Properties

Label 1520.429
Modulus $1520$
Conductor $1520$
Order $12$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1520, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,9,6,8]))
 
pari: [g,chi] = znchar(Mod(429,1520))
 

Basic properties

Modulus: \(1520\)
Conductor: \(1520\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
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 1520.cy

\(\chi_{1520}(349,\cdot)\) \(\chi_{1520}(429,\cdot)\) \(\chi_{1520}(1109,\cdot)\) \(\chi_{1520}(1189,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.12.2279495244707790848000000.1

Values on generators

\((191,1141,1217,401)\) → \((1,-i,-1,e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(1\)\(e\left(\frac{5}{6}\right)\)\(-i\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(i\)\(e\left(\frac{7}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1520 }(429,a) \;\) at \(\;a = \) e.g. 2