Properties

Label 429.410
Modulus $429$
Conductor $429$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,48,55]))
 
pari: [g,chi] = znchar(Mod(410,429))
 

Basic properties

Modulus: \(429\)
Conductor: \(429\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 429.bv

\(\chi_{429}(20,\cdot)\) \(\chi_{429}(59,\cdot)\) \(\chi_{429}(71,\cdot)\) \(\chi_{429}(80,\cdot)\) \(\chi_{429}(119,\cdot)\) \(\chi_{429}(137,\cdot)\) \(\chi_{429}(158,\cdot)\) \(\chi_{429}(236,\cdot)\) \(\chi_{429}(245,\cdot)\) \(\chi_{429}(284,\cdot)\) \(\chi_{429}(323,\cdot)\) \(\chi_{429}(344,\cdot)\) \(\chi_{429}(383,\cdot)\) \(\chi_{429}(401,\cdot)\) \(\chi_{429}(410,\cdot)\) \(\chi_{429}(422,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((287,79,67)\) → \((-1,e\left(\frac{4}{5}\right),e\left(\frac{11}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 429 }(410, a) \) \(1\)\(1\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{59}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 429 }(410,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 429 }(410,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 429 }(410,·),\chi_{ 429 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 429 }(410,·)) \;\) at \(\; a,b = \) e.g. 1,2