Properties

Label 731.451
Modulus $731$
Conductor $731$
Order $56$
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(56))
 
M = H._module
 
chi = DirichletCharacter(H, M([7,48]))
 
pari: [g,chi] = znchar(Mod(451,731))
 

Basic properties

Modulus: \(731\)
Conductor: \(731\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(56\)
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.be

\(\chi_{731}(59,\cdot)\) \(\chi_{731}(121,\cdot)\) \(\chi_{731}(127,\cdot)\) \(\chi_{731}(145,\cdot)\) \(\chi_{731}(213,\cdot)\) \(\chi_{731}(219,\cdot)\) \(\chi_{731}(236,\cdot)\) \(\chi_{731}(274,\cdot)\) \(\chi_{731}(342,\cdot)\) \(\chi_{731}(348,\cdot)\) \(\chi_{731}(355,\cdot)\) \(\chi_{731}(365,\cdot)\) \(\chi_{731}(434,\cdot)\) \(\chi_{731}(451,\cdot)\) \(\chi_{731}(484,\cdot)\) \(\chi_{731}(508,\cdot)\) \(\chi_{731}(563,\cdot)\) \(\chi_{731}(570,\cdot)\) \(\chi_{731}(580,\cdot)\) \(\chi_{731}(637,\cdot)\) \(\chi_{731}(661,\cdot)\) \(\chi_{731}(699,\cdot)\) \(\chi_{731}(723,\cdot)\) \(\chi_{731}(729,\cdot)\)

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: $\Q(\zeta_{56})$
Fixed field: Number field defined by a degree 56 polynomial

Values on generators

\((173,562)\) → \((e\left(\frac{1}{8}\right),e\left(\frac{6}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 731 }(451, a) \) \(1\)\(1\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{55}{56}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{3}{56}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{53}{56}\right)\)\(e\left(\frac{33}{56}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 731 }(451,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 731 }(451,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 731 }(451,·),\chi_{ 731 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 731 }(451,·)) \;\) at \(\; a,b = \) e.g. 1,2