Properties

Label 349.169
Modulus $349$
Conductor $349$
Order $174$
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(174))
 
M = H._module
 
chi = DirichletCharacter(H, M([97]))
 
pari: [g,chi] = znchar(Mod(169,349))
 

Basic properties

Modulus: \(349\)
Conductor: \(349\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(174\)
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.k

\(\chi_{349}(3,\cdot)\) \(\chi_{349}(4,\cdot)\) \(\chi_{349}(5,\cdot)\) \(\chi_{349}(22,\cdot)\) \(\chi_{349}(29,\cdot)\) \(\chi_{349}(42,\cdot)\) \(\chi_{349}(49,\cdot)\) \(\chi_{349}(56,\cdot)\) \(\chi_{349}(57,\cdot)\) \(\chi_{349}(70,\cdot)\) \(\chi_{349}(73,\cdot)\) \(\chi_{349}(76,\cdot)\) \(\chi_{349}(78,\cdot)\) \(\chi_{349}(83,\cdot)\) \(\chi_{349}(91,\cdot)\) \(\chi_{349}(93,\cdot)\) \(\chi_{349}(95,\cdot)\) \(\chi_{349}(104,\cdot)\) \(\chi_{349}(109,\cdot)\) \(\chi_{349}(124,\cdot)\) \(\chi_{349}(130,\cdot)\) \(\chi_{349}(142,\cdot)\) \(\chi_{349}(153,\cdot)\) \(\chi_{349}(155,\cdot)\) \(\chi_{349}(157,\cdot)\) \(\chi_{349}(164,\cdot)\) \(\chi_{349}(169,\cdot)\) \(\chi_{349}(191,\cdot)\) \(\chi_{349}(198,\cdot)\) \(\chi_{349}(201,\cdot)\) ...

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: $\Q(\zeta_{87})$
Fixed field: Number field defined by a degree 174 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{97}{174}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 349 }(169, a) \) \(1\)\(1\)\(e\left(\frac{97}{174}\right)\)\(e\left(\frac{43}{87}\right)\)\(e\left(\frac{10}{87}\right)\)\(e\left(\frac{4}{87}\right)\)\(e\left(\frac{3}{58}\right)\)\(e\left(\frac{61}{174}\right)\)\(e\left(\frac{39}{58}\right)\)\(e\left(\frac{86}{87}\right)\)\(e\left(\frac{35}{58}\right)\)\(e\left(\frac{41}{58}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 349 }(169,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 349 }(169,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 349 }(169,·),\chi_{ 349 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 349 }(169,·)) \;\) at \(\; a,b = \) e.g. 1,2