from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6080, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,33,24,16]))
pari: [g,chi] = znchar(Mod(349,6080))
Basic properties
| Modulus: | \(6080\) | |
| Conductor: | \(6080\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | 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 6080.gr
\(\chi_{6080}(349,\cdot)\) \(\chi_{6080}(429,\cdot)\) \(\chi_{6080}(1109,\cdot)\) \(\chi_{6080}(1189,\cdot)\) \(\chi_{6080}(1869,\cdot)\) \(\chi_{6080}(1949,\cdot)\) \(\chi_{6080}(2629,\cdot)\) \(\chi_{6080}(2709,\cdot)\) \(\chi_{6080}(3389,\cdot)\) \(\chi_{6080}(3469,\cdot)\) \(\chi_{6080}(4149,\cdot)\) \(\chi_{6080}(4229,\cdot)\) \(\chi_{6080}(4909,\cdot)\) \(\chi_{6080}(4989,\cdot)\) \(\chi_{6080}(5669,\cdot)\) \(\chi_{6080}(5749,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((191,5701,1217,1921)\) → \((1,e\left(\frac{11}{16}\right),-1,e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 6080 }(349, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{11}{48}\right)\) |
sage: chi.jacobi_sum(n)