Properties

Label 239.237
Modulus $239$
Conductor $239$
Order $238$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(239, base_ring=CyclotomicField(238))
 
M = H._module
 
chi = DirichletCharacter(H, M([185]))
 
pari: [g,chi] = znchar(Mod(237,239))
 

Basic properties

Modulus: \(239\)
Conductor: \(239\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(238\)
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 239.h

\(\chi_{239}(7,\cdot)\) \(\chi_{239}(13,\cdot)\) \(\chi_{239}(14,\cdot)\) \(\chi_{239}(19,\cdot)\) \(\chi_{239}(21,\cdot)\) \(\chi_{239}(26,\cdot)\) \(\chi_{239}(35,\cdot)\) \(\chi_{239}(37,\cdot)\) \(\chi_{239}(39,\cdot)\) \(\chi_{239}(41,\cdot)\) \(\chi_{239}(42,\cdot)\) \(\chi_{239}(43,\cdot)\) \(\chi_{239}(46,\cdot)\) \(\chi_{239}(47,\cdot)\) \(\chi_{239}(53,\cdot)\) \(\chi_{239}(56,\cdot)\) \(\chi_{239}(57,\cdot)\) \(\chi_{239}(59,\cdot)\) \(\chi_{239}(63,\cdot)\) \(\chi_{239}(65,\cdot)\) \(\chi_{239}(69,\cdot)\) \(\chi_{239}(70,\cdot)\) \(\chi_{239}(74,\cdot)\) \(\chi_{239}(77,\cdot)\) \(\chi_{239}(78,\cdot)\) \(\chi_{239}(79,\cdot)\) \(\chi_{239}(82,\cdot)\) \(\chi_{239}(84,\cdot)\) \(\chi_{239}(86,\cdot)\) \(\chi_{239}(89,\cdot)\) ...

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: $\Q(\zeta_{119})$
Fixed field: Number field defined by a degree 238 polynomial (not computed)

Values on generators

\(7\) → \(e\left(\frac{185}{238}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 239 }(237, a) \) \(-1\)\(1\)\(e\left(\frac{36}{119}\right)\)\(e\left(\frac{62}{119}\right)\)\(e\left(\frac{72}{119}\right)\)\(e\left(\frac{32}{119}\right)\)\(e\left(\frac{14}{17}\right)\)\(e\left(\frac{185}{238}\right)\)\(e\left(\frac{108}{119}\right)\)\(e\left(\frac{5}{119}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{13}{119}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 239 }(237,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 239 }(237,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 239 }(237,·),\chi_{ 239 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 239 }(237,·)) \;\) at \(\; a,b = \) e.g. 1,2