from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1755, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([34,27,12]))
pari: [g,chi] = znchar(Mod(68,1755))
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.fg
\(\chi_{1755}(68,\cdot)\) \(\chi_{1755}(302,\cdot)\) \(\chi_{1755}(308,\cdot)\) \(\chi_{1755}(542,\cdot)\) \(\chi_{1755}(653,\cdot)\) \(\chi_{1755}(887,\cdot)\) \(\chi_{1755}(893,\cdot)\) \(\chi_{1755}(1127,\cdot)\) \(\chi_{1755}(1238,\cdot)\) \(\chi_{1755}(1472,\cdot)\) \(\chi_{1755}(1478,\cdot)\) \(\chi_{1755}(1712,\cdot)\)
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
\((326,352,1081)\) → \((e\left(\frac{17}{18}\right),-i,e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 1755 }(68, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(-1\) | \(e\left(\frac{23}{36}\right)\) |
sage: chi.jacobi_sum(n)