Properties

Label 3344.27
Modulus $3344$
Conductor $3344$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,15,24,10]))
 
pari: [g,chi] = znchar(Mod(27,3344))
 

Basic properties

Modulus: \(3344\)
Conductor: \(3344\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 3344.em

\(\chi_{3344}(27,\cdot)\) \(\chi_{3344}(179,\cdot)\) \(\chi_{3344}(411,\cdot)\) \(\chi_{3344}(867,\cdot)\) \(\chi_{3344}(939,\cdot)\) \(\chi_{3344}(1171,\cdot)\) \(\chi_{3344}(1323,\cdot)\) \(\chi_{3344}(1395,\cdot)\) \(\chi_{3344}(1699,\cdot)\) \(\chi_{3344}(1851,\cdot)\) \(\chi_{3344}(2083,\cdot)\) \(\chi_{3344}(2539,\cdot)\) \(\chi_{3344}(2611,\cdot)\) \(\chi_{3344}(2843,\cdot)\) \(\chi_{3344}(2995,\cdot)\) \(\chi_{3344}(3067,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((2927,837,2433,705)\) → \((-1,i,e\left(\frac{2}{5}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(21\)\(23\)\(25\)
\( \chi_{ 3344 }(27, a) \) \(1\)\(1\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3344 }(27,a) \;\) at \(\;a = \) e.g. 2