from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6045, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,35,58]))
pari: [g,chi] = znchar(Mod(362,6045))
Basic properties
Modulus: | \(6045\) | |
Conductor: | \(6045\) | 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 6045.pc
\(\chi_{6045}(362,\cdot)\) \(\chi_{6045}(548,\cdot)\) \(\chi_{6045}(1532,\cdot)\) \(\chi_{6045}(1553,\cdot)\) \(\chi_{6045}(1718,\cdot)\) \(\chi_{6045}(1727,\cdot)\) \(\chi_{6045}(1748,\cdot)\) \(\chi_{6045}(1913,\cdot)\) \(\chi_{6045}(2528,\cdot)\) \(\chi_{6045}(3062,\cdot)\) \(\chi_{6045}(4622,\cdot)\) \(\chi_{6045}(4817,\cdot)\) \(\chi_{6045}(4847,\cdot)\) \(\chi_{6045}(5033,\cdot)\) \(\chi_{6045}(5597,\cdot)\) \(\chi_{6045}(6038,\cdot)\)
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
\((4031,4837,1861,2731)\) → \((-1,i,e\left(\frac{7}{12}\right),e\left(\frac{29}{30}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
\( \chi_{ 6045 }(362, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) |
sage: chi.jacobi_sum(n)