Properties

Label 731.211
Modulus $731$
Conductor $731$
Order $112$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(112))
 
M = H._module
 
chi = DirichletCharacter(H, M([77,88]))
 
pari: [g,chi] = znchar(Mod(211,731))
 

Basic properties

Modulus: \(731\)
Conductor: \(731\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(112\)
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 731.bj

\(\chi_{731}(22,\cdot)\) \(\chi_{731}(27,\cdot)\) \(\chi_{731}(39,\cdot)\) \(\chi_{731}(45,\cdot)\) \(\chi_{731}(65,\cdot)\) \(\chi_{731}(75,\cdot)\) \(\chi_{731}(82,\cdot)\) \(\chi_{731}(88,\cdot)\) \(\chi_{731}(108,\cdot)\) \(\chi_{731}(113,\cdot)\) \(\chi_{731}(125,\cdot)\) \(\chi_{731}(131,\cdot)\) \(\chi_{731}(156,\cdot)\) \(\chi_{731}(180,\cdot)\) \(\chi_{731}(194,\cdot)\) \(\chi_{731}(199,\cdot)\) \(\chi_{731}(211,\cdot)\) \(\chi_{731}(260,\cdot)\) \(\chi_{731}(266,\cdot)\) \(\chi_{731}(303,\cdot)\) \(\chi_{731}(309,\cdot)\) \(\chi_{731}(328,\cdot)\) \(\chi_{731}(333,\cdot)\) \(\chi_{731}(346,\cdot)\) \(\chi_{731}(352,\cdot)\) \(\chi_{731}(371,\cdot)\) \(\chi_{731}(414,\cdot)\) \(\chi_{731}(419,\cdot)\) \(\chi_{731}(432,\cdot)\) \(\chi_{731}(452,\cdot)\) ...

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: $\Q(\zeta_{112})$
Fixed field: Number field defined by a degree 112 polynomial (not computed)

Values on generators

\((173,562)\) → \((e\left(\frac{11}{16}\right),e\left(\frac{11}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 731 }(211, a) \) \(1\)\(1\)\(e\left(\frac{47}{56}\right)\)\(e\left(\frac{53}{112}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{9}{112}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{29}{56}\right)\)\(e\left(\frac{53}{56}\right)\)\(e\left(\frac{103}{112}\right)\)\(e\left(\frac{43}{112}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 731 }(211,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 731 }(211,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 731 }(211,·),\chi_{ 731 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 731 }(211,·)) \;\) at \(\; a,b = \) e.g. 1,2