Properties

Label 291.272
Modulus $291$
Conductor $291$
Order $32$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(291, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([16,11]))
 
pari: [g,chi] = znchar(Mod(272,291))
 

Basic properties

Modulus: \(291\)
Conductor: \(291\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
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 291.s

\(\chi_{291}(20,\cdot)\) \(\chi_{291}(77,\cdot)\) \(\chi_{291}(116,\cdot)\) \(\chi_{291}(125,\cdot)\) \(\chi_{291}(131,\cdot)\) \(\chi_{291}(143,\cdot)\) \(\chi_{291}(149,\cdot)\) \(\chi_{291}(152,\cdot)\) \(\chi_{291}(164,\cdot)\) \(\chi_{291}(224,\cdot)\) \(\chi_{291}(236,\cdot)\) \(\chi_{291}(239,\cdot)\) \(\chi_{291}(245,\cdot)\) \(\chi_{291}(257,\cdot)\) \(\chi_{291}(263,\cdot)\) \(\chi_{291}(272,\cdot)\)

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: \(\Q(\zeta_{32})\)
Fixed field: 32.32.1674417821664703472295706925363164706188610888296487588786531799142113.1

Values on generators

\((98,199)\) → \((-1,e\left(\frac{11}{32}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 291 }(272, a) \) \(1\)\(1\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{27}{32}\right)\)\(e\left(\frac{21}{32}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{1}{32}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{19}{32}\right)\)\(e\left(\frac{27}{32}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 291 }(272,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 291 }(272,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 291 }(272,·),\chi_{ 291 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 291 }(272,·)) \;\) at \(\; a,b = \) e.g. 1,2