Properties

Label 781.542
Modulus $781$
Conductor $781$
Order $35$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(781, base_ring=CyclotomicField(70))
 
M = H._module
 
chi = DirichletCharacter(H, M([56,10]))
 
pari: [g,chi] = znchar(Mod(542,781))
 

Basic properties

Modulus: \(781\)
Conductor: \(781\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(35\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 781.bj

\(\chi_{781}(20,\cdot)\) \(\chi_{781}(37,\cdot)\) \(\chi_{781}(48,\cdot)\) \(\chi_{781}(91,\cdot)\) \(\chi_{781}(103,\cdot)\) \(\chi_{781}(108,\cdot)\) \(\chi_{781}(119,\cdot)\) \(\chi_{781}(174,\cdot)\) \(\chi_{781}(179,\cdot)\) \(\chi_{781}(190,\cdot)\) \(\chi_{781}(245,\cdot)\) \(\chi_{781}(258,\cdot)\) \(\chi_{781}(400,\cdot)\) \(\chi_{781}(456,\cdot)\) \(\chi_{781}(471,\cdot)\) \(\chi_{781}(542,\cdot)\) \(\chi_{781}(588,\cdot)\) \(\chi_{781}(598,\cdot)\) \(\chi_{781}(669,\cdot)\) \(\chi_{781}(676,\cdot)\) \(\chi_{781}(687,\cdot)\) \(\chi_{781}(730,\cdot)\) \(\chi_{781}(740,\cdot)\) \(\chi_{781}(742,\cdot)\)

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 35 polynomial

Values on generators

\((640,78)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{1}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 781 }(542, a) \) \(1\)\(1\)\(e\left(\frac{23}{35}\right)\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{11}{35}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{27}{35}\right)\)\(e\left(\frac{26}{35}\right)\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{8}{35}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{3}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 781 }(542,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 781 }(542,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 781 }(542,·),\chi_{ 781 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 781 }(542,·)) \;\) at \(\; a,b = \) e.g. 1,2