Properties

Label 806.95
Modulus $806$
Conductor $403$
Order $30$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([5,24]))
 
pari: [g,chi] = znchar(Mod(95,806))
 

Basic properties

Modulus: \(806\)
Conductor: \(403\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{403}(95,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 806.bv

\(\chi_{806}(95,\cdot)\) \(\chi_{806}(101,\cdot)\) \(\chi_{806}(225,\cdot)\) \(\chi_{806}(283,\cdot)\) \(\chi_{806}(407,\cdot)\) \(\chi_{806}(543,\cdot)\) \(\chi_{806}(667,\cdot)\) \(\chi_{806}(777,\cdot)\)

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

\((249,313)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{4}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 806 }(95, a) \) \(1\)\(1\)\(e\left(\frac{7}{15}\right)\)\(-1\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{4}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 806 }(95,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 806 }(95,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 806 }(95,·),\chi_{ 806 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 806 }(95,·)) \;\) at \(\; a,b = \) e.g. 1,2