from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1734, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([0,8]))
pari: [g,chi] = znchar(Mod(103,1734))
Basic properties
Modulus: | \(1734\) | |
Conductor: | \(289\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(17\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{289}(103,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1734.k
\(\chi_{1734}(103,\cdot)\) \(\chi_{1734}(205,\cdot)\) \(\chi_{1734}(307,\cdot)\) \(\chi_{1734}(409,\cdot)\) \(\chi_{1734}(511,\cdot)\) \(\chi_{1734}(613,\cdot)\) \(\chi_{1734}(715,\cdot)\) \(\chi_{1734}(817,\cdot)\) \(\chi_{1734}(919,\cdot)\) \(\chi_{1734}(1021,\cdot)\) \(\chi_{1734}(1123,\cdot)\) \(\chi_{1734}(1225,\cdot)\) \(\chi_{1734}(1327,\cdot)\) \(\chi_{1734}(1429,\cdot)\) \(\chi_{1734}(1531,\cdot)\) \(\chi_{1734}(1633,\cdot)\)
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 17 polynomial |
Values on generators
\((1157,1159)\) → \((1,e\left(\frac{4}{17}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 1734 }(103, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) |
sage: chi.jacobi_sum(n)