Properties

Label 725.716
Modulus $725$
Conductor $725$
Order $35$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(70))
 
M = H._module
 
chi = DirichletCharacter(H, M([14,60]))
 
pari: [g,chi] = znchar(Mod(716,725))
 

Basic properties

Modulus: \(725\)
Conductor: \(725\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(35\)
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 725.be

\(\chi_{725}(16,\cdot)\) \(\chi_{725}(36,\cdot)\) \(\chi_{725}(81,\cdot)\) \(\chi_{725}(111,\cdot)\) \(\chi_{725}(136,\cdot)\) \(\chi_{725}(141,\cdot)\) \(\chi_{725}(161,\cdot)\) \(\chi_{725}(181,\cdot)\) \(\chi_{725}(256,\cdot)\) \(\chi_{725}(281,\cdot)\) \(\chi_{725}(286,\cdot)\) \(\chi_{725}(306,\cdot)\) \(\chi_{725}(371,\cdot)\) \(\chi_{725}(431,\cdot)\) \(\chi_{725}(471,\cdot)\) \(\chi_{725}(516,\cdot)\) \(\chi_{725}(546,\cdot)\) \(\chi_{725}(571,\cdot)\) \(\chi_{725}(596,\cdot)\) \(\chi_{725}(616,\cdot)\) \(\chi_{725}(661,\cdot)\) \(\chi_{725}(691,\cdot)\) \(\chi_{725}(716,\cdot)\) \(\chi_{725}(721,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 35 polynomial

Values on generators

\((552,176)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{6}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 725 }(716, a) \) \(1\)\(1\)\(e\left(\frac{2}{35}\right)\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{26}{35}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{6}{35}\right)\)\(e\left(\frac{13}{35}\right)\)\(e\left(\frac{22}{35}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{8}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 725 }(716,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 725 }(716,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 725 }(716,·),\chi_{ 725 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 725 }(716,·)) \;\) at \(\; a,b = \) e.g. 1,2