Properties

Label 967.197
Modulus $967$
Conductor $967$
Order $138$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(967, base_ring=CyclotomicField(138))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([121]))
 
pari: [g,chi] = znchar(Mod(197,967))
 

Basic properties

Modulus: \(967\)
Conductor: \(967\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(138\)
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 967.l

\(\chi_{967}(20,\cdot)\) \(\chi_{967}(52,\cdot)\) \(\chi_{967}(54,\cdot)\) \(\chi_{967}(55,\cdot)\) \(\chi_{967}(92,\cdot)\) \(\chi_{967}(93,\cdot)\) \(\chi_{967}(96,\cdot)\) \(\chi_{967}(147,\cdot)\) \(\chi_{967}(197,\cdot)\) \(\chi_{967}(210,\cdot)\) \(\chi_{967}(211,\cdot)\) \(\chi_{967}(239,\cdot)\) \(\chi_{967}(249,\cdot)\) \(\chi_{967}(382,\cdot)\) \(\chi_{967}(413,\cdot)\) \(\chi_{967}(428,\cdot)\) \(\chi_{967}(443,\cdot)\) \(\chi_{967}(454,\cdot)\) \(\chi_{967}(473,\cdot)\) \(\chi_{967}(522,\cdot)\) \(\chi_{967}(546,\cdot)\) \(\chi_{967}(567,\cdot)\) \(\chi_{967}(590,\cdot)\) \(\chi_{967}(615,\cdot)\) \(\chi_{967}(626,\cdot)\) \(\chi_{967}(632,\cdot)\) \(\chi_{967}(646,\cdot)\) \(\chi_{967}(687,\cdot)\) \(\chi_{967}(726,\cdot)\) \(\chi_{967}(742,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{69})$
Fixed field: Number field defined by a degree 138 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{121}{138}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{64}{69}\right)\)\(e\left(\frac{37}{46}\right)\)\(e\left(\frac{59}{69}\right)\)\(e\left(\frac{121}{138}\right)\)\(e\left(\frac{101}{138}\right)\)\(e\left(\frac{103}{138}\right)\)\(e\left(\frac{18}{23}\right)\)\(e\left(\frac{14}{23}\right)\)\(e\left(\frac{37}{46}\right)\)\(e\left(\frac{14}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 967 }(197,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 967 }(197,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 967 }(197,·),\chi_{ 967 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 967 }(197,·)) \;\) at \(\; a,b = \) e.g. 1,2