Properties

Label 361.18
Modulus $361$
Conductor $361$
Order $38$
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(38))
 
M = H._module
 
chi = DirichletCharacter(H, M([31]))
 
pari: [g,chi] = znchar(Mod(18,361))
 

Basic properties

Modulus: \(361\)
Conductor: \(361\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(38\)
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.h

\(\chi_{361}(18,\cdot)\) \(\chi_{361}(37,\cdot)\) \(\chi_{361}(56,\cdot)\) \(\chi_{361}(75,\cdot)\) \(\chi_{361}(94,\cdot)\) \(\chi_{361}(113,\cdot)\) \(\chi_{361}(132,\cdot)\) \(\chi_{361}(151,\cdot)\) \(\chi_{361}(170,\cdot)\) \(\chi_{361}(189,\cdot)\) \(\chi_{361}(208,\cdot)\) \(\chi_{361}(227,\cdot)\) \(\chi_{361}(246,\cdot)\) \(\chi_{361}(265,\cdot)\) \(\chi_{361}(284,\cdot)\) \(\chi_{361}(303,\cdot)\) \(\chi_{361}(322,\cdot)\) \(\chi_{361}(341,\cdot)\)

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: \(\Q(\zeta_{19})\)
Fixed field: 38.0.2233638411813024816853081773648251688534529753590642239923912316757382599022775822751448518259.1

Values on generators

\(2\) → \(e\left(\frac{31}{38}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 361 }(18, a) \) \(-1\)\(1\)\(e\left(\frac{31}{38}\right)\)\(e\left(\frac{15}{38}\right)\)\(e\left(\frac{12}{19}\right)\)\(e\left(\frac{5}{19}\right)\)\(e\left(\frac{4}{19}\right)\)\(e\left(\frac{7}{19}\right)\)\(e\left(\frac{17}{38}\right)\)\(e\left(\frac{15}{19}\right)\)\(e\left(\frac{3}{38}\right)\)\(e\left(\frac{4}{19}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 361 }(18,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 361 }(18,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 361 }(18,·),\chi_{ 361 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 361 }(18,·)) \;\) at \(\; a,b = \) e.g. 1,2