Properties

Label 169.114
Modulus $169$
Conductor $169$
Order $78$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(169, base_ring=CyclotomicField(78))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([17]))
 
pari: [g,chi] = znchar(Mod(114,169))
 

Basic properties

Modulus: \(169\)
Conductor: \(169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
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 169.k

\(\chi_{169}(4,\cdot)\) \(\chi_{169}(10,\cdot)\) \(\chi_{169}(17,\cdot)\) \(\chi_{169}(30,\cdot)\) \(\chi_{169}(36,\cdot)\) \(\chi_{169}(43,\cdot)\) \(\chi_{169}(49,\cdot)\) \(\chi_{169}(56,\cdot)\) \(\chi_{169}(62,\cdot)\) \(\chi_{169}(69,\cdot)\) \(\chi_{169}(75,\cdot)\) \(\chi_{169}(82,\cdot)\) \(\chi_{169}(88,\cdot)\) \(\chi_{169}(95,\cdot)\) \(\chi_{169}(101,\cdot)\) \(\chi_{169}(108,\cdot)\) \(\chi_{169}(114,\cdot)\) \(\chi_{169}(121,\cdot)\) \(\chi_{169}(127,\cdot)\) \(\chi_{169}(134,\cdot)\) \(\chi_{169}(140,\cdot)\) \(\chi_{169}(153,\cdot)\) \(\chi_{169}(160,\cdot)\) \(\chi_{169}(166,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\(2\) → \(e\left(\frac{17}{78}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{17}{78}\right)\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{17}{39}\right)\)\(e\left(\frac{25}{26}\right)\)\(e\left(\frac{19}{78}\right)\)\(e\left(\frac{25}{78}\right)\)\(e\left(\frac{17}{26}\right)\)\(e\left(\frac{2}{39}\right)\)\(e\left(\frac{7}{39}\right)\)\(e\left(\frac{35}{78}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 169 }(114,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 169 }(114,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 169 }(114,·),\chi_{ 169 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 169 }(114,·)) \;\) at \(\; a,b = \) e.g. 1,2