Properties

Label 2015.109
Modulus $2015$
Conductor $2015$
Order $20$
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([10,15,4]))
 
pari: [g,chi] = znchar(Mod(109,2015))
 

Basic properties

Modulus: \(2015\)
Conductor: \(2015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
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.eq

\(\chi_{2015}(109,\cdot)\) \(\chi_{2015}(684,\cdot)\) \(\chi_{2015}(814,\cdot)\) \(\chi_{2015}(1149,\cdot)\) \(\chi_{2015}(1279,\cdot)\) \(\chi_{2015}(1399,\cdot)\) \(\chi_{2015}(1659,\cdot)\) \(\chi_{2015}(1864,\cdot)\)

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

\((807,1861,716)\) → \((-1,-i,e\left(\frac{1}{5}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\(-1\)\(1\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(-i\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{2}{5}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{20})\)
Fixed field: Number field defined by a degree 20 polynomial