from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6045, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,10,36]))
pari: [g,chi] = znchar(Mod(407,6045))
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.nt
\(\chi_{6045}(407,\cdot)\) \(\chi_{6045}(1583,\cdot)\) \(\chi_{6045}(2513,\cdot)\) \(\chi_{6045}(2558,\cdot)\) \(\chi_{6045}(2792,\cdot)\) \(\chi_{6045}(3488,\cdot)\) \(\chi_{6045}(3722,\cdot)\) \(\chi_{6045}(3728,\cdot)\) \(\chi_{6045}(3767,\cdot)\) \(\chi_{6045}(4313,\cdot)\) \(\chi_{6045}(4658,\cdot)\) \(\chi_{6045}(4697,\cdot)\) \(\chi_{6045}(4937,\cdot)\) \(\chi_{6045}(5243,\cdot)\) \(\chi_{6045}(5522,\cdot)\) \(\chi_{6045}(5867,\cdot)\)
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
\((4031,4837,1861,2731)\) → \((-1,i,e\left(\frac{1}{6}\right),e\left(\frac{3}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 6045 }(407, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) |
sage: chi.jacobi_sum(n)