sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(87, base_ring=CyclotomicField(28))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([14,25]))
pari: [g,chi] = znchar(Mod(11,87))
Basic properties
| Modulus: | \(87\) | |
| Conductor: | \(87\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | 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 87.k
\(\chi_{87}(2,\cdot)\) \(\chi_{87}(8,\cdot)\) \(\chi_{87}(11,\cdot)\) \(\chi_{87}(14,\cdot)\) \(\chi_{87}(26,\cdot)\) \(\chi_{87}(32,\cdot)\) \(\chi_{87}(44,\cdot)\) \(\chi_{87}(47,\cdot)\) \(\chi_{87}(50,\cdot)\) \(\chi_{87}(56,\cdot)\) \(\chi_{87}(68,\cdot)\) \(\chi_{87}(77,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{28})\) |
| Fixed field: | \(\Q(\zeta_{87})^+\) |
Values on generators
\((59,31)\) → \((-1,e\left(\frac{25}{28}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \(1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)