Properties

Label 172.95
Modulus $172$
Conductor $172$
Order $42$
Real no
Primitive yes
Minimal yes
Parity odd

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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([21,2]))
 
pari: [g,chi] = znchar(Mod(95,172))
 

Basic properties

Modulus: \(172\)
Conductor: \(172\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 172.o

\(\chi_{172}(15,\cdot)\) \(\chi_{172}(23,\cdot)\) \(\chi_{172}(31,\cdot)\) \(\chi_{172}(67,\cdot)\) \(\chi_{172}(83,\cdot)\) \(\chi_{172}(95,\cdot)\) \(\chi_{172}(99,\cdot)\) \(\chi_{172}(103,\cdot)\) \(\chi_{172}(111,\cdot)\) \(\chi_{172}(139,\cdot)\) \(\chi_{172}(143,\cdot)\) \(\chi_{172}(167,\cdot)\)

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: 42.0.959396304051793463814262846982490027578741814649477038563926538598268329263104.1

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 172 }(95, a) \) \(-1\)\(1\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{5}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 172 }(95,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 172 }(95,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 172 }(95,·),\chi_{ 172 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 172 }(95,·)) \;\) at \(\; a,b = \) e.g. 1,2