from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6045, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,50,4]))
pari: [g,chi] = znchar(Mod(257,6045))
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.nx
\(\chi_{6045}(257,\cdot)\) \(\chi_{6045}(413,\cdot)\) \(\chi_{6045}(452,\cdot)\) \(\chi_{6045}(758,\cdot)\) \(\chi_{6045}(1538,\cdot)\) \(\chi_{6045}(1622,\cdot)\) \(\chi_{6045}(1733,\cdot)\) \(\chi_{6045}(1967,\cdot)\) \(\chi_{6045}(1973,\cdot)\) \(\chi_{6045}(2747,\cdot)\) \(\chi_{6045}(2942,\cdot)\) \(\chi_{6045}(3182,\cdot)\) \(\chi_{6045}(3293,\cdot)\) \(\chi_{6045}(4502,\cdot)\) \(\chi_{6045}(5093,\cdot)\) \(\chi_{6045}(5288,\cdot)\)
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
\((4031,4837,1861,2731)\) → \((-1,i,e\left(\frac{5}{6}\right),e\left(\frac{1}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 6045 }(257, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{20}\right)\) |
sage: chi.jacobi_sum(n)