Properties

Label 619.29
Modulus $619$
Conductor $619$
Order $206$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(619, base_ring=CyclotomicField(206))
 
M = H._module
 
chi = DirichletCharacter(H, M([101]))
 
pari: [g,chi] = znchar(Mod(29,619))
 

Basic properties

Modulus: \(619\)
Conductor: \(619\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(206\)
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 619.f

\(\chi_{619}(3,\cdot)\) \(\chi_{619}(8,\cdot)\) \(\chi_{619}(13,\cdot)\) \(\chi_{619}(14,\cdot)\) \(\chi_{619}(17,\cdot)\) \(\chi_{619}(27,\cdot)\) \(\chi_{619}(29,\cdot)\) \(\chi_{619}(43,\cdot)\) \(\chi_{619}(44,\cdot)\) \(\chi_{619}(47,\cdot)\) \(\chi_{619}(50,\cdot)\) \(\chi_{619}(60,\cdot)\) \(\chi_{619}(72,\cdot)\) \(\chi_{619}(74,\cdot)\) \(\chi_{619}(77,\cdot)\) \(\chi_{619}(93,\cdot)\) \(\chi_{619}(95,\cdot)\) \(\chi_{619}(105,\cdot)\) \(\chi_{619}(113,\cdot)\) \(\chi_{619}(114,\cdot)\) \(\chi_{619}(117,\cdot)\) \(\chi_{619}(126,\cdot)\) \(\chi_{619}(139,\cdot)\) \(\chi_{619}(153,\cdot)\) \(\chi_{619}(160,\cdot)\) \(\chi_{619}(192,\cdot)\) \(\chi_{619}(202,\cdot)\) \(\chi_{619}(213,\cdot)\) \(\chi_{619}(218,\cdot)\) \(\chi_{619}(219,\cdot)\) ...

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: $\Q(\zeta_{103})$
Fixed field: Number field defined by a degree 206 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{101}{206}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 619 }(29, a) \) \(-1\)\(1\)\(e\left(\frac{101}{206}\right)\)\(e\left(\frac{159}{206}\right)\)\(e\left(\frac{101}{103}\right)\)\(e\left(\frac{84}{103}\right)\)\(e\left(\frac{27}{103}\right)\)\(e\left(\frac{49}{103}\right)\)\(e\left(\frac{97}{206}\right)\)\(e\left(\frac{56}{103}\right)\)\(e\left(\frac{63}{206}\right)\)\(e\left(\frac{67}{206}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 619 }(29,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 619 }(29,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 619 }(29,·),\chi_{ 619 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 619 }(29,·)) \;\) at \(\; a,b = \) e.g. 1,2