from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6069, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,17,26]))
pari: [g,chi] = znchar(Mod(188,6069))
Basic properties
Modulus: | \(6069\) | |
Conductor: | \(6069\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(34\) | 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 6069.bp
\(\chi_{6069}(188,\cdot)\) \(\chi_{6069}(545,\cdot)\) \(\chi_{6069}(902,\cdot)\) \(\chi_{6069}(1259,\cdot)\) \(\chi_{6069}(1616,\cdot)\) \(\chi_{6069}(1973,\cdot)\) \(\chi_{6069}(2330,\cdot)\) \(\chi_{6069}(2687,\cdot)\) \(\chi_{6069}(3044,\cdot)\) \(\chi_{6069}(3401,\cdot)\) \(\chi_{6069}(4115,\cdot)\) \(\chi_{6069}(4472,\cdot)\) \(\chi_{6069}(4829,\cdot)\) \(\chi_{6069}(5186,\cdot)\) \(\chi_{6069}(5543,\cdot)\) \(\chi_{6069}(5900,\cdot)\)
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: | \(\Q(\zeta_{17})\) |
Fixed field: | Number field defined by a degree 34 polynomial |
Values on generators
\((2024,4336,3760)\) → \((-1,-1,e\left(\frac{13}{17}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(19\) | \(20\) |
\( \chi_{ 6069 }(188, a) \) | \(1\) | \(1\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{12}{17}\right)\) |
sage: chi.jacobi_sum(n)