Properties

Label 403.239
Modulus $403$
Conductor $403$
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(403, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([45,34]))
 
pari: [g,chi] = znchar(Mod(239,403))
 

Basic properties

Modulus: \(403\)
Conductor: \(403\)
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 403.cd

\(\chi_{403}(21,\cdot)\) \(\chi_{403}(34,\cdot)\) \(\chi_{403}(44,\cdot)\) \(\chi_{403}(73,\cdot)\) \(\chi_{403}(83,\cdot)\) \(\chi_{403}(86,\cdot)\) \(\chi_{403}(96,\cdot)\) \(\chi_{403}(135,\cdot)\) \(\chi_{403}(148,\cdot)\) \(\chi_{403}(177,\cdot)\) \(\chi_{403}(203,\cdot)\) \(\chi_{403}(229,\cdot)\) \(\chi_{403}(239,\cdot)\) \(\chi_{403}(265,\cdot)\) \(\chi_{403}(291,\cdot)\) \(\chi_{403}(385,\cdot)\)

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

\((249,313)\) → \((-i,e\left(\frac{17}{30}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 403 }(239, a) \) \(1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{17}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 403 }(239,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 403 }(239,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 403 }(239,·),\chi_{ 403 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 403 }(239,·)) \;\) at \(\; a,b = \) e.g. 1,2