from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(432, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,28]))
pari: [g,chi] = znchar(Mod(157,432))
Basic properties
Modulus: | \(432\) | |
Conductor: | \(432\) | 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 432.bg
\(\chi_{432}(13,\cdot)\) \(\chi_{432}(61,\cdot)\) \(\chi_{432}(85,\cdot)\) \(\chi_{432}(133,\cdot)\) \(\chi_{432}(157,\cdot)\) \(\chi_{432}(205,\cdot)\) \(\chi_{432}(229,\cdot)\) \(\chi_{432}(277,\cdot)\) \(\chi_{432}(301,\cdot)\) \(\chi_{432}(349,\cdot)\) \(\chi_{432}(373,\cdot)\) \(\chi_{432}(421,\cdot)\)
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: | 36.36.614667125325361522818798575155151578949632894783197825857500612833312768.1 |
Values on generators
\((271,325,353)\) → \((1,-i,e\left(\frac{7}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 432 }(157, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) |
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)