Properties

Label 1620.979
Modulus $1620$
Conductor $1620$
Order $54$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(54))
 
M = H._module
 
chi = DirichletCharacter(H, M([27,16,27]))
 
pari: [g,chi] = znchar(Mod(979,1620))
 

Basic properties

Modulus: \(1620\)
Conductor: \(1620\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
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 1620.bp

\(\chi_{1620}(79,\cdot)\) \(\chi_{1620}(139,\cdot)\) \(\chi_{1620}(259,\cdot)\) \(\chi_{1620}(319,\cdot)\) \(\chi_{1620}(439,\cdot)\) \(\chi_{1620}(499,\cdot)\) \(\chi_{1620}(619,\cdot)\) \(\chi_{1620}(679,\cdot)\) \(\chi_{1620}(799,\cdot)\) \(\chi_{1620}(859,\cdot)\) \(\chi_{1620}(979,\cdot)\) \(\chi_{1620}(1039,\cdot)\) \(\chi_{1620}(1159,\cdot)\) \(\chi_{1620}(1219,\cdot)\) \(\chi_{1620}(1339,\cdot)\) \(\chi_{1620}(1399,\cdot)\) \(\chi_{1620}(1519,\cdot)\) \(\chi_{1620}(1579,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 54 polynomial

Values on generators

\((811,1541,1297)\) → \((-1,e\left(\frac{8}{27}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1620 }(979, a) \) \(-1\)\(1\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{19}{54}\right)\)\(e\left(\frac{47}{54}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{7}{27}\right)\)\(e\left(\frac{26}{27}\right)\)\(e\left(\frac{23}{54}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{19}{27}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1620 }(979,a) \;\) at \(\;a = \) e.g. 2