Properties

Label 261.170
Modulus $261$
Conductor $87$
Order $14$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(14))
 
M = H._module
 
chi = DirichletCharacter(H, M([7,8]))
 
pari: [g,chi] = znchar(Mod(170,261))
 

Basic properties

Modulus: \(261\)
Conductor: \(87\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{87}(83,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 261.p

\(\chi_{261}(53,\cdot)\) \(\chi_{261}(107,\cdot)\) \(\chi_{261}(152,\cdot)\) \(\chi_{261}(161,\cdot)\) \(\chi_{261}(170,\cdot)\) \(\chi_{261}(197,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: 14.0.773792930870360792667.1

Values on generators

\((146,118)\) → \((-1,e\left(\frac{4}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 261 }(170, a) \) \(-1\)\(1\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{2}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 261 }(170,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 261 }(170,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 261 }(170,·),\chi_{ 261 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 261 }(170,·)) \;\) at \(\; a,b = \) e.g. 1,2