Properties

Label 169.60
Modulus $169$
Conductor $169$
Order $52$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(52))
 
M = H._module
 
chi = DirichletCharacter(H, M([45]))
 
pari: [g,chi] = znchar(Mod(60,169))
 

Basic properties

Modulus: \(169\)
Conductor: \(169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(52\)
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 169.j

\(\chi_{169}(5,\cdot)\) \(\chi_{169}(8,\cdot)\) \(\chi_{169}(18,\cdot)\) \(\chi_{169}(21,\cdot)\) \(\chi_{169}(31,\cdot)\) \(\chi_{169}(34,\cdot)\) \(\chi_{169}(44,\cdot)\) \(\chi_{169}(47,\cdot)\) \(\chi_{169}(57,\cdot)\) \(\chi_{169}(60,\cdot)\) \(\chi_{169}(73,\cdot)\) \(\chi_{169}(83,\cdot)\) \(\chi_{169}(86,\cdot)\) \(\chi_{169}(96,\cdot)\) \(\chi_{169}(109,\cdot)\) \(\chi_{169}(112,\cdot)\) \(\chi_{169}(122,\cdot)\) \(\chi_{169}(125,\cdot)\) \(\chi_{169}(135,\cdot)\) \(\chi_{169}(138,\cdot)\) \(\chi_{169}(148,\cdot)\) \(\chi_{169}(151,\cdot)\) \(\chi_{169}(161,\cdot)\) \(\chi_{169}(164,\cdot)\)

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: $\Q(\zeta_{52})$
Fixed field: Number field defined by a degree 52 polynomial

Values on generators

\(2\) → \(e\left(\frac{45}{52}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 169 }(60, a) \) \(-1\)\(1\)\(e\left(\frac{45}{52}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{19}{26}\right)\)\(e\left(\frac{41}{52}\right)\)\(e\left(\frac{9}{52}\right)\)\(e\left(\frac{31}{52}\right)\)\(e\left(\frac{31}{52}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{17}{26}\right)\)\(e\left(\frac{7}{52}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 169 }(60,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 169 }(60,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 169 }(60,·),\chi_{ 169 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 169 }(60,·)) \;\) at \(\; a,b = \) e.g. 1,2