Properties

Label 2015.463
Modulus $2015$
Conductor $2015$
Order $20$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2015, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,5,6]))
 
pari: [g,chi] = znchar(Mod(463,2015))
 

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.ei

\(\chi_{2015}(122,\cdot)\) \(\chi_{2015}(333,\cdot)\) \(\chi_{2015}(463,\cdot)\) \(\chi_{2015}(1162,\cdot)\) \(\chi_{2015}(1422,\cdot)\) \(\chi_{2015}(1503,\cdot)\) \(\chi_{2015}(1763,\cdot)\) \(\chi_{2015}(2007,\cdot)\)

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

Related number fields

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

Values on generators

\((807,1861,716)\) → \((-i,i,e\left(\frac{3}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\( \chi_{ 2015 }(463, a) \) \(-1\)\(1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(-i\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{1}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2015 }(463,a) \;\) at \(\;a = \) e.g. 2