Properties

Label 712.291
Modulus $712$
Conductor $712$
Order $88$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(712, base_ring=CyclotomicField(88))
 
M = H._module
 
chi = DirichletCharacter(H, M([44,44,49]))
 
pari: [g,chi] = znchar(Mod(291,712))
 

Basic properties

Modulus: \(712\)
Conductor: \(712\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(88\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 712.bf

\(\chi_{712}(3,\cdot)\) \(\chi_{712}(19,\cdot)\) \(\chi_{712}(27,\cdot)\) \(\chi_{712}(35,\cdot)\) \(\chi_{712}(43,\cdot)\) \(\chi_{712}(51,\cdot)\) \(\chi_{712}(59,\cdot)\) \(\chi_{712}(75,\cdot)\) \(\chi_{712}(83,\cdot)\) \(\chi_{712}(115,\cdot)\) \(\chi_{712}(147,\cdot)\) \(\chi_{712}(155,\cdot)\) \(\chi_{712}(163,\cdot)\) \(\chi_{712}(171,\cdot)\) \(\chi_{712}(211,\cdot)\) \(\chi_{712}(219,\cdot)\) \(\chi_{712}(243,\cdot)\) \(\chi_{712}(291,\cdot)\) \(\chi_{712}(315,\cdot)\) \(\chi_{712}(323,\cdot)\) \(\chi_{712}(363,\cdot)\) \(\chi_{712}(371,\cdot)\) \(\chi_{712}(379,\cdot)\) \(\chi_{712}(387,\cdot)\) \(\chi_{712}(419,\cdot)\) \(\chi_{712}(451,\cdot)\) \(\chi_{712}(459,\cdot)\) \(\chi_{712}(475,\cdot)\) \(\chi_{712}(483,\cdot)\) \(\chi_{712}(491,\cdot)\) ...

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: $\Q(\zeta_{88})$
Fixed field: Number field defined by a degree 88 polynomial

Values on generators

\((535,357,537)\) → \((-1,-1,e\left(\frac{49}{88}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 712 }(291, a) \) \(1\)\(1\)\(e\left(\frac{49}{88}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{53}{88}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{27}{88}\right)\)\(e\left(\frac{3}{88}\right)\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{43}{88}\right)\)\(e\left(\frac{7}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 712 }(291,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 712 }(291,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 712 }(291,·),\chi_{ 712 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 712 }(291,·)) \;\) at \(\; a,b = \) e.g. 1,2