Properties

Label 1339.4
Modulus $1339$
Conductor $1339$
Order $102$
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(102))
 
M = H._module
 
chi = DirichletCharacter(H, M([17,88]))
 
pari: [g,chi] = znchar(Mod(4,1339))
 

Basic properties

Modulus: \(1339\)
Conductor: \(1339\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(102\)
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.bs

\(\chi_{1339}(4,\cdot)\) \(\chi_{1339}(17,\cdot)\) \(\chi_{1339}(49,\cdot)\) \(\chi_{1339}(82,\cdot)\) \(\chi_{1339}(231,\cdot)\) \(\chi_{1339}(238,\cdot)\) \(\chi_{1339}(264,\cdot)\) \(\chi_{1339}(316,\cdot)\) \(\chi_{1339}(335,\cdot)\) \(\chi_{1339}(342,\cdot)\) \(\chi_{1339}(407,\cdot)\) \(\chi_{1339}(517,\cdot)\) \(\chi_{1339}(543,\cdot)\) \(\chi_{1339}(556,\cdot)\) \(\chi_{1339}(634,\cdot)\) \(\chi_{1339}(647,\cdot)\) \(\chi_{1339}(654,\cdot)\) \(\chi_{1339}(673,\cdot)\) \(\chi_{1339}(686,\cdot)\) \(\chi_{1339}(771,\cdot)\) \(\chi_{1339}(784,\cdot)\) \(\chi_{1339}(842,\cdot)\) \(\chi_{1339}(862,\cdot)\) \(\chi_{1339}(907,\cdot)\) \(\chi_{1339}(946,\cdot)\) \(\chi_{1339}(979,\cdot)\) \(\chi_{1339}(1018,\cdot)\) \(\chi_{1339}(1024,\cdot)\) \(\chi_{1339}(1089,\cdot)\) \(\chi_{1339}(1122,\cdot)\) ...

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: $\Q(\zeta_{51})$
Fixed field: Number field defined by a degree 102 polynomial (not computed)

Values on generators

\((1237,417)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{44}{51}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1339 }(4, a) \) \(1\)\(1\)\(e\left(\frac{13}{102}\right)\)\(e\left(\frac{16}{51}\right)\)\(e\left(\frac{13}{51}\right)\)\(e\left(\frac{37}{102}\right)\)\(e\left(\frac{15}{34}\right)\)\(e\left(\frac{29}{102}\right)\)\(e\left(\frac{13}{34}\right)\)\(e\left(\frac{32}{51}\right)\)\(e\left(\frac{25}{51}\right)\)\(e\left(\frac{27}{34}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1339 }(4,a) \;\) at \(\;a = \) e.g. 2