Properties

Label 349.304
Modulus $349$
Conductor $349$
Order $29$
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([32]))
 
pari: [g,chi] = znchar(Mod(304,349))
 

Basic properties

Modulus: \(349\)
Conductor: \(349\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(29\)
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.g

\(\chi_{349}(31,\cdot)\) \(\chi_{349}(41,\cdot)\) \(\chi_{349}(66,\cdot)\) \(\chi_{349}(67,\cdot)\) \(\chi_{349}(88,\cdot)\) \(\chi_{349}(110,\cdot)\) \(\chi_{349}(118,\cdot)\) \(\chi_{349}(126,\cdot)\) \(\chi_{349}(168,\cdot)\) \(\chi_{349}(171,\cdot)\) \(\chi_{349}(210,\cdot)\) \(\chi_{349}(224,\cdot)\) \(\chi_{349}(228,\cdot)\) \(\chi_{349}(234,\cdot)\) \(\chi_{349}(249,\cdot)\) \(\chi_{349}(257,\cdot)\) \(\chi_{349}(263,\cdot)\) \(\chi_{349}(269,\cdot)\) \(\chi_{349}(274,\cdot)\) \(\chi_{349}(280,\cdot)\) \(\chi_{349}(285,\cdot)\) \(\chi_{349}(289,\cdot)\) \(\chi_{349}(301,\cdot)\) \(\chi_{349}(304,\cdot)\) \(\chi_{349}(312,\cdot)\) \(\chi_{349}(313,\cdot)\) \(\chi_{349}(322,\cdot)\) \(\chi_{349}(332,\cdot)\)

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 29 polynomial

Values on generators

\(2\) → \(e\left(\frac{16}{29}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 349 }(304, a) \) \(1\)\(1\)\(e\left(\frac{16}{29}\right)\)\(e\left(\frac{10}{29}\right)\)\(e\left(\frac{3}{29}\right)\)\(e\left(\frac{7}{29}\right)\)\(e\left(\frac{26}{29}\right)\)\(e\left(\frac{28}{29}\right)\)\(e\left(\frac{19}{29}\right)\)\(e\left(\frac{20}{29}\right)\)\(e\left(\frac{23}{29}\right)\)\(e\left(\frac{17}{29}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 349 }(304,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 349 }(304,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 349 }(304,·),\chi_{ 349 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 349 }(304,·)) \;\) at \(\; a,b = \) e.g. 1,2