Properties

Label 349.12
Modulus $349$
Conductor $349$
Order $87$
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([14]))
 
pari: [g,chi] = znchar(Mod(12,349))
 

Basic properties

Modulus: \(349\)
Conductor: \(349\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(87\)
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.i

\(\chi_{349}(9,\cdot)\) \(\chi_{349}(12,\cdot)\) \(\chi_{349}(14,\cdot)\) \(\chi_{349}(15,\cdot)\) \(\chi_{349}(16,\cdot)\) \(\chi_{349}(19,\cdot)\) \(\chi_{349}(20,\cdot)\) \(\chi_{349}(23,\cdot)\) \(\chi_{349}(25,\cdot)\) \(\chi_{349}(26,\cdot)\) \(\chi_{349}(51,\cdot)\) \(\chi_{349}(68,\cdot)\) \(\chi_{349}(77,\cdot)\) \(\chi_{349}(81,\cdot)\) \(\chi_{349}(85,\cdot)\) \(\chi_{349}(87,\cdot)\) \(\chi_{349}(94,\cdot)\) \(\chi_{349}(106,\cdot)\) \(\chi_{349}(108,\cdot)\) \(\chi_{349}(111,\cdot)\) \(\chi_{349}(116,\cdot)\) \(\chi_{349}(135,\cdot)\) \(\chi_{349}(143,\cdot)\) \(\chi_{349}(144,\cdot)\) \(\chi_{349}(145,\cdot)\) \(\chi_{349}(147,\cdot)\) \(\chi_{349}(148,\cdot)\) \(\chi_{349}(151,\cdot)\) \(\chi_{349}(158,\cdot)\) \(\chi_{349}(180,\cdot)\) ...

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 87 polynomial

Values on generators

\(2\) → \(e\left(\frac{7}{87}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 349 }(12, a) \) \(1\)\(1\)\(e\left(\frac{7}{87}\right)\)\(e\left(\frac{8}{87}\right)\)\(e\left(\frac{14}{87}\right)\)\(e\left(\frac{23}{87}\right)\)\(e\left(\frac{5}{29}\right)\)\(e\left(\frac{34}{87}\right)\)\(e\left(\frac{7}{29}\right)\)\(e\left(\frac{16}{87}\right)\)\(e\left(\frac{10}{29}\right)\)\(e\left(\frac{20}{29}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 349 }(12,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 349 }(12,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 349 }(12,·),\chi_{ 349 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 349 }(12,·)) \;\) at \(\; a,b = \) e.g. 1,2