Properties

Label 53.14
Modulus $53$
Conductor $53$
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(53, base_ring=CyclotomicField(52))
 
M = H._module
 
chi = DirichletCharacter(H, M([15]))
 
pari: [g,chi] = znchar(Mod(14,53))
 

Basic properties

Modulus: \(53\)
Conductor: \(53\)
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 53.f

\(\chi_{53}(2,\cdot)\) \(\chi_{53}(3,\cdot)\) \(\chi_{53}(5,\cdot)\) \(\chi_{53}(8,\cdot)\) \(\chi_{53}(12,\cdot)\) \(\chi_{53}(14,\cdot)\) \(\chi_{53}(18,\cdot)\) \(\chi_{53}(19,\cdot)\) \(\chi_{53}(20,\cdot)\) \(\chi_{53}(21,\cdot)\) \(\chi_{53}(22,\cdot)\) \(\chi_{53}(26,\cdot)\) \(\chi_{53}(27,\cdot)\) \(\chi_{53}(31,\cdot)\) \(\chi_{53}(32,\cdot)\) \(\chi_{53}(33,\cdot)\) \(\chi_{53}(34,\cdot)\) \(\chi_{53}(35,\cdot)\) \(\chi_{53}(39,\cdot)\) \(\chi_{53}(41,\cdot)\) \(\chi_{53}(45,\cdot)\) \(\chi_{53}(48,\cdot)\) \(\chi_{53}(50,\cdot)\) \(\chi_{53}(51,\cdot)\)

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

\(2\) → \(e\left(\frac{15}{52}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 53 }(14, a) \) \(-1\)\(1\)\(e\left(\frac{15}{52}\right)\)\(e\left(\frac{47}{52}\right)\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{29}{52}\right)\)\(e\left(\frac{5}{26}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{45}{52}\right)\)\(e\left(\frac{21}{26}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{19}{26}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 53 }(14,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 53 }(14,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 53 }(14,·),\chi_{ 53 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 53 }(14,·)) \;\) at \(\; a,b = \) e.g. 1,2