Properties

Modulus 2015
Conductor 2015
Order 12
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 2015.ea

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2015)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3,7,10]))
 
pari: [g,chi] = znchar(Mod(37,2015))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 2015
Conductor = 2015
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 12
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 2015.ea
Orbit index = 105

Galois orbit

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{2015}(37,\cdot)\) \(\chi_{2015}(223,\cdot)\) \(\chi_{2015}(708,\cdot)\) \(\chi_{2015}(1762,\cdot)\)

Values on generators

\((807,1861,716)\) → \((i,e\left(\frac{7}{12}\right),e\left(\frac{5}{6}\right))\)

Values

-112346789111214
\(-1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(-i\)\(1\)\(-1\)\(e\left(\frac{5}{6}\right)\)\(i\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{6}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{12})\)