from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(146, base_ring=CyclotomicField(72))
M = H._module
chi = DirichletCharacter(H, M([31]))
pari: [g,chi] = znchar(Mod(47,146))
Basic properties
Modulus: | \(146\) | |
Conductor: | \(73\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(72\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{73}(47,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 146.l
\(\chi_{146}(5,\cdot)\) \(\chi_{146}(11,\cdot)\) \(\chi_{146}(13,\cdot)\) \(\chi_{146}(15,\cdot)\) \(\chi_{146}(29,\cdot)\) \(\chi_{146}(31,\cdot)\) \(\chi_{146}(33,\cdot)\) \(\chi_{146}(39,\cdot)\) \(\chi_{146}(45,\cdot)\) \(\chi_{146}(47,\cdot)\) \(\chi_{146}(53,\cdot)\) \(\chi_{146}(59,\cdot)\) \(\chi_{146}(87,\cdot)\) \(\chi_{146}(93,\cdot)\) \(\chi_{146}(99,\cdot)\) \(\chi_{146}(101,\cdot)\) \(\chi_{146}(107,\cdot)\) \(\chi_{146}(113,\cdot)\) \(\chi_{146}(115,\cdot)\) \(\chi_{146}(117,\cdot)\) \(\chi_{146}(131,\cdot)\) \(\chi_{146}(133,\cdot)\) \(\chi_{146}(135,\cdot)\) \(\chi_{146}(141,\cdot)\)
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
\(5\) → \(e\left(\frac{31}{72}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 146 }(47, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{19}{24}\right)\) |
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)