Properties

Label 145.27
Modulus $145$
Conductor $145$
Order $28$
Real no
Primitive yes
Minimal yes
Parity even

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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([7,15]))
 
pari: [g,chi] = znchar(Mod(27,145))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 145.t

\(\chi_{145}(3,\cdot)\) \(\chi_{145}(27,\cdot)\) \(\chi_{145}(37,\cdot)\) \(\chi_{145}(43,\cdot)\) \(\chi_{145}(47,\cdot)\) \(\chi_{145}(48,\cdot)\) \(\chi_{145}(97,\cdot)\) \(\chi_{145}(98,\cdot)\) \(\chi_{145}(102,\cdot)\) \(\chi_{145}(108,\cdot)\) \(\chi_{145}(118,\cdot)\) \(\chi_{145}(142,\cdot)\)

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: 28.28.1455848000512373226044338588471370773272037506103515625.2

Values on generators

\((117,31)\) → \((i,e\left(\frac{15}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 145 }(27, a) \) \(1\)\(1\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{11}{28}\right)\)\(1\)\(e\left(\frac{11}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 145 }(27,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 145 }(27,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 145 }(27,·),\chi_{ 145 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 145 }(27,·)) \;\) at \(\; a,b = \) e.g. 1,2