Properties

Label 722.31
Modulus $722$
Conductor $361$
Order $114$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(722, base_ring=CyclotomicField(114))
 
M = H._module
 
chi = DirichletCharacter(H, M([101]))
 
pari: [g,chi] = znchar(Mod(31,722))
 

Basic properties

Modulus: \(722\)
Conductor: \(361\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(114\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{361}(31,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 722.j

\(\chi_{722}(27,\cdot)\) \(\chi_{722}(31,\cdot)\) \(\chi_{722}(65,\cdot)\) \(\chi_{722}(103,\cdot)\) \(\chi_{722}(107,\cdot)\) \(\chi_{722}(141,\cdot)\) \(\chi_{722}(145,\cdot)\) \(\chi_{722}(179,\cdot)\) \(\chi_{722}(183,\cdot)\) \(\chi_{722}(217,\cdot)\) \(\chi_{722}(221,\cdot)\) \(\chi_{722}(255,\cdot)\) \(\chi_{722}(259,\cdot)\) \(\chi_{722}(297,\cdot)\) \(\chi_{722}(331,\cdot)\) \(\chi_{722}(335,\cdot)\) \(\chi_{722}(369,\cdot)\) \(\chi_{722}(373,\cdot)\) \(\chi_{722}(407,\cdot)\) \(\chi_{722}(411,\cdot)\) \(\chi_{722}(445,\cdot)\) \(\chi_{722}(449,\cdot)\) \(\chi_{722}(483,\cdot)\) \(\chi_{722}(487,\cdot)\) \(\chi_{722}(521,\cdot)\) \(\chi_{722}(525,\cdot)\) \(\chi_{722}(559,\cdot)\) \(\chi_{722}(563,\cdot)\) \(\chi_{722}(597,\cdot)\) \(\chi_{722}(601,\cdot)\) ...

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: $\Q(\zeta_{57})$
Fixed field: Number field defined by a degree 114 polynomial (not computed)

Values on generators

\(363\) → \(e\left(\frac{101}{114}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 722 }(31, a) \) \(-1\)\(1\)\(e\left(\frac{17}{114}\right)\)\(e\left(\frac{31}{57}\right)\)\(e\left(\frac{17}{19}\right)\)\(e\left(\frac{17}{57}\right)\)\(e\left(\frac{7}{19}\right)\)\(e\left(\frac{55}{114}\right)\)\(e\left(\frac{79}{114}\right)\)\(e\left(\frac{13}{57}\right)\)\(e\left(\frac{5}{114}\right)\)\(e\left(\frac{20}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 722 }(31,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 722 }(31,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 722 }(31,·),\chi_{ 722 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 722 }(31,·)) \;\) at \(\; a,b = \) e.g. 1,2