Properties

Label 2856.803
Modulus $2856$
Conductor $2856$
Order $12$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2856, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,6,6,10,9]))
 
pari: [g,chi] = znchar(Mod(803,2856))
 

Basic properties

Modulus: \(2856\)
Conductor: \(2856\)
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 2856.eb

\(\chi_{2856}(395,\cdot)\) \(\chi_{2856}(803,\cdot)\) \(\chi_{2856}(1067,\cdot)\) \(\chi_{2856}(1475,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.0.6401594233356076787154812928.1

Values on generators

\((2143,1429,953,409,2689)\) → \((-1,-1,-1,e\left(\frac{5}{6}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 2856 }(803, a) \) \(-1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(-i\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2856 }(803,a) \;\) at \(\;a = \) e.g. 2