from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1755, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([22,27,30]))
pari: [g,chi] = znchar(Mod(23,1755))
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.fr
\(\chi_{1755}(23,\cdot)\) \(\chi_{1755}(173,\cdot)\) \(\chi_{1755}(257,\cdot)\) \(\chi_{1755}(407,\cdot)\) \(\chi_{1755}(608,\cdot)\) \(\chi_{1755}(758,\cdot)\) \(\chi_{1755}(842,\cdot)\) \(\chi_{1755}(992,\cdot)\) \(\chi_{1755}(1193,\cdot)\) \(\chi_{1755}(1343,\cdot)\) \(\chi_{1755}(1427,\cdot)\) \(\chi_{1755}(1577,\cdot)\)
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
\((326,352,1081)\) → \((e\left(\frac{11}{18}\right),-i,e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
\( \chi_{ 1755 }(23, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(1\) | \(e\left(\frac{35}{36}\right)\) |
sage: chi.jacobi_sum(n)