Properties

Label 1113.38
Modulus $1113$
Conductor $1113$
Order $78$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1113, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([39,13,57]))
 
pari: [g,chi] = znchar(Mod(38,1113))
 

Basic properties

Modulus: \(1113\)
Conductor: \(1113\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
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 1113.bp

\(\chi_{1113}(17,\cdot)\) \(\chi_{1113}(38,\cdot)\) \(\chi_{1113}(59,\cdot)\) \(\chi_{1113}(110,\cdot)\) \(\chi_{1113}(131,\cdot)\) \(\chi_{1113}(143,\cdot)\) \(\chi_{1113}(269,\cdot)\) \(\chi_{1113}(290,\cdot)\) \(\chi_{1113}(467,\cdot)\) \(\chi_{1113}(488,\cdot)\) \(\chi_{1113}(626,\cdot)\) \(\chi_{1113}(647,\cdot)\) \(\chi_{1113}(698,\cdot)\) \(\chi_{1113}(782,\cdot)\) \(\chi_{1113}(824,\cdot)\) \(\chi_{1113}(857,\cdot)\) \(\chi_{1113}(908,\cdot)\) \(\chi_{1113}(941,\cdot)\) \(\chi_{1113}(971,\cdot)\) \(\chi_{1113}(983,\cdot)\) \(\chi_{1113}(992,\cdot)\) \(\chi_{1113}(1013,\cdot)\) \(\chi_{1113}(1067,\cdot)\) \(\chi_{1113}(1097,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((743,955,1009)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{19}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 1113 }(38, a) \) \(1\)\(1\)\(e\left(\frac{22}{39}\right)\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{53}{78}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{19}{78}\right)\)\(e\left(\frac{43}{78}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{10}{39}\right)\)\(e\left(\frac{38}{39}\right)\)\(e\left(\frac{34}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1113 }(38,a) \;\) at \(\;a = \) e.g. 2