Properties

Label 828.5
Modulus $828$
Conductor $207$
Order $66$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(828, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,55,3]))
 
pari: [g,chi] = znchar(Mod(5,828))
 

Basic properties

Modulus: \(828\)
Conductor: \(207\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{207}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 828.be

\(\chi_{828}(5,\cdot)\) \(\chi_{828}(65,\cdot)\) \(\chi_{828}(113,\cdot)\) \(\chi_{828}(149,\cdot)\) \(\chi_{828}(221,\cdot)\) \(\chi_{828}(245,\cdot)\) \(\chi_{828}(281,\cdot)\) \(\chi_{828}(293,\cdot)\) \(\chi_{828}(329,\cdot)\) \(\chi_{828}(365,\cdot)\) \(\chi_{828}(389,\cdot)\) \(\chi_{828}(401,\cdot)\) \(\chi_{828}(425,\cdot)\) \(\chi_{828}(497,\cdot)\) \(\chi_{828}(569,\cdot)\) \(\chi_{828}(605,\cdot)\) \(\chi_{828}(617,\cdot)\) \(\chi_{828}(641,\cdot)\) \(\chi_{828}(677,\cdot)\) \(\chi_{828}(797,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((415,461,649)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{1}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 828 }(5, a) \) \(1\)\(1\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{13}{66}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{14}{33}\right)\)\(e\left(\frac{43}{66}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{9}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 828 }(5,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 828 }(5,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 828 }(5,·),\chi_{ 828 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 828 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2