Properties

Label 1113.8
Modulus $1113$
Conductor $159$
Order $52$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1113, base_ring=CyclotomicField(52))
 
M = H._module
 
chi = DirichletCharacter(H, M([26,0,3]))
 
pari: [g,chi] = znchar(Mod(8,1113))
 

Basic properties

Modulus: \(1113\)
Conductor: \(159\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(52\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{159}(8,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1113.bh

\(\chi_{1113}(8,\cdot)\) \(\chi_{1113}(50,\cdot)\) \(\chi_{1113}(71,\cdot)\) \(\chi_{1113}(92,\cdot)\) \(\chi_{1113}(239,\cdot)\) \(\chi_{1113}(260,\cdot)\) \(\chi_{1113}(323,\cdot)\) \(\chi_{1113}(344,\cdot)\) \(\chi_{1113}(491,\cdot)\) \(\chi_{1113}(512,\cdot)\) \(\chi_{1113}(533,\cdot)\) \(\chi_{1113}(575,\cdot)\) \(\chi_{1113}(617,\cdot)\) \(\chi_{1113}(638,\cdot)\) \(\chi_{1113}(701,\cdot)\) \(\chi_{1113}(722,\cdot)\) \(\chi_{1113}(764,\cdot)\) \(\chi_{1113}(827,\cdot)\) \(\chi_{1113}(869,\cdot)\) \(\chi_{1113}(932,\cdot)\) \(\chi_{1113}(974,\cdot)\) \(\chi_{1113}(995,\cdot)\) \(\chi_{1113}(1058,\cdot)\) \(\chi_{1113}(1079,\cdot)\)

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

\((743,955,1009)\) → \((-1,1,e\left(\frac{3}{52}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 1113 }(8, a) \) \(1\)\(1\)\(e\left(\frac{29}{52}\right)\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{11}{52}\right)\)\(e\left(\frac{35}{52}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{7}{52}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1113 }(8,a) \;\) at \(\;a = \) e.g. 2