Properties

Label 1183.36
Modulus $1183$
Conductor $169$
Order $78$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,47]))
 
pari: [g,chi] = znchar(Mod(36,1183))
 

Basic properties

Modulus: \(1183\)
Conductor: \(169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{169}(36,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1183.bt

\(\chi_{1183}(36,\cdot)\) \(\chi_{1183}(43,\cdot)\) \(\chi_{1183}(127,\cdot)\) \(\chi_{1183}(134,\cdot)\) \(\chi_{1183}(218,\cdot)\) \(\chi_{1183}(225,\cdot)\) \(\chi_{1183}(309,\cdot)\) \(\chi_{1183}(400,\cdot)\) \(\chi_{1183}(407,\cdot)\) \(\chi_{1183}(491,\cdot)\) \(\chi_{1183}(498,\cdot)\) \(\chi_{1183}(582,\cdot)\) \(\chi_{1183}(589,\cdot)\) \(\chi_{1183}(673,\cdot)\) \(\chi_{1183}(680,\cdot)\) \(\chi_{1183}(764,\cdot)\) \(\chi_{1183}(771,\cdot)\) \(\chi_{1183}(855,\cdot)\) \(\chi_{1183}(862,\cdot)\) \(\chi_{1183}(946,\cdot)\) \(\chi_{1183}(953,\cdot)\) \(\chi_{1183}(1044,\cdot)\) \(\chi_{1183}(1128,\cdot)\) \(\chi_{1183}(1135,\cdot)\)

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

\((339,1016)\) → \((1,e\left(\frac{47}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 1183 }(36, a) \) \(1\)\(1\)\(e\left(\frac{47}{78}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{8}{39}\right)\)\(e\left(\frac{11}{26}\right)\)\(e\left(\frac{25}{78}\right)\)\(e\left(\frac{21}{26}\right)\)\(e\left(\frac{17}{39}\right)\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{5}{78}\right)\)\(e\left(\frac{12}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1183 }(36,a) \;\) at \(\;a = \) e.g. 2