Properties

Label 3240.59
Modulus $3240$
Conductor $3240$
Order $54$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3240, base_ring=CyclotomicField(54))
 
M = H._module
 
chi = DirichletCharacter(H, M([27,27,41,27]))
 
pari: [g,chi] = znchar(Mod(59,3240))
 

Basic properties

Modulus: \(3240\)
Conductor: \(3240\)
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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3240.de

\(\chi_{3240}(59,\cdot)\) \(\chi_{3240}(299,\cdot)\) \(\chi_{3240}(419,\cdot)\) \(\chi_{3240}(659,\cdot)\) \(\chi_{3240}(779,\cdot)\) \(\chi_{3240}(1019,\cdot)\) \(\chi_{3240}(1139,\cdot)\) \(\chi_{3240}(1379,\cdot)\) \(\chi_{3240}(1499,\cdot)\) \(\chi_{3240}(1739,\cdot)\) \(\chi_{3240}(1859,\cdot)\) \(\chi_{3240}(2099,\cdot)\) \(\chi_{3240}(2219,\cdot)\) \(\chi_{3240}(2459,\cdot)\) \(\chi_{3240}(2579,\cdot)\) \(\chi_{3240}(2819,\cdot)\) \(\chi_{3240}(2939,\cdot)\) \(\chi_{3240}(3179,\cdot)\)

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

\((2431,1621,3161,1297)\) → \((-1,-1,e\left(\frac{41}{54}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3240 }(59, a) \) \(1\)\(1\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{47}{54}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{19}{54}\right)\)\(e\left(\frac{16}{27}\right)\)\(e\left(\frac{37}{54}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{13}{54}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3240 }(59,a) \;\) at \(\;a = \) e.g. 2