Properties

Label 531.34
Modulus $531$
Conductor $531$
Order $174$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, base_ring=CyclotomicField(174))
 
M = H._module
 
chi = DirichletCharacter(H, M([116,123]))
 
pari: [g,chi] = znchar(Mod(34,531))
 

Basic properties

Modulus: \(531\)
Conductor: \(531\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(174\)
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 531.o

\(\chi_{531}(13,\cdot)\) \(\chi_{531}(31,\cdot)\) \(\chi_{531}(34,\cdot)\) \(\chi_{531}(40,\cdot)\) \(\chi_{531}(43,\cdot)\) \(\chi_{531}(52,\cdot)\) \(\chi_{531}(61,\cdot)\) \(\chi_{531}(67,\cdot)\) \(\chi_{531}(70,\cdot)\) \(\chi_{531}(97,\cdot)\) \(\chi_{531}(103,\cdot)\) \(\chi_{531}(106,\cdot)\) \(\chi_{531}(115,\cdot)\) \(\chi_{531}(124,\cdot)\) \(\chi_{531}(142,\cdot)\) \(\chi_{531}(148,\cdot)\) \(\chi_{531}(151,\cdot)\) \(\chi_{531}(157,\cdot)\) \(\chi_{531}(160,\cdot)\) \(\chi_{531}(187,\cdot)\) \(\chi_{531}(211,\cdot)\) \(\chi_{531}(214,\cdot)\) \(\chi_{531}(220,\cdot)\) \(\chi_{531}(229,\cdot)\) \(\chi_{531}(232,\cdot)\) \(\chi_{531}(238,\cdot)\) \(\chi_{531}(247,\cdot)\) \(\chi_{531}(250,\cdot)\) \(\chi_{531}(259,\cdot)\) \(\chi_{531}(268,\cdot)\) ...

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: $\Q(\zeta_{87})$
Fixed field: Number field defined by a degree 174 polynomial (not computed)

Values on generators

\((119,415)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{41}{58}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 531 }(34, a) \) \(-1\)\(1\)\(e\left(\frac{65}{174}\right)\)\(e\left(\frac{65}{87}\right)\)\(e\left(\frac{50}{87}\right)\)\(e\left(\frac{34}{87}\right)\)\(e\left(\frac{7}{58}\right)\)\(e\left(\frac{55}{58}\right)\)\(e\left(\frac{59}{174}\right)\)\(e\left(\frac{25}{174}\right)\)\(e\left(\frac{133}{174}\right)\)\(e\left(\frac{43}{87}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 531 }(34,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 531 }(34,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 531 }(34,·),\chi_{ 531 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 531 }(34,·)) \;\) at \(\; a,b = \) e.g. 1,2