from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6039, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,30,43]))
pari: [g,chi] = znchar(Mod(43,6039))
Basic properties
Modulus: | \(6039\) | |
Conductor: | \(6039\) | 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 6039.ok
\(\chi_{6039}(43,\cdot)\) \(\chi_{6039}(274,\cdot)\) \(\chi_{6039}(1264,\cdot)\) \(\chi_{6039}(1726,\cdot)\) \(\chi_{6039}(1759,\cdot)\) \(\chi_{6039}(2320,\cdot)\) \(\chi_{6039}(2353,\cdot)\) \(\chi_{6039}(2617,\cdot)\) \(\chi_{6039}(3409,\cdot)\) \(\chi_{6039}(3442,\cdot)\) \(\chi_{6039}(4036,\cdot)\) \(\chi_{6039}(4399,\cdot)\) \(\chi_{6039}(4531,\cdot)\) \(\chi_{6039}(5191,\cdot)\) \(\chi_{6039}(5488,\cdot)\) \(\chi_{6039}(5521,\cdot)\)
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
\((1343,5491,5248)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{43}{60}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
\( \chi_{ 6039 }(43, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) |
sage: chi.jacobi_sum(n)