Properties

Label 1960.83
Modulus $1960$
Conductor $1960$
Order $28$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1960, base_ring=CyclotomicField(28))
 
M = H._module
 
chi = DirichletCharacter(H, M([14,14,21,6]))
 
pari: [g,chi] = znchar(Mod(83,1960))
 

Basic properties

Modulus: \(1960\)
Conductor: \(1960\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
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 1960.cp

\(\chi_{1960}(27,\cdot)\) \(\chi_{1960}(83,\cdot)\) \(\chi_{1960}(307,\cdot)\) \(\chi_{1960}(363,\cdot)\) \(\chi_{1960}(643,\cdot)\) \(\chi_{1960}(867,\cdot)\) \(\chi_{1960}(923,\cdot)\) \(\chi_{1960}(1147,\cdot)\) \(\chi_{1960}(1203,\cdot)\) \(\chi_{1960}(1427,\cdot)\) \(\chi_{1960}(1483,\cdot)\) \(\chi_{1960}(1707,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: 28.0.3771654561118105678109014156786321272080955342848000000000000000000000.1

Values on generators

\((1471,981,1177,1081)\) → \((-1,-1,-i,e\left(\frac{3}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 1960 }(83, a) \) \(-1\)\(1\)\(e\left(\frac{13}{28}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{3}{28}\right)\)\(1\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{6}{7}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1960 }(83,a) \;\) at \(\;a = \) e.g. 2