Properties

Label 1444.1151
Modulus $1444$
Conductor $76$
Order $6$
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(1444, base_ring=CyclotomicField(6))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3,4]))
 
pari: [g,chi] = znchar(Mod(1151,1444))
 

Basic properties

Modulus: \(1444\)
Conductor: \(76\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{76}(11,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1444.g

\(\chi_{1444}(1151,\cdot)\) \(\chi_{1444}(1375,\cdot)\)

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: \(\Q(\sqrt{-3}) \)
Fixed field: 6.0.8340544.1

Values on generators

\((723,1085)\) → \((-1,e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\(-1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(-1\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1444 }(1151,a) \;\) at \(\;a = \) e.g. 2