Properties

Label 113.52
Modulus $113$
Conductor $113$
Order $56$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(113, base_ring=CyclotomicField(56))
 
M = H._module
 
chi = DirichletCharacter(H, M([23]))
 
pari: [g,chi] = znchar(Mod(52,113))
 

Basic properties

Modulus: \(113\)
Conductor: \(113\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(56\)
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 113.i

\(\chi_{113}(9,\cdot)\) \(\chi_{113}(11,\cdot)\) \(\chi_{113}(13,\cdot)\) \(\chi_{113}(22,\cdot)\) \(\chi_{113}(25,\cdot)\) \(\chi_{113}(26,\cdot)\) \(\chi_{113}(31,\cdot)\) \(\chi_{113}(36,\cdot)\) \(\chi_{113}(41,\cdot)\) \(\chi_{113}(50,\cdot)\) \(\chi_{113}(51,\cdot)\) \(\chi_{113}(52,\cdot)\) \(\chi_{113}(61,\cdot)\) \(\chi_{113}(62,\cdot)\) \(\chi_{113}(63,\cdot)\) \(\chi_{113}(72,\cdot)\) \(\chi_{113}(77,\cdot)\) \(\chi_{113}(82,\cdot)\) \(\chi_{113}(87,\cdot)\) \(\chi_{113}(88,\cdot)\) \(\chi_{113}(91,\cdot)\) \(\chi_{113}(100,\cdot)\) \(\chi_{113}(102,\cdot)\) \(\chi_{113}(104,\cdot)\)

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: $\Q(\zeta_{56})$
Fixed field: Number field defined by a degree 56 polynomial

Values on generators

\(3\) → \(e\left(\frac{23}{56}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 113 }(52, a) \) \(1\)\(1\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{23}{56}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{5}{56}\right)\)\(e\left(\frac{19}{56}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{1}{56}\right)\)\(e\left(\frac{11}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 113 }(52,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 113 }(52,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 113 }(52,·),\chi_{ 113 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 113 }(52,·)) \;\) at \(\; a,b = \) e.g. 1,2