Properties

Label 1339.6
Modulus $1339$
Conductor $1339$
Order $204$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1339, base_ring=CyclotomicField(204))
 
M = H._module
 
chi = DirichletCharacter(H, M([85,166]))
 
pari: [g,chi] = znchar(Mod(6,1339))
 

Basic properties

Modulus: \(1339\)
Conductor: \(1339\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(204\)
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 1339.cd

\(\chi_{1339}(6,\cdot)\) \(\chi_{1339}(11,\cdot)\) \(\chi_{1339}(20,\cdot)\) \(\chi_{1339}(67,\cdot)\) \(\chi_{1339}(84,\cdot)\) \(\chi_{1339}(85,\cdot)\) \(\chi_{1339}(115,\cdot)\) \(\chi_{1339}(123,\cdot)\) \(\chi_{1339}(154,\cdot)\) \(\chi_{1339}(188,\cdot)\) \(\chi_{1339}(241,\cdot)\) \(\chi_{1339}(254,\cdot)\) \(\chi_{1339}(271,\cdot)\) \(\chi_{1339}(280,\cdot)\) \(\chi_{1339}(293,\cdot)\) \(\chi_{1339}(314,\cdot)\) \(\chi_{1339}(344,\cdot)\) \(\chi_{1339}(349,\cdot)\) \(\chi_{1339}(357,\cdot)\) \(\chi_{1339}(362,\cdot)\) \(\chi_{1339}(371,\cdot)\) \(\chi_{1339}(379,\cdot)\) \(\chi_{1339}(383,\cdot)\) \(\chi_{1339}(384,\cdot)\) \(\chi_{1339}(396,\cdot)\) \(\chi_{1339}(405,\cdot)\) \(\chi_{1339}(410,\cdot)\) \(\chi_{1339}(457,\cdot)\) \(\chi_{1339}(474,\cdot)\) \(\chi_{1339}(479,\cdot)\) ...

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: $\Q(\zeta_{204})$
Fixed field: Number field defined by a degree 204 polynomial (not computed)

Values on generators

\((1237,417)\) → \((e\left(\frac{5}{12}\right),e\left(\frac{83}{102}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1339 }(6, a) \) \(1\)\(1\)\(e\left(\frac{15}{68}\right)\)\(e\left(\frac{41}{102}\right)\)\(e\left(\frac{15}{34}\right)\)\(e\left(\frac{115}{204}\right)\)\(e\left(\frac{127}{204}\right)\)\(e\left(\frac{57}{68}\right)\)\(e\left(\frac{45}{68}\right)\)\(e\left(\frac{41}{51}\right)\)\(e\left(\frac{40}{51}\right)\)\(e\left(\frac{113}{204}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1339 }(6,a) \;\) at \(\;a = \) e.g. 2