Properties

Label 119.76
Modulus $119$
Conductor $119$
Order $8$
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(119, base_ring=CyclotomicField(8))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([4,5]))
 
pari: [g,chi] = znchar(Mod(76,119))
 

Basic properties

Modulus: \(119\)
Conductor: \(119\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
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 119.l

\(\chi_{119}(76,\cdot)\) \(\chi_{119}(83,\cdot)\) \(\chi_{119}(104,\cdot)\) \(\chi_{119}(111,\cdot)\)

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: 8.0.985223153873.1

Values on generators

\((52,71)\) → \((-1,e\left(\frac{5}{8}\right))\)

Values

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 119 }(76,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{119}(76,\cdot)) = \sum_{r\in \Z/119\Z} \chi_{119}(76,r) e\left(\frac{2r}{119}\right) = -7.106025825+-8.2767383053i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 119 }(76,·),\chi_{ 119 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{119}(76,\cdot),\chi_{119}(1,\cdot)) = \sum_{r\in \Z/119\Z} \chi_{119}(76,r) \chi_{119}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 119 }(76,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{119}(76,·)) = \sum_{r \in \Z/119\Z} \chi_{119}(76,r) e\left(\frac{1 r + 2 r^{-1}}{119}\right) = 3.9926699273+3.9926699273i \)