from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(916, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([0,14]))
pari: [g,chi] = znchar(Mod(245,916))
Basic properties
Modulus: | \(916\) | |
Conductor: | \(229\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(19\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{229}(16,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 916.m
\(\chi_{916}(17,\cdot)\) \(\chi_{916}(53,\cdot)\) \(\chi_{916}(57,\cdot)\) \(\chi_{916}(61,\cdot)\) \(\chi_{916}(121,\cdot)\) \(\chi_{916}(161,\cdot)\) \(\chi_{916}(165,\cdot)\) \(\chi_{916}(225,\cdot)\) \(\chi_{916}(245,\cdot)\) \(\chi_{916}(273,\cdot)\) \(\chi_{916}(289,\cdot)\) \(\chi_{916}(333,\cdot)\) \(\chi_{916}(485,\cdot)\) \(\chi_{916}(501,\cdot)\) \(\chi_{916}(661,\cdot)\) \(\chi_{916}(729,\cdot)\) \(\chi_{916}(901,\cdot)\) \(\chi_{916}(905,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
Fixed field: | Number field defined by a degree 19 polynomial |
Values on generators
\((459,693)\) → \((1,e\left(\frac{7}{19}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 916 }(245, a) \) | \(1\) | \(1\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) |
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)