Properties

Label 900.859
Modulus $900$
Conductor $900$
Order $30$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,10,21]))
 
pari: [g,chi] = znchar(Mod(859,900))
 

Basic properties

Modulus: \(900\)
Conductor: \(900\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 900.bl

\(\chi_{900}(79,\cdot)\) \(\chi_{900}(139,\cdot)\) \(\chi_{900}(259,\cdot)\) \(\chi_{900}(319,\cdot)\) \(\chi_{900}(439,\cdot)\) \(\chi_{900}(619,\cdot)\) \(\chi_{900}(679,\cdot)\) \(\chi_{900}(859,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Values on generators

\((451,101,577)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 900 }(859, a) \) \(-1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{7}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 900 }(859,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 900 }(859,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 900 }(859,·),\chi_{ 900 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 900 }(859,·)) \;\) at \(\; a,b = \) e.g. 1,2