Properties

Conductor 1
Order 1
Real Yes
Primitive No
Parity Even

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(36)
sage: chi = H[1]
pari: [g,chi] = znchar(Mod(1,36))

Inducing primitive character

sage: sage: chi.primitive_character()
pari: znconreyconductor(g,chi,&chi0)
pari: chi0

\(\chi_{1}(1,\cdot)\)

Values on generators

sage: chi(k) for k in H.gens()
pari: [ chareval(g,chi,x) | x <- g.gen ] \\ value in Q/Z

\((29,19)\) → \((1,1)\)

Values

157111317192325293135
111111111111
value at  e.g. 2

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 1
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 1
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Even
Real = Yes
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = No

Galois orbit

sage: chi.sage_character().galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

Related number fields

Field of values \(\Q\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 36 }(1,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 36 }(1,·),\chi_{ 36 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 36 }(1,·)) \;\) at \(\; a,b = \) e.g. 1,2
a:
b: