Properties

Label 172.81
Modulus $172$
Conductor $43$
Order $21$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(172, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,4]))
 
pari: [g,chi] = znchar(Mod(81,172))
 

Basic properties

Modulus: \(172\)
Conductor: \(43\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{43}(38,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 172.m

\(\chi_{172}(9,\cdot)\) \(\chi_{172}(13,\cdot)\) \(\chi_{172}(17,\cdot)\) \(\chi_{172}(25,\cdot)\) \(\chi_{172}(53,\cdot)\) \(\chi_{172}(57,\cdot)\) \(\chi_{172}(81,\cdot)\) \(\chi_{172}(101,\cdot)\) \(\chi_{172}(109,\cdot)\) \(\chi_{172}(117,\cdot)\) \(\chi_{172}(153,\cdot)\) \(\chi_{172}(169,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 21 polynomial

Values on generators

\((87,89)\) → \((1,e\left(\frac{2}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 172 }(81, a) \) \(1\)\(1\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{3}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 172 }(81,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 172 }(81,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 172 }(81,·),\chi_{ 172 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 172 }(81,·)) \;\) at \(\; a,b = \) e.g. 1,2