Properties

Label 845.418
Modulus 845845
Conductor 6565
Order 1212
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,1]))
 
pari: [g,chi] = znchar(Mod(418,845))
 

Basic properties

Modulus: 845845
Conductor: 6565
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 1212
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from χ65(28,)\chi_{65}(28,\cdot)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 845.t

χ845(188,)\chi_{845}(188,\cdot) χ845(418,)\chi_{845}(418,\cdot) χ845(427,)\chi_{845}(427,\cdot) χ845(657,)\chi_{845}(657,\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(ζ12)\Q(\zeta_{12})
Fixed field: 12.12.3500313269603515625.1

Values on generators

(677,171)(677,171)(i,e(112))(-i,e\left(\frac{1}{12}\right))

First values

aa 1-11122334466778899111112121414
χ845(418,a) \chi_{ 845 }(418, a) 1111e(56)e\left(\frac{5}{6}\right)e(712)e\left(\frac{7}{12}\right)e(23)e\left(\frac{2}{3}\right)e(512)e\left(\frac{5}{12}\right)e(23)e\left(\frac{2}{3}\right)1-1e(16)e\left(\frac{1}{6}\right)e(712)e\left(\frac{7}{12}\right)ii1-1
sage: chi.jacobi_sum(n)
 
χ845(418,a)   \chi_{ 845 }(418,a) \; at   a=\;a = e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
τa(χ845(418,))   \tau_{ a }( \chi_{ 845 }(418,·) )\; at   a=\;a = e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
J(χ845(418,),χ845(n,))   J(\chi_{ 845 }(418,·),\chi_{ 845 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
K(a,b,χ845(418,))  K(a,b,\chi_{ 845 }(418,·)) \; at   a,b=\; a,b = e.g. 1,2