from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1859, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([21,10]))
pari: [g,chi] = znchar(Mod(315,1859))
Basic properties
Modulus: | \(1859\) | |
Conductor: | \(143\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{143}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1859.x
\(\chi_{1859}(315,\cdot)\) \(\chi_{1859}(360,\cdot)\) \(\chi_{1859}(822,\cdot)\) \(\chi_{1859}(1036,\cdot)\) \(\chi_{1859}(1205,\cdot)\) \(\chi_{1859}(1498,\cdot)\) \(\chi_{1859}(1667,\cdot)\) \(\chi_{1859}(1712,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | 30.0.249154964698353870876083085574129912384252301255171.1 |
Values on generators
\((508,1354)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 1859 }(315, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)