Properties

Label 361.13
Modulus $361$
Conductor $361$
Order $342$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(361, base_ring=CyclotomicField(342))
 
M = H._module
 
chi = DirichletCharacter(H, M([329]))
 
pari: [g,chi] = znchar(Mod(13,361))
 

Basic properties

Modulus: \(361\)
Conductor: \(361\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(342\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 361.l

\(\chi_{361}(2,\cdot)\) \(\chi_{361}(3,\cdot)\) \(\chi_{361}(10,\cdot)\) \(\chi_{361}(13,\cdot)\) \(\chi_{361}(14,\cdot)\) \(\chi_{361}(15,\cdot)\) \(\chi_{361}(21,\cdot)\) \(\chi_{361}(22,\cdot)\) \(\chi_{361}(29,\cdot)\) \(\chi_{361}(32,\cdot)\) \(\chi_{361}(33,\cdot)\) \(\chi_{361}(34,\cdot)\) \(\chi_{361}(40,\cdot)\) \(\chi_{361}(41,\cdot)\) \(\chi_{361}(48,\cdot)\) \(\chi_{361}(51,\cdot)\) \(\chi_{361}(52,\cdot)\) \(\chi_{361}(53,\cdot)\) \(\chi_{361}(59,\cdot)\) \(\chi_{361}(60,\cdot)\) \(\chi_{361}(67,\cdot)\) \(\chi_{361}(70,\cdot)\) \(\chi_{361}(71,\cdot)\) \(\chi_{361}(72,\cdot)\) \(\chi_{361}(78,\cdot)\) \(\chi_{361}(79,\cdot)\) \(\chi_{361}(86,\cdot)\) \(\chi_{361}(89,\cdot)\) \(\chi_{361}(90,\cdot)\) \(\chi_{361}(91,\cdot)\) ...

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: $\Q(\zeta_{171})$
Fixed field: Number field defined by a degree 342 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{329}{342}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 361 }(13, a) \) \(-1\)\(1\)\(e\left(\frac{329}{342}\right)\)\(e\left(\frac{245}{342}\right)\)\(e\left(\frac{158}{171}\right)\)\(e\left(\frac{31}{171}\right)\)\(e\left(\frac{116}{171}\right)\)\(e\left(\frac{17}{57}\right)\)\(e\left(\frac{101}{114}\right)\)\(e\left(\frac{74}{171}\right)\)\(e\left(\frac{49}{342}\right)\)\(e\left(\frac{7}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 361 }(13,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 361 }(13,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 361 }(13,·),\chi_{ 361 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 361 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2