Properties

Label 2015.38
Modulus $2015$
Conductor $2015$
Order $60$
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([45,30,56]))
 
pari: [g,chi] = znchar(Mod(38,2015))
 

Basic properties

Modulus: \(2015\)
Conductor: \(2015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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.hb

\(\chi_{2015}(38,\cdot)\) \(\chi_{2015}(103,\cdot)\) \(\chi_{2015}(142,\cdot)\) \(\chi_{2015}(298,\cdot)\) \(\chi_{2015}(493,\cdot)\) \(\chi_{2015}(727,\cdot)\) \(\chi_{2015}(753,\cdot)\) \(\chi_{2015}(857,\cdot)\) \(\chi_{2015}(948,\cdot)\) \(\chi_{2015}(1247,\cdot)\) \(\chi_{2015}(1312,\cdot)\) \(\chi_{2015}(1507,\cdot)\) \(\chi_{2015}(1533,\cdot)\) \(\chi_{2015}(1663,\cdot)\) \(\chi_{2015}(1702,\cdot)\) \(\chi_{2015}(1962,\cdot)\)

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

\((807,1861,716)\) → \((-i,-1,e\left(\frac{14}{15}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\(-1\)\(1\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{1}{30}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial