Properties

Label 119.36
Modulus $119$
Conductor $17$
Order $8$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(119, base_ring=CyclotomicField(8))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,7]))
 
pari: [g,chi] = znchar(Mod(36,119))
 

Basic properties

Modulus: \(119\)
Conductor: \(17\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{17}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 119.k

\(\chi_{119}(8,\cdot)\) \(\chi_{119}(15,\cdot)\) \(\chi_{119}(36,\cdot)\) \(\chi_{119}(43,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: \(\Q(\zeta_{17})^+\)

Values on generators

\((52,71)\) → \((1,e\left(\frac{7}{8}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(i\)\(e\left(\frac{7}{8}\right)\)\(-1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(-i\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{3}{8}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 119 }(36,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{119}(36,\cdot)) = \sum_{r\in \Z/119\Z} \chi_{119}(36,r) e\left(\frac{2r}{119}\right) = 4.1186430292+-0.1917800773i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 119 }(36,·),\chi_{ 119 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{119}(36,\cdot),\chi_{119}(1,\cdot)) = \sum_{r\in \Z/119\Z} \chi_{119}(36,r) \chi_{119}(1,1-r) = -5 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 119 }(36,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{119}(36,·)) = \sum_{r \in \Z/119\Z} \chi_{119}(36,r) e\left(\frac{1 r + 2 r^{-1}}{119}\right) = -4.5441018422+-4.5441018422i \)