from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6039, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,54,43]))
pari: [g,chi] = znchar(Mod(226,6039))
Basic properties
Modulus: | \(6039\) | |
Conductor: | \(671\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{671}(226,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6039.ng
\(\chi_{6039}(226,\cdot)\) \(\chi_{6039}(514,\cdot)\) \(\chi_{6039}(1405,\cdot)\) \(\chi_{6039}(1531,\cdot)\) \(\chi_{6039}(1630,\cdot)\) \(\chi_{6039}(1909,\cdot)\) \(\chi_{6039}(2206,\cdot)\) \(\chi_{6039}(3043,\cdot)\) \(\chi_{6039}(3142,\cdot)\) \(\chi_{6039}(3934,\cdot)\) \(\chi_{6039}(4033,\cdot)\) \(\chi_{6039}(4573,\cdot)\) \(\chi_{6039}(4897,\cdot)\) \(\chi_{6039}(4996,\cdot)\) \(\chi_{6039}(5464,\cdot)\) \(\chi_{6039}(5968,\cdot)\)
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
\((1343,5491,5248)\) → \((1,e\left(\frac{9}{10}\right),e\left(\frac{43}{60}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
\( \chi_{ 6039 }(226, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) |
sage: chi.jacobi_sum(n)