Properties

Label 2028.131
Modulus $2028$
Conductor $2028$
Order $26$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2028, base_ring=CyclotomicField(26))
 
M = H._module
 
chi = DirichletCharacter(H, M([13,13,24]))
 
pari: [g,chi] = znchar(Mod(131,2028))
 

Basic properties

Modulus: \(2028\)
Conductor: \(2028\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(26\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2028.be

\(\chi_{2028}(131,\cdot)\) \(\chi_{2028}(287,\cdot)\) \(\chi_{2028}(443,\cdot)\) \(\chi_{2028}(599,\cdot)\) \(\chi_{2028}(755,\cdot)\) \(\chi_{2028}(911,\cdot)\) \(\chi_{2028}(1067,\cdot)\) \(\chi_{2028}(1223,\cdot)\) \(\chi_{2028}(1379,\cdot)\) \(\chi_{2028}(1535,\cdot)\) \(\chi_{2028}(1847,\cdot)\) \(\chi_{2028}(2003,\cdot)\)

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: \(\Q(\zeta_{13})\)
Fixed field: Number field defined by a degree 26 polynomial

Values on generators

\((1015,677,1861)\) → \((-1,-1,e\left(\frac{12}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 2028 }(131, a) \) \(1\)\(1\)\(e\left(\frac{21}{26}\right)\)\(e\left(\frac{7}{26}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{7}{26}\right)\)\(-1\)\(1\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{11}{26}\right)\)\(e\left(\frac{23}{26}\right)\)\(e\left(\frac{1}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2028 }(131,a) \;\) at \(\;a = \) e.g. 2