Properties

Label 145.17
Modulus $145$
Conductor $145$
Order $4$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(145, base_ring=CyclotomicField(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1,3]))
 
pari: [g,chi] = znchar(Mod(17,145))
 

Basic properties

Modulus: \(145\)
Conductor: \(145\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
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 145.j

\(\chi_{145}(17,\cdot)\) \(\chi_{145}(128,\cdot)\)

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: \(\Q(\sqrt{-1}) \)
Fixed field: 4.4.3048625.1

Values on generators

\((117,31)\) → \((i,-i)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\(1\)\(1\)\(1\)\(-1\)\(1\)\(-1\)\(i\)\(1\)\(1\)\(-i\)\(-1\)\(i\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 145 }(17,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{145}(17,\cdot)) = \sum_{r\in \Z/145\Z} \chi_{145}(17,r) e\left(\frac{2r}{145}\right) = 11.255539774+4.2793485948i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 145 }(17,·),\chi_{ 145 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{145}(17,\cdot),\chi_{145}(1,\cdot)) = \sum_{r\in \Z/145\Z} \chi_{145}(17,r) \chi_{145}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 145 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{145}(17,·)) = \sum_{r \in \Z/145\Z} \chi_{145}(17,r) e\left(\frac{1 r + 2 r^{-1}}{145}\right) = 16.7697733886 \)