Properties

Label 1944.43
Modulus $1944$
Conductor $1944$
Order $162$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1944, base_ring=CyclotomicField(162))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([81,81,130]))
 
pari: [g,chi] = znchar(Mod(43,1944))
 

Basic properties

Modulus: \(1944\)
Conductor: \(1944\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(162\)
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 1944.bh

\(\chi_{1944}(43,\cdot)\) \(\chi_{1944}(67,\cdot)\) \(\chi_{1944}(115,\cdot)\) \(\chi_{1944}(139,\cdot)\) \(\chi_{1944}(187,\cdot)\) \(\chi_{1944}(211,\cdot)\) \(\chi_{1944}(259,\cdot)\) \(\chi_{1944}(283,\cdot)\) \(\chi_{1944}(331,\cdot)\) \(\chi_{1944}(355,\cdot)\) \(\chi_{1944}(403,\cdot)\) \(\chi_{1944}(427,\cdot)\) \(\chi_{1944}(475,\cdot)\) \(\chi_{1944}(499,\cdot)\) \(\chi_{1944}(547,\cdot)\) \(\chi_{1944}(571,\cdot)\) \(\chi_{1944}(619,\cdot)\) \(\chi_{1944}(643,\cdot)\) \(\chi_{1944}(691,\cdot)\) \(\chi_{1944}(715,\cdot)\) \(\chi_{1944}(763,\cdot)\) \(\chi_{1944}(787,\cdot)\) \(\chi_{1944}(835,\cdot)\) \(\chi_{1944}(859,\cdot)\) \(\chi_{1944}(907,\cdot)\) \(\chi_{1944}(931,\cdot)\) \(\chi_{1944}(979,\cdot)\) \(\chi_{1944}(1003,\cdot)\) \(\chi_{1944}(1051,\cdot)\) \(\chi_{1944}(1075,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{81})$
Fixed field: Number field defined by a degree 162 polynomial (not computed)

Values on generators

\((487,973,1217)\) → \((-1,-1,e\left(\frac{65}{81}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{155}{162}\right)\)\(e\left(\frac{109}{162}\right)\)\(e\left(\frac{8}{81}\right)\)\(e\left(\frac{149}{162}\right)\)\(e\left(\frac{13}{27}\right)\)\(e\left(\frac{5}{27}\right)\)\(e\left(\frac{107}{162}\right)\)\(e\left(\frac{74}{81}\right)\)\(e\left(\frac{31}{162}\right)\)\(e\left(\frac{89}{162}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1944 }(43,a) \;\) at \(\;a = \) e.g. 2