from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1001, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,42,15]))
pari: [g,chi] = znchar(Mod(359,1001))
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.ej
\(\chi_{1001}(18,\cdot)\) \(\chi_{1001}(151,\cdot)\) \(\chi_{1001}(200,\cdot)\) \(\chi_{1001}(226,\cdot)\) \(\chi_{1001}(359,\cdot)\) \(\chi_{1001}(382,\cdot)\) \(\chi_{1001}(424,\cdot)\) \(\chi_{1001}(541,\cdot)\) \(\chi_{1001}(590,\cdot)\) \(\chi_{1001}(655,\cdot)\) \(\chi_{1001}(723,\cdot)\) \(\chi_{1001}(772,\cdot)\) \(\chi_{1001}(788,\cdot)\) \(\chi_{1001}(954,\cdot)\) \(\chi_{1001}(970,\cdot)\) \(\chi_{1001}(996,\cdot)\)
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
\((430,365,925)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{7}{10}\right),i)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(15\) |
\( \chi_{ 1001 }(359, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{20}\right)\) |
sage: chi.jacobi_sum(n)