Properties

Label 169.77
Modulus $169$
Conductor $169$
Order $26$
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(26))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9]))
 
pari: [g,chi] = znchar(Mod(77,169))
 

Basic properties

Modulus: \(169\)
Conductor: \(169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(26\)
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.h

\(\chi_{169}(12,\cdot)\) \(\chi_{169}(25,\cdot)\) \(\chi_{169}(38,\cdot)\) \(\chi_{169}(51,\cdot)\) \(\chi_{169}(64,\cdot)\) \(\chi_{169}(77,\cdot)\) \(\chi_{169}(90,\cdot)\) \(\chi_{169}(103,\cdot)\) \(\chi_{169}(116,\cdot)\) \(\chi_{169}(129,\cdot)\) \(\chi_{169}(142,\cdot)\) \(\chi_{169}(155,\cdot)\)

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: \(\Q(\zeta_{13})\)
Fixed field: 26.26.3830224792147131369362629348887201408953937846517364173.1

Values on generators

\(2\) → \(e\left(\frac{9}{26}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{7}{26}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{17}{26}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 169 }(77,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 169 }(77,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 169 }(77,·),\chi_{ 169 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 169 }(77,·)) \;\) at \(\; a,b = \) e.g. 1,2