Properties

Label 1944.379
Modulus $1944$
Conductor $216$
Order $18$
Real no
Primitive no
Minimal no
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(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9,9,4]))
 
pari: [g,chi] = znchar(Mod(379,1944))
 

Basic properties

Modulus: \(1944\)
Conductor: \(216\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{216}(43,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1944.r

\(\chi_{1944}(379,\cdot)\) \(\chi_{1944}(595,\cdot)\) \(\chi_{1944}(1027,\cdot)\) \(\chi_{1944}(1243,\cdot)\) \(\chi_{1944}(1675,\cdot)\) \(\chi_{1944}(1891,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: 18.0.132173713091594538512566714368.2

Values on generators

\((487,973,1217)\) → \((-1,-1,e\left(\frac{2}{9}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{17}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1944 }(379,a) \;\) at \(\;a = \) e.g. 2