Properties

Label 71.42
Modulus $71$
Conductor $71$
Order $70$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(71, base_ring=CyclotomicField(70))
 
M = H._module
 
chi = DirichletCharacter(H, M([33]))
 
pari: [g,chi] = znchar(Mod(42,71))
 

Basic properties

Modulus: \(71\)
Conductor: \(71\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(70\)
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 71.h

\(\chi_{71}(7,\cdot)\) \(\chi_{71}(11,\cdot)\) \(\chi_{71}(13,\cdot)\) \(\chi_{71}(21,\cdot)\) \(\chi_{71}(22,\cdot)\) \(\chi_{71}(28,\cdot)\) \(\chi_{71}(31,\cdot)\) \(\chi_{71}(33,\cdot)\) \(\chi_{71}(35,\cdot)\) \(\chi_{71}(42,\cdot)\) \(\chi_{71}(44,\cdot)\) \(\chi_{71}(47,\cdot)\) \(\chi_{71}(52,\cdot)\) \(\chi_{71}(53,\cdot)\) \(\chi_{71}(55,\cdot)\) \(\chi_{71}(56,\cdot)\) \(\chi_{71}(59,\cdot)\) \(\chi_{71}(61,\cdot)\) \(\chi_{71}(62,\cdot)\) \(\chi_{71}(63,\cdot)\) \(\chi_{71}(65,\cdot)\) \(\chi_{71}(67,\cdot)\) \(\chi_{71}(68,\cdot)\) \(\chi_{71}(69,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\(7\) → \(e\left(\frac{33}{70}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 71 }(42, a) \) \(-1\)\(1\)\(e\left(\frac{29}{35}\right)\)\(e\left(\frac{9}{35}\right)\)\(e\left(\frac{23}{35}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{35}\right)\)\(e\left(\frac{33}{70}\right)\)\(e\left(\frac{17}{35}\right)\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{1}{35}\right)\)\(e\left(\frac{43}{70}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 71 }(42,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 71 }(42,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 71 }(42,·),\chi_{ 71 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 71 }(42,·)) \;\) at \(\; a,b = \) e.g. 1,2