from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6336, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,15,10,18]))
pari: [g,chi] = znchar(Mod(4529,6336))
Basic properties
| Modulus: | \(6336\) | |
| Conductor: | \(1584\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1584}(965,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6336.fd
\(\chi_{6336}(497,\cdot)\) \(\chi_{6336}(689,\cdot)\) \(\chi_{6336}(1073,\cdot)\) \(\chi_{6336}(1361,\cdot)\) \(\chi_{6336}(1553,\cdot)\) \(\chi_{6336}(2129,\cdot)\) \(\chi_{6336}(2417,\cdot)\) \(\chi_{6336}(2801,\cdot)\) \(\chi_{6336}(3665,\cdot)\) \(\chi_{6336}(3857,\cdot)\) \(\chi_{6336}(4241,\cdot)\) \(\chi_{6336}(4529,\cdot)\) \(\chi_{6336}(4721,\cdot)\) \(\chi_{6336}(5297,\cdot)\) \(\chi_{6336}(5585,\cdot)\) \(\chi_{6336}(5969,\cdot)\)
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
\((4159,4357,3521,1729)\) → \((1,i,e\left(\frac{1}{6}\right),e\left(\frac{3}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 6336 }(4529, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{20}\right)\) |
sage: chi.jacobi_sum(n)