Properties

Label 713.242
Modulus $713$
Conductor $713$
Order $33$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(713, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([60,22]))
 
pari: [g,chi] = znchar(Mod(242,713))
 

Basic properties

Modulus: \(713\)
Conductor: \(713\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
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 713.u

\(\chi_{713}(25,\cdot)\) \(\chi_{713}(36,\cdot)\) \(\chi_{713}(87,\cdot)\) \(\chi_{713}(98,\cdot)\) \(\chi_{713}(118,\cdot)\) \(\chi_{713}(211,\cdot)\) \(\chi_{713}(242,\cdot)\) \(\chi_{713}(284,\cdot)\) \(\chi_{713}(315,\cdot)\) \(\chi_{713}(335,\cdot)\) \(\chi_{713}(377,\cdot)\) \(\chi_{713}(397,\cdot)\) \(\chi_{713}(439,\cdot)\) \(\chi_{713}(501,\cdot)\) \(\chi_{713}(532,\cdot)\) \(\chi_{713}(583,\cdot)\) \(\chi_{713}(614,\cdot)\) \(\chi_{713}(625,\cdot)\) \(\chi_{713}(656,\cdot)\) \(\chi_{713}(676,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 33 polynomial

Values on generators

\((373,530)\) → \((e\left(\frac{10}{11}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 713 }(242, a) \) \(1\)\(1\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{13}{33}\right)\)\(e\left(\frac{28}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 713 }(242,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 713 }(242,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 713 }(242,·),\chi_{ 713 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 713 }(242,·)) \;\) at \(\; a,b = \) e.g. 1,2