from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1001, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,36,55]))
pari: [g,chi] = znchar(Mod(20,1001))
Basic properties
| Modulus: | \(1001\) | |
| Conductor: | \(1001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1001.ed
\(\chi_{1001}(20,\cdot)\) \(\chi_{1001}(97,\cdot)\) \(\chi_{1001}(202,\cdot)\) \(\chi_{1001}(223,\cdot)\) \(\chi_{1001}(258,\cdot)\) \(\chi_{1001}(279,\cdot)\) \(\chi_{1001}(405,\cdot)\) \(\chi_{1001}(531,\cdot)\) \(\chi_{1001}(566,\cdot)\) \(\chi_{1001}(587,\cdot)\) \(\chi_{1001}(643,\cdot)\) \(\chi_{1001}(713,\cdot)\) \(\chi_{1001}(839,\cdot)\) \(\chi_{1001}(895,\cdot)\) \(\chi_{1001}(916,\cdot)\) \(\chi_{1001}(951,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((430,365,925)\) → \((-1,e\left(\frac{3}{5}\right),e\left(\frac{11}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(15\) |
| \( \chi_{ 1001 }(20, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{7}{60}\right)\) |
sage: chi.jacobi_sum(n)