Properties

Label 683.7
Modulus $683$
Conductor $683$
Order $682$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(683, base_ring=CyclotomicField(682))
 
M = H._module
 
chi = DirichletCharacter(H, M([265]))
 
pari: [g,chi] = znchar(Mod(7,683))
 

Basic properties

Modulus: \(683\)
Conductor: \(683\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(682\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 683.h

\(\chi_{683}(5,\cdot)\) \(\chi_{683}(6,\cdot)\) \(\chi_{683}(7,\cdot)\) \(\chi_{683}(11,\cdot)\) \(\chi_{683}(13,\cdot)\) \(\chi_{683}(15,\cdot)\) \(\chi_{683}(17,\cdot)\) \(\chi_{683}(18,\cdot)\) \(\chi_{683}(20,\cdot)\) \(\chi_{683}(21,\cdot)\) \(\chi_{683}(23,\cdot)\) \(\chi_{683}(24,\cdot)\) \(\chi_{683}(28,\cdot)\) \(\chi_{683}(31,\cdot)\) \(\chi_{683}(33,\cdot)\) \(\chi_{683}(38,\cdot)\) \(\chi_{683}(39,\cdot)\) \(\chi_{683}(41,\cdot)\) \(\chi_{683}(43,\cdot)\) \(\chi_{683}(44,\cdot)\) \(\chi_{683}(45,\cdot)\) \(\chi_{683}(47,\cdot)\) \(\chi_{683}(50,\cdot)\) \(\chi_{683}(51,\cdot)\) \(\chi_{683}(52,\cdot)\) \(\chi_{683}(54,\cdot)\) \(\chi_{683}(58,\cdot)\) \(\chi_{683}(60,\cdot)\) \(\chi_{683}(63,\cdot)\) \(\chi_{683}(68,\cdot)\) ...

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: $\Q(\zeta_{341})$
Fixed field: Number field defined by a degree 682 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{265}{682}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 683 }(7, a) \) \(-1\)\(1\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{7}{31}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{265}{682}\right)\)\(e\left(\frac{371}{682}\right)\)\(e\left(\frac{661}{682}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{14}{31}\right)\)\(e\left(\frac{241}{341}\right)\)\(e\left(\frac{195}{682}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 683 }(7,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 683 }(7,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 683 }(7,·),\chi_{ 683 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 683 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2