Properties

Label 1444.1111
Modulus $1444$
Conductor $76$
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(1444, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9,8]))
 
pari: [g,chi] = znchar(Mod(1111,1444))
 

Basic properties

Modulus: \(1444\)
Conductor: \(76\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{76}(47,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1444.l

\(\chi_{1444}(99,\cdot)\) \(\chi_{1444}(415,\cdot)\) \(\chi_{1444}(423,\cdot)\) \(\chi_{1444}(595,\cdot)\) \(\chi_{1444}(967,\cdot)\) \(\chi_{1444}(1111,\cdot)\)

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: 18.0.75613185918270483380568064.1

Values on generators

\((723,1085)\) → \((-1,e\left(\frac{4}{9}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\(-1\)\(1\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{7}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1444 }(1111,a) \;\) at \(\;a = \) e.g. 2