Properties

Label 463.78
Modulus $463$
Conductor $463$
Order $77$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(463, base_ring=CyclotomicField(154))
 
M = H._module
 
chi = DirichletCharacter(H, M([30]))
 
pari: [g,chi] = znchar(Mod(78,463))
 

Basic properties

Modulus: \(463\)
Conductor: \(463\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(77\)
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 463.m

\(\chi_{463}(8,\cdot)\) \(\chi_{463}(18,\cdot)\) \(\chi_{463}(47,\cdot)\) \(\chi_{463}(49,\cdot)\) \(\chi_{463}(57,\cdot)\) \(\chi_{463}(58,\cdot)\) \(\chi_{463}(64,\cdot)\) \(\chi_{463}(65,\cdot)\) \(\chi_{463}(66,\cdot)\) \(\chi_{463}(70,\cdot)\) \(\chi_{463}(78,\cdot)\) \(\chi_{463}(84,\cdot)\) \(\chi_{463}(86,\cdot)\) \(\chi_{463}(97,\cdot)\) \(\chi_{463}(100,\cdot)\) \(\chi_{463}(111,\cdot)\) \(\chi_{463}(120,\cdot)\) \(\chi_{463}(123,\cdot)\) \(\chi_{463}(124,\cdot)\) \(\chi_{463}(144,\cdot)\) \(\chi_{463}(146,\cdot)\) \(\chi_{463}(149,\cdot)\) \(\chi_{463}(159,\cdot)\) \(\chi_{463}(161,\cdot)\) \(\chi_{463}(189,\cdot)\) \(\chi_{463}(209,\cdot)\) \(\chi_{463}(226,\cdot)\) \(\chi_{463}(242,\cdot)\) \(\chi_{463}(244,\cdot)\) \(\chi_{463}(262,\cdot)\) ...

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: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 77 polynomial

Values on generators

\(3\) → \(e\left(\frac{15}{77}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 463 }(78, a) \) \(1\)\(1\)\(e\left(\frac{48}{77}\right)\)\(e\left(\frac{15}{77}\right)\)\(e\left(\frac{19}{77}\right)\)\(e\left(\frac{27}{77}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{62}{77}\right)\)\(e\left(\frac{67}{77}\right)\)\(e\left(\frac{30}{77}\right)\)\(e\left(\frac{75}{77}\right)\)\(e\left(\frac{29}{77}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 463 }(78,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 463 }(78,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 463 }(78,·),\chi_{ 463 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 463 }(78,·)) \;\) at \(\; a,b = \) e.g. 1,2