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Properties

Label 195.38
Modulus $195$
Conductor $195$
Order $4$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(195, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([2,3,2]))
 
Copy content pari:[g,chi] = znchar(Mod(38,195))
 

Basic properties

Modulus: \(195\)
Conductor: \(195\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(4\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 195.s

\(\chi_{195}(38,\cdot)\) \(\chi_{195}(77,\cdot)\)

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

Related number fields

Field of values: \(\mathbb{Q}(i)\)
Fixed field: 4.4.190125.1

Values on generators

\((131,157,106)\) → \((-1,-i,-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(14\)\(16\)\(17\)\(19\)\(22\)
\( \chi_{ 195 }(38, a) \) \(1\)\(1\)\(-i\)\(-1\)\(i\)\(i\)\(1\)\(1\)\(1\)\(i\)\(1\)\(-i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 195 }(38,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 195 }(38,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 195 }(38,·),\chi_{ 195 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 195 }(38,·)) \;\) at \(\; a,b = \) e.g. 1,2