Properties

Label 169.64
Modulus $169$
Conductor $169$
Order $26$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(26))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(64,169))
 

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()
 
order = charorder(g,chi)
 
[ 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

\(2\) → \(e\left(\frac{1}{26}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 169 }(64, a) \) \(1\)\(1\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{21}{26}\right)\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{25}{26}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 169 }(64,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 169 }(64,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 169 }(64,·),\chi_{ 169 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 169 }(64,·)) \;\) at \(\; a,b = \) e.g. 1,2