from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1001, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([25,12,10]))
pari: [g,chi] = znchar(Mod(159,1001))
Basic properties
Modulus: | \(1001\) | |
Conductor: | \(1001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | 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 1001.cy
\(\chi_{1001}(159,\cdot)\) \(\chi_{1001}(269,\cdot)\) \(\chi_{1001}(432,\cdot)\) \(\chi_{1001}(542,\cdot)\) \(\chi_{1001}(614,\cdot)\) \(\chi_{1001}(724,\cdot)\) \(\chi_{1001}(796,\cdot)\) \(\chi_{1001}(906,\cdot)\)
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: | Number field defined by a degree 30 polynomial |
Values on generators
\((430,365,925)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{2}{5}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(15\) |
\( \chi_{ 1001 }(159, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{15}\right)\) |
sage: chi.jacobi_sum(n)