from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1755, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([2,18,9]))
pari: [g,chi] = znchar(Mod(164,1755))
Basic properties
Modulus: | \(1755\) | |
Conductor: | \(1755\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | 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 1755.fv
\(\chi_{1755}(164,\cdot)\) \(\chi_{1755}(239,\cdot)\) \(\chi_{1755}(434,\cdot)\) \(\chi_{1755}(554,\cdot)\) \(\chi_{1755}(749,\cdot)\) \(\chi_{1755}(824,\cdot)\) \(\chi_{1755}(1019,\cdot)\) \(\chi_{1755}(1139,\cdot)\) \(\chi_{1755}(1334,\cdot)\) \(\chi_{1755}(1409,\cdot)\) \(\chi_{1755}(1604,\cdot)\) \(\chi_{1755}(1724,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((326,352,1081)\) → \((e\left(\frac{1}{18}\right),-1,i)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
\( \chi_{ 1755 }(164, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) |
sage: chi.jacobi_sum(n)