Properties

Label 349.139
Modulus $349$
Conductor $349$
Order $58$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(349, base_ring=CyclotomicField(58))
 
M = H._module
 
chi = DirichletCharacter(H, M([39]))
 
pari: [g,chi] = znchar(Mod(139,349))
 

Basic properties

Modulus: \(349\)
Conductor: \(349\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(58\)
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 349.h

\(\chi_{349}(17,\cdot)\) \(\chi_{349}(27,\cdot)\) \(\chi_{349}(36,\cdot)\) \(\chi_{349}(37,\cdot)\) \(\chi_{349}(45,\cdot)\) \(\chi_{349}(48,\cdot)\) \(\chi_{349}(60,\cdot)\) \(\chi_{349}(64,\cdot)\) \(\chi_{349}(69,\cdot)\) \(\chi_{349}(75,\cdot)\) \(\chi_{349}(80,\cdot)\) \(\chi_{349}(86,\cdot)\) \(\chi_{349}(92,\cdot)\) \(\chi_{349}(100,\cdot)\) \(\chi_{349}(115,\cdot)\) \(\chi_{349}(121,\cdot)\) \(\chi_{349}(125,\cdot)\) \(\chi_{349}(139,\cdot)\) \(\chi_{349}(178,\cdot)\) \(\chi_{349}(181,\cdot)\) \(\chi_{349}(223,\cdot)\) \(\chi_{349}(231,\cdot)\) \(\chi_{349}(239,\cdot)\) \(\chi_{349}(261,\cdot)\) \(\chi_{349}(282,\cdot)\) \(\chi_{349}(283,\cdot)\) \(\chi_{349}(308,\cdot)\) \(\chi_{349}(318,\cdot)\)

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: $\Q(\zeta_{29})$
Fixed field: Number field defined by a degree 58 polynomial

Values on generators

\(2\) → \(e\left(\frac{39}{58}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 349 }(139, a) \) \(1\)\(1\)\(e\left(\frac{39}{58}\right)\)\(e\left(\frac{14}{29}\right)\)\(e\left(\frac{10}{29}\right)\)\(e\left(\frac{4}{29}\right)\)\(e\left(\frac{9}{58}\right)\)\(e\left(\frac{3}{58}\right)\)\(e\left(\frac{1}{58}\right)\)\(e\left(\frac{28}{29}\right)\)\(e\left(\frac{47}{58}\right)\)\(e\left(\frac{7}{58}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 349 }(139,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 349 }(139,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 349 }(139,·),\chi_{ 349 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 349 }(139,·)) \;\) at \(\; a,b = \) e.g. 1,2