Properties

Label 731.659
Modulus $731$
Conductor $731$
Order $84$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,40]))
 
pari: [g,chi] = znchar(Mod(659,731))
 

Basic properties

Modulus: \(731\)
Conductor: \(731\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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 731.bh

\(\chi_{731}(13,\cdot)\) \(\chi_{731}(38,\cdot)\) \(\chi_{731}(81,\cdot)\) \(\chi_{731}(225,\cdot)\) \(\chi_{731}(268,\cdot)\) \(\chi_{731}(310,\cdot)\) \(\chi_{731}(353,\cdot)\) \(\chi_{731}(361,\cdot)\) \(\chi_{731}(404,\cdot)\) \(\chi_{731}(412,\cdot)\) \(\chi_{731}(455,\cdot)\) \(\chi_{731}(497,\cdot)\) \(\chi_{731}(531,\cdot)\) \(\chi_{731}(540,\cdot)\) \(\chi_{731}(574,\cdot)\) \(\chi_{731}(582,\cdot)\) \(\chi_{731}(599,\cdot)\) \(\chi_{731}(616,\cdot)\) \(\chi_{731}(625,\cdot)\) \(\chi_{731}(633,\cdot)\) \(\chi_{731}(642,\cdot)\) \(\chi_{731}(659,\cdot)\) \(\chi_{731}(676,\cdot)\) \(\chi_{731}(701,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((173,562)\) → \((i,e\left(\frac{10}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 731 }(659, a) \) \(1\)\(1\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{61}{84}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{13}{84}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{43}{84}\right)\)\(e\left(\frac{1}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 731 }(659,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 731 }(659,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 731 }(659,·),\chi_{ 731 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 731 }(659,·)) \;\) at \(\; a,b = \) e.g. 1,2