Properties

Label 784.45
Modulus $784$
Conductor $784$
Order $84$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,63,62]))
 
pari: [g,chi] = znchar(Mod(45,784))
 

Basic properties

Modulus: \(784\)
Conductor: \(784\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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 784.bs

\(\chi_{784}(5,\cdot)\) \(\chi_{784}(45,\cdot)\) \(\chi_{784}(61,\cdot)\) \(\chi_{784}(101,\cdot)\) \(\chi_{784}(157,\cdot)\) \(\chi_{784}(173,\cdot)\) \(\chi_{784}(213,\cdot)\) \(\chi_{784}(229,\cdot)\) \(\chi_{784}(269,\cdot)\) \(\chi_{784}(285,\cdot)\) \(\chi_{784}(341,\cdot)\) \(\chi_{784}(381,\cdot)\) \(\chi_{784}(397,\cdot)\) \(\chi_{784}(437,\cdot)\) \(\chi_{784}(453,\cdot)\) \(\chi_{784}(493,\cdot)\) \(\chi_{784}(549,\cdot)\) \(\chi_{784}(565,\cdot)\) \(\chi_{784}(605,\cdot)\) \(\chi_{784}(621,\cdot)\) \(\chi_{784}(661,\cdot)\) \(\chi_{784}(677,\cdot)\) \(\chi_{784}(733,\cdot)\) \(\chi_{784}(773,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((687,197,689)\) → \((1,-i,e\left(\frac{31}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 784 }(45, a) \) \(-1\)\(1\)\(e\left(\frac{83}{84}\right)\)\(e\left(\frac{13}{84}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{23}{84}\right)\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{13}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 784 }(45,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 784 }(45,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 784 }(45,·),\chi_{ 784 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 784 }(45,·)) \;\) at \(\; a,b = \) e.g. 1,2