Properties

Label 967.128
Modulus $967$
Conductor $967$
Order $69$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(967, base_ring=CyclotomicField(138))
 
M = H._module
 
chi = DirichletCharacter(H, M([98]))
 
pari: [g,chi] = znchar(Mod(128,967))
 

Basic properties

Modulus: \(967\)
Conductor: \(967\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(69\)
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 967.k

\(\chi_{967}(15,\cdot)\) \(\chi_{967}(39,\cdot)\) \(\chi_{967}(53,\cdot)\) \(\chi_{967}(61,\cdot)\) \(\chi_{967}(68,\cdot)\) \(\chi_{967}(73,\cdot)\) \(\chi_{967}(113,\cdot)\) \(\chi_{967}(124,\cdot)\) \(\chi_{967}(128,\cdot)\) \(\chi_{967}(129,\cdot)\) \(\chi_{967}(145,\cdot)\) \(\chi_{967}(190,\cdot)\) \(\chi_{967}(198,\cdot)\) \(\chi_{967}(202,\cdot)\) \(\chi_{967}(225,\cdot)\) \(\chi_{967}(241,\cdot)\) \(\chi_{967}(280,\cdot)\) \(\chi_{967}(321,\cdot)\) \(\chi_{967}(335,\cdot)\) \(\chi_{967}(341,\cdot)\) \(\chi_{967}(352,\cdot)\) \(\chi_{967}(377,\cdot)\) \(\chi_{967}(400,\cdot)\) \(\chi_{967}(421,\cdot)\) \(\chi_{967}(445,\cdot)\) \(\chi_{967}(494,\cdot)\) \(\chi_{967}(513,\cdot)\) \(\chi_{967}(524,\cdot)\) \(\chi_{967}(539,\cdot)\) \(\chi_{967}(554,\cdot)\) ...

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: $\Q(\zeta_{69})$
Fixed field: Number field defined by a degree 69 polynomial

Values on generators

\(5\) → \(e\left(\frac{49}{69}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 967 }(128, a) \) \(1\)\(1\)\(e\left(\frac{41}{69}\right)\)\(e\left(\frac{7}{23}\right)\)\(e\left(\frac{13}{69}\right)\)\(e\left(\frac{49}{69}\right)\)\(e\left(\frac{62}{69}\right)\)\(e\left(\frac{40}{69}\right)\)\(e\left(\frac{18}{23}\right)\)\(e\left(\frac{14}{23}\right)\)\(e\left(\frac{7}{23}\right)\)\(e\left(\frac{14}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 967 }(128,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 967 }(128,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 967 }(128,·),\chi_{ 967 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 967 }(128,·)) \;\) at \(\; a,b = \) e.g. 1,2