Properties

Label 1148.677
Modulus $1148$
Conductor $287$
Order $60$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1148, base_ring=CyclotomicField(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,50,21]))
 
pari: [g,chi] = znchar(Mod(677,1148))
 

Basic properties

Modulus: \(1148\)
Conductor: \(287\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{287}(103,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1148.ch

\(\chi_{1148}(5,\cdot)\) \(\chi_{1148}(33,\cdot)\) \(\chi_{1148}(61,\cdot)\) \(\chi_{1148}(185,\cdot)\) \(\chi_{1148}(213,\cdot)\) \(\chi_{1148}(241,\cdot)\) \(\chi_{1148}(285,\cdot)\) \(\chi_{1148}(453,\cdot)\) \(\chi_{1148}(661,\cdot)\) \(\chi_{1148}(677,\cdot)\) \(\chi_{1148}(689,\cdot)\) \(\chi_{1148}(705,\cdot)\) \(\chi_{1148}(717,\cdot)\) \(\chi_{1148}(733,\cdot)\) \(\chi_{1148}(941,\cdot)\) \(\chi_{1148}(1109,\cdot)\)

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

\((575,493,785)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{7}{20}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(-1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{11}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1148 }(677,a) \;\) at \(\;a = \) e.g. 2