Properties

Label 1620.31
Modulus $1620$
Conductor $324$
Order $54$
Real no
Primitive no
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([27,20,0]))
 
pari: [g,chi] = znchar(Mod(31,1620))
 

Basic properties

Modulus: \(1620\)
Conductor: \(324\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{324}(31,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1620.bn

\(\chi_{1620}(31,\cdot)\) \(\chi_{1620}(151,\cdot)\) \(\chi_{1620}(211,\cdot)\) \(\chi_{1620}(331,\cdot)\) \(\chi_{1620}(391,\cdot)\) \(\chi_{1620}(511,\cdot)\) \(\chi_{1620}(571,\cdot)\) \(\chi_{1620}(691,\cdot)\) \(\chi_{1620}(751,\cdot)\) \(\chi_{1620}(871,\cdot)\) \(\chi_{1620}(931,\cdot)\) \(\chi_{1620}(1051,\cdot)\) \(\chi_{1620}(1111,\cdot)\) \(\chi_{1620}(1231,\cdot)\) \(\chi_{1620}(1291,\cdot)\) \(\chi_{1620}(1411,\cdot)\) \(\chi_{1620}(1471,\cdot)\) \(\chi_{1620}(1591,\cdot)\)

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

\((811,1541,1297)\) → \((-1,e\left(\frac{10}{27}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1620 }(31, a) \) \(-1\)\(1\)\(e\left(\frac{23}{54}\right)\)\(e\left(\frac{17}{54}\right)\)\(e\left(\frac{26}{27}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{31}{54}\right)\)\(e\left(\frac{19}{27}\right)\)\(e\left(\frac{49}{54}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{17}{27}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1620 }(31,a) \;\) at \(\;a = \) e.g. 2