Properties

Label 2015.997
Modulus $2015$
Conductor $2015$
Order $12$
Real no
Primitive yes
Minimal yes
Parity odd

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([3,8,8]))
 
pari: [g,chi] = znchar(Mod(997,2015))
 

Basic properties

Modulus: \(2015\)
Conductor: \(2015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2015.dm

\(\chi_{2015}(997,\cdot)\) \(\chi_{2015}(1017,\cdot)\) \(\chi_{2015}(1803,\cdot)\) \(\chi_{2015}(1823,\cdot)\)

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

Values on generators

\((807,1861,716)\) → \((i,e\left(\frac{2}{3}\right),e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\(-1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(1\)\(i\)\(-i\)\(e\left(\frac{1}{6}\right)\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{6}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: Number field defined by a degree 12 polynomial