Properties

Label 291.38
Modulus $291$
Conductor $291$
Order $96$
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(96))
 
M = H._module
 
chi = DirichletCharacter(H, M([48,19]))
 
pari: [g,chi] = znchar(Mod(38,291))
 

Basic properties

Modulus: \(291\)
Conductor: \(291\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(96\)
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.x

\(\chi_{291}(5,\cdot)\) \(\chi_{291}(14,\cdot)\) \(\chi_{291}(17,\cdot)\) \(\chi_{291}(23,\cdot)\) \(\chi_{291}(26,\cdot)\) \(\chi_{291}(29,\cdot)\) \(\chi_{291}(38,\cdot)\) \(\chi_{291}(41,\cdot)\) \(\chi_{291}(56,\cdot)\) \(\chi_{291}(59,\cdot)\) \(\chi_{291}(68,\cdot)\) \(\chi_{291}(71,\cdot)\) \(\chi_{291}(74,\cdot)\) \(\chi_{291}(80,\cdot)\) \(\chi_{291}(83,\cdot)\) \(\chi_{291}(92,\cdot)\) \(\chi_{291}(104,\cdot)\) \(\chi_{291}(107,\cdot)\) \(\chi_{291}(110,\cdot)\) \(\chi_{291}(134,\cdot)\) \(\chi_{291}(137,\cdot)\) \(\chi_{291}(155,\cdot)\) \(\chi_{291}(173,\cdot)\) \(\chi_{291}(179,\cdot)\) \(\chi_{291}(209,\cdot)\) \(\chi_{291}(215,\cdot)\) \(\chi_{291}(233,\cdot)\) \(\chi_{291}(251,\cdot)\) \(\chi_{291}(254,\cdot)\) \(\chi_{291}(278,\cdot)\) ...

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: $\Q(\zeta_{96})$
Fixed field: Number field defined by a degree 96 polynomial

Values on generators

\((98,199)\) → \((-1,e\left(\frac{19}{96}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 291 }(38, a) \) \(1\)\(1\)\(e\left(\frac{11}{48}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{67}{96}\right)\)\(e\left(\frac{13}{96}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{89}{96}\right)\)\(e\left(\frac{25}{48}\right)\)\(e\left(\frac{91}{96}\right)\)\(e\left(\frac{35}{96}\right)\)\(e\left(\frac{11}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 291 }(38,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 291 }(38,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 291 }(38,·),\chi_{ 291 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 291 }(38,·)) \;\) at \(\; a,b = \) e.g. 1,2