from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(73, base_ring=CyclotomicField(72))
M = H._module
chi = DirichletCharacter(H, M([19]))
pari: [g,chi] = znchar(Mod(62,73))
Basic properties
Modulus: | \(73\) | |
Conductor: | \(73\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(72\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 73.l
\(\chi_{73}(5,\cdot)\) \(\chi_{73}(11,\cdot)\) \(\chi_{73}(13,\cdot)\) \(\chi_{73}(14,\cdot)\) \(\chi_{73}(15,\cdot)\) \(\chi_{73}(20,\cdot)\) \(\chi_{73}(26,\cdot)\) \(\chi_{73}(28,\cdot)\) \(\chi_{73}(29,\cdot)\) \(\chi_{73}(31,\cdot)\) \(\chi_{73}(33,\cdot)\) \(\chi_{73}(34,\cdot)\) \(\chi_{73}(39,\cdot)\) \(\chi_{73}(40,\cdot)\) \(\chi_{73}(42,\cdot)\) \(\chi_{73}(44,\cdot)\) \(\chi_{73}(45,\cdot)\) \(\chi_{73}(47,\cdot)\) \(\chi_{73}(53,\cdot)\) \(\chi_{73}(58,\cdot)\) \(\chi_{73}(59,\cdot)\) \(\chi_{73}(60,\cdot)\) \(\chi_{73}(62,\cdot)\) \(\chi_{73}(68,\cdot)\)
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: | $\Q(\zeta_{72})$ |
Fixed field: | Number field defined by a degree 72 polynomial |
Values on generators
\(5\) → \(e\left(\frac{19}{72}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 73 }(62, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{37}{72}\right)\) |
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)