Properties

Label 916.245
Modulus $916$
Conductor $229$
Order $19$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
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)
 
\( \chi_{ 916 }(245,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 916 }(245,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 916 }(245,·),\chi_{ 916 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 916 }(245,·)) \;\) at \(\; a,b = \) e.g. 1,2