Properties

Label 525.158
Modulus $525$
Conductor $525$
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(525, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,9,40]))
 
pari: [g,chi] = znchar(Mod(158,525))
 

Basic properties

Modulus: \(525\)
Conductor: \(525\)
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 525.bs

\(\chi_{525}(2,\cdot)\) \(\chi_{525}(23,\cdot)\) \(\chi_{525}(53,\cdot)\) \(\chi_{525}(128,\cdot)\) \(\chi_{525}(137,\cdot)\) \(\chi_{525}(158,\cdot)\) \(\chi_{525}(212,\cdot)\) \(\chi_{525}(233,\cdot)\) \(\chi_{525}(242,\cdot)\) \(\chi_{525}(263,\cdot)\) \(\chi_{525}(317,\cdot)\) \(\chi_{525}(338,\cdot)\) \(\chi_{525}(347,\cdot)\) \(\chi_{525}(422,\cdot)\) \(\chi_{525}(452,\cdot)\) \(\chi_{525}(473,\cdot)\)

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

\((176,127,451)\) → \((-1,e\left(\frac{3}{20}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 525 }(158, a) \) \(1\)\(1\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{29}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 525 }(158,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 525 }(158,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 525 }(158,·),\chi_{ 525 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 525 }(158,·)) \;\) at \(\; a,b = \) e.g. 1,2