Properties

Label 145.69
Modulus $145$
Conductor $145$
Order $28$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(145, base_ring=CyclotomicField(28))
 
M = H._module
 
chi = DirichletCharacter(H, M([14,25]))
 
pari: [g,chi] = znchar(Mod(69,145))
 

Basic properties

Modulus: \(145\)
Conductor: \(145\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
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 145.s

\(\chi_{145}(14,\cdot)\) \(\chi_{145}(19,\cdot)\) \(\chi_{145}(39,\cdot)\) \(\chi_{145}(44,\cdot)\) \(\chi_{145}(69,\cdot)\) \(\chi_{145}(79,\cdot)\) \(\chi_{145}(84,\cdot)\) \(\chi_{145}(89,\cdot)\) \(\chi_{145}(114,\cdot)\) \(\chi_{145}(119,\cdot)\) \(\chi_{145}(124,\cdot)\) \(\chi_{145}(134,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: 28.0.18634854406558377293367533932433545897882080078125.1

Values on generators

\((117,31)\) → \((-1,e\left(\frac{25}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 145 }(69, a) \) \(-1\)\(1\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{9}{28}\right)\)\(-i\)\(e\left(\frac{4}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 145 }(69,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 145 }(69,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 145 }(69,·),\chi_{ 145 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 145 }(69,·)) \;\) at \(\; a,b = \) e.g. 1,2