{375}\right)\) | \(e\left(\frac{1747}{2000}\right)\) | \(e\left(\frac{2863}{3000}\right)\) |
sage: chi.jacobi_sum(n)