from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(148, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,1]))
pari: [g,chi] = znchar(Mod(39,148))
Basic properties
| Modulus: | \(148\) | |
| Conductor: | \(148\) | 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 148.q
\(\chi_{148}(15,\cdot)\) \(\chi_{148}(19,\cdot)\) \(\chi_{148}(35,\cdot)\) \(\chi_{148}(39,\cdot)\) \(\chi_{148}(55,\cdot)\) \(\chi_{148}(59,\cdot)\) \(\chi_{148}(79,\cdot)\) \(\chi_{148}(87,\cdot)\) \(\chi_{148}(91,\cdot)\) \(\chi_{148}(131,\cdot)\) \(\chi_{148}(135,\cdot)\) \(\chi_{148}(143,\cdot)\)
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: | \(\Q(\zeta_{148})^+\) |
Values on generators
\((75,113)\) → \((-1,e\left(\frac{1}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 148 }(39, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) |
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)