from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6300, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,10,51,50]))
pari: [g,chi] = znchar(Mod(47,6300))
Basic properties
| Modulus: | \(6300\) | |
| Conductor: | \(6300\) | 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 6300.ib
\(\chi_{6300}(47,\cdot)\) \(\chi_{6300}(563,\cdot)\) \(\chi_{6300}(803,\cdot)\) \(\chi_{6300}(1067,\cdot)\) \(\chi_{6300}(1823,\cdot)\) \(\chi_{6300}(2063,\cdot)\) \(\chi_{6300}(2327,\cdot)\) \(\chi_{6300}(2567,\cdot)\) \(\chi_{6300}(3083,\cdot)\) \(\chi_{6300}(3323,\cdot)\) \(\chi_{6300}(3587,\cdot)\) \(\chi_{6300}(3827,\cdot)\) \(\chi_{6300}(4583,\cdot)\) \(\chi_{6300}(4847,\cdot)\) \(\chi_{6300}(5087,\cdot)\) \(\chi_{6300}(5603,\cdot)\)
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
\((3151,2801,3277,3601)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{17}{20}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6300 }(47, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{12}\right)\) |
sage: chi.jacobi_sum(n)