Properties

Label 19.10
Modulus $19$
Conductor $19$
Order $18$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(19)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([17]))
 
pari: [g,chi] = znchar(Mod(10,19))
 

Basic properties

Modulus: \(19\)
Conductor: \(19\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
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 19.f

\(\chi_{19}(2,\cdot)\) \(\chi_{19}(3,\cdot)\) \(\chi_{19}(10,\cdot)\) \(\chi_{19}(13,\cdot)\) \(\chi_{19}(14,\cdot)\) \(\chi_{19}(15,\cdot)\)

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

Values on generators

\(2\) → \(e\left(\frac{17}{18}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{3}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{9})\)
Fixed field: \(\Q(\zeta_{19})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 19 }(10,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{19}(10,\cdot)) = \sum_{r\in \Z/19\Z} \chi_{19}(10,r) e\left(\frac{2r}{19}\right) = -2.2663117537+3.7234165809i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 19 }(10,·),\chi_{ 19 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{19}(10,\cdot),\chi_{19}(1,\cdot)) = \sum_{r\in \Z/19\Z} \chi_{19}(10,r) \chi_{19}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 19 }(10,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{19}(10,·)) = \sum_{r \in \Z/19\Z} \chi_{19}(10,r) e\left(\frac{1 r + 2 r^{-1}}{19}\right) = -0.3965933587+-2.2491927047i \)