Properties

Label 538.369
Modulus $538$
Conductor $269$
Order $134$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(134))
 
M = H._module
 
chi = DirichletCharacter(H, M([75]))
 
pari: [g,chi] = znchar(Mod(369,538))
 

Basic properties

Modulus: \(538\)
Conductor: \(269\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(134\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{269}(100,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 538.e

\(\chi_{538}(9,\cdot)\) \(\chi_{538}(11,\cdot)\) \(\chi_{538}(13,\cdot)\) \(\chi_{538}(43,\cdot)\) \(\chi_{538}(45,\cdot)\) \(\chi_{538}(49,\cdot)\) \(\chi_{538}(51,\cdot)\) \(\chi_{538}(55,\cdot)\) \(\chi_{538}(65,\cdot)\) \(\chi_{538}(73,\cdot)\) \(\chi_{538}(79,\cdot)\) \(\chi_{538}(89,\cdot)\) \(\chi_{538}(97,\cdot)\) \(\chi_{538}(103,\cdot)\) \(\chi_{538}(127,\cdot)\) \(\chi_{538}(133,\cdot)\) \(\chi_{538}(149,\cdot)\) \(\chi_{538}(151,\cdot)\) \(\chi_{538}(189,\cdot)\) \(\chi_{538}(191,\cdot)\) \(\chi_{538}(199,\cdot)\) \(\chi_{538}(203,\cdot)\) \(\chi_{538}(207,\cdot)\) \(\chi_{538}(211,\cdot)\) \(\chi_{538}(215,\cdot)\) \(\chi_{538}(217,\cdot)\) \(\chi_{538}(225,\cdot)\) \(\chi_{538}(231,\cdot)\) \(\chi_{538}(233,\cdot)\) \(\chi_{538}(245,\cdot)\) ...

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: $\Q(\zeta_{67})$
Fixed field: Number field defined by a degree 134 polynomial (not computed)

Values on generators

\(271\) → \(e\left(\frac{75}{134}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 538 }(369, a) \) \(1\)\(1\)\(e\left(\frac{1}{134}\right)\)\(e\left(\frac{28}{67}\right)\)\(e\left(\frac{85}{134}\right)\)\(e\left(\frac{1}{67}\right)\)\(e\left(\frac{49}{67}\right)\)\(e\left(\frac{32}{67}\right)\)\(e\left(\frac{57}{134}\right)\)\(e\left(\frac{103}{134}\right)\)\(e\left(\frac{109}{134}\right)\)\(e\left(\frac{43}{67}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 538 }(369,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 538 }(369,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 538 }(369,·),\chi_{ 538 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 538 }(369,·)) \;\) at \(\; a,b = \) e.g. 1,2