Properties

Label 722.47
Modulus $722$
Conductor $361$
Order $171$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(722, base_ring=CyclotomicField(342))
 
M = H._module
 
chi = DirichletCharacter(H, M([26]))
 
pari: [g,chi] = znchar(Mod(47,722))
 

Basic properties

Modulus: \(722\)
Conductor: \(361\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(171\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{361}(47,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 722.k

\(\chi_{722}(5,\cdot)\) \(\chi_{722}(9,\cdot)\) \(\chi_{722}(17,\cdot)\) \(\chi_{722}(23,\cdot)\) \(\chi_{722}(25,\cdot)\) \(\chi_{722}(35,\cdot)\) \(\chi_{722}(43,\cdot)\) \(\chi_{722}(47,\cdot)\) \(\chi_{722}(55,\cdot)\) \(\chi_{722}(61,\cdot)\) \(\chi_{722}(63,\cdot)\) \(\chi_{722}(73,\cdot)\) \(\chi_{722}(81,\cdot)\) \(\chi_{722}(85,\cdot)\) \(\chi_{722}(93,\cdot)\) \(\chi_{722}(101,\cdot)\) \(\chi_{722}(111,\cdot)\) \(\chi_{722}(119,\cdot)\) \(\chi_{722}(123,\cdot)\) \(\chi_{722}(131,\cdot)\) \(\chi_{722}(137,\cdot)\) \(\chi_{722}(139,\cdot)\) \(\chi_{722}(149,\cdot)\) \(\chi_{722}(157,\cdot)\) \(\chi_{722}(161,\cdot)\) \(\chi_{722}(169,\cdot)\) \(\chi_{722}(175,\cdot)\) \(\chi_{722}(177,\cdot)\) \(\chi_{722}(187,\cdot)\) \(\chi_{722}(195,\cdot)\) ...

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: $\Q(\zeta_{171})$
Fixed field: Number field defined by a degree 171 polynomial (not computed)

Values on generators

\(363\) → \(e\left(\frac{13}{171}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 722 }(47, a) \) \(1\)\(1\)\(e\left(\frac{97}{171}\right)\)\(e\left(\frac{109}{171}\right)\)\(e\left(\frac{23}{57}\right)\)\(e\left(\frac{23}{171}\right)\)\(e\left(\frac{43}{57}\right)\)\(e\left(\frac{2}{171}\right)\)\(e\left(\frac{35}{171}\right)\)\(e\left(\frac{31}{171}\right)\)\(e\left(\frac{166}{171}\right)\)\(e\left(\frac{17}{171}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 722 }(47,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 722 }(47,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 722 }(47,·),\chi_{ 722 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 722 }(47,·)) \;\) at \(\; a,b = \) e.g. 1,2