Properties

Label 605.386
Modulus $605$
Conductor $121$
Order $11$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,14]))
 
pari: [g,chi] = znchar(Mod(386,605))
 

Basic properties

Modulus: \(605\)
Conductor: \(121\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(11\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{121}(23,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 605.k

\(\chi_{605}(56,\cdot)\) \(\chi_{605}(111,\cdot)\) \(\chi_{605}(166,\cdot)\) \(\chi_{605}(221,\cdot)\) \(\chi_{605}(276,\cdot)\) \(\chi_{605}(331,\cdot)\) \(\chi_{605}(386,\cdot)\) \(\chi_{605}(441,\cdot)\) \(\chi_{605}(496,\cdot)\) \(\chi_{605}(551,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: 11.11.672749994932560009201.1

Values on generators

\((122,486)\) → \((1,e\left(\frac{7}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 605 }(386, a) \) \(1\)\(1\)\(e\left(\frac{7}{11}\right)\)\(1\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(1\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{1}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 605 }(386,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 605 }(386,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 605 }(386,·),\chi_{ 605 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 605 }(386,·)) \;\) at \(\; a,b = \) e.g. 1,2