Properties

Label 1148.59
Modulus $1148$
Conductor $1148$
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(1148, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([15,5,12]))
 
pari: [g,chi] = znchar(Mod(59,1148))
 

Basic properties

Modulus: \(1148\)
Conductor: \(1148\)
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 1148.bu

\(\chi_{1148}(59,\cdot)\) \(\chi_{1148}(215,\cdot)\) \(\chi_{1148}(283,\cdot)\) \(\chi_{1148}(467,\cdot)\) \(\chi_{1148}(551,\cdot)\) \(\chi_{1148}(775,\cdot)\) \(\chi_{1148}(871,\cdot)\) \(\chi_{1148}(1123,\cdot)\)

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

\((575,493,785)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{2}{5}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{4}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1148 }(59,a) \;\) at \(\;a = \) e.g. 2