from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6039, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,36,29]))
pari: [g,chi] = znchar(Mod(152,6039))
Basic properties
Modulus: | \(6039\) | |
Conductor: | \(2013\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2013}(152,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6039.qa
\(\chi_{6039}(152,\cdot)\) \(\chi_{6039}(251,\cdot)\) \(\chi_{6039}(323,\cdot)\) \(\chi_{6039}(368,\cdot)\) \(\chi_{6039}(620,\cdot)\) \(\chi_{6039}(1664,\cdot)\) \(\chi_{6039}(1763,\cdot)\) \(\chi_{6039}(3536,\cdot)\) \(\chi_{6039}(4337,\cdot)\) \(\chi_{6039}(4382,\cdot)\) \(\chi_{6039}(4427,\cdot)\) \(\chi_{6039}(4436,\cdot)\) \(\chi_{6039}(4679,\cdot)\) \(\chi_{6039}(5300,\cdot)\) \(\chi_{6039}(5399,\cdot)\) \(\chi_{6039}(5516,\cdot)\)
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
\((1343,5491,5248)\) → \((-1,e\left(\frac{3}{5}\right),e\left(\frac{29}{60}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
\( \chi_{ 6039 }(152, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(-i\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{37}{60}\right)\) |
sage: chi.jacobi_sum(n)