Properties

Label 1480.349
Modulus $1480$
Conductor $1480$
Order $18$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1480, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,9,9,2]))
 
pari: [g,chi] = znchar(Mod(349,1480))
 

Basic properties

Modulus: \(1480\)
Conductor: \(1480\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1480.dp

\(\chi_{1480}(229,\cdot)\) \(\chi_{1480}(349,\cdot)\) \(\chi_{1480}(589,\cdot)\) \(\chi_{1480}(749,\cdot)\) \(\chi_{1480}(789,\cdot)\) \(\chi_{1480}(1069,\cdot)\)

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: Number field defined by a degree 18 polynomial

Values on generators

\((1111,741,297,1001)\) → \((1,-1,-1,e\left(\frac{1}{9}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1480 }(349, a) \) \(1\)\(1\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1480 }(349,a) \;\) at \(\;a = \) e.g. 2