from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6045, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,30,55,32]))
pari: [g,chi] = znchar(Mod(59,6045))
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.na
\(\chi_{6045}(59,\cdot)\) \(\chi_{6045}(1064,\cdot)\) \(\chi_{6045}(1094,\cdot)\) \(\chi_{6045}(1289,\cdot)\) \(\chi_{6045}(1619,\cdot)\) \(\chi_{6045}(2459,\cdot)\) \(\chi_{6045}(2624,\cdot)\) \(\chi_{6045}(3269,\cdot)\) \(\chi_{6045}(4019,\cdot)\) \(\chi_{6045}(4049,\cdot)\) \(\chi_{6045}(4244,\cdot)\) \(\chi_{6045}(5129,\cdot)\) \(\chi_{6045}(5744,\cdot)\) \(\chi_{6045}(5804,\cdot)\) \(\chi_{6045}(5909,\cdot)\) \(\chi_{6045}(5939,\cdot)\)
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
\((4031,4837,1861,2731)\) → \((-1,-1,e\left(\frac{11}{12}\right),e\left(\frac{8}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 6045 }(59, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) |
sage: chi.jacobi_sum(n)