from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6017, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,32]))
pari: [g,chi] = znchar(Mod(2861,6017))
Basic properties
Modulus: | \(6017\) | |
Conductor: | \(547\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{547}(126,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6017.s
\(\chi_{6017}(716,\cdot)\) \(\chi_{6017}(1211,\cdot)\) \(\chi_{6017}(2146,\cdot)\) \(\chi_{6017}(2201,\cdot)\) \(\chi_{6017}(2311,\cdot)\) \(\chi_{6017}(2366,\cdot)\) \(\chi_{6017}(2696,\cdot)\) \(\chi_{6017}(2861,\cdot)\) \(\chi_{6017}(3642,\cdot)\) \(\chi_{6017}(4390,\cdot)\) \(\chi_{6017}(5666,\cdot)\) \(\chi_{6017}(5897,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | Number field defined by a degree 21 polynomial |
Values on generators
\((3830,2190)\) → \((1,e\left(\frac{16}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 6017 }(2861, a) \) | \(1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{8}{21}\right)\) |
sage: chi.jacobi_sum(n)