Properties

Label 1287.47
Modulus $1287$
Conductor $1287$
Order $60$
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(1287, base_ring=CyclotomicField(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,48,15]))
 
pari: [g,chi] = znchar(Mod(47,1287))
 

Basic properties

Modulus: \(1287\)
Conductor: \(1287\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 1287.ea

\(\chi_{1287}(5,\cdot)\) \(\chi_{1287}(47,\cdot)\) \(\chi_{1287}(86,\cdot)\) \(\chi_{1287}(203,\cdot)\) \(\chi_{1287}(317,\cdot)\) \(\chi_{1287}(356,\cdot)\) \(\chi_{1287}(434,\cdot)\) \(\chi_{1287}(515,\cdot)\) \(\chi_{1287}(554,\cdot)\) \(\chi_{1287}(632,\cdot)\) \(\chi_{1287}(707,\cdot)\) \(\chi_{1287}(785,\cdot)\) \(\chi_{1287}(905,\cdot)\) \(\chi_{1287}(983,\cdot)\) \(\chi_{1287}(1136,\cdot)\) \(\chi_{1287}(1175,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((1145,937,496)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{4}{5}\right),i)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(14\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{13}{20}\right)\)\(-1\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{13}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1287 }(47,a) \;\) at \(\;a = \) e.g. 2