Properties

Label 1287.49
Modulus $1287$
Conductor $1287$
Order $30$
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(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,12,25]))
 
pari: [g,chi] = znchar(Mod(49,1287))
 

Basic properties

Modulus: \(1287\)
Conductor: \(1287\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
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.dg

\(\chi_{1287}(49,\cdot)\) \(\chi_{1287}(394,\cdot)\) \(\chi_{1287}(400,\cdot)\) \(\chi_{1287}(511,\cdot)\) \(\chi_{1287}(751,\cdot)\) \(\chi_{1287}(862,\cdot)\) \(\chi_{1287}(1213,\cdot)\) \(\chi_{1287}(1219,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Values on generators

\((1145,937,496)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{2}{5}\right),e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(14\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{11}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1287 }(49,a) \;\) at \(\;a = \) e.g. 2