Properties

Label 576.275
Modulus $576$
Conductor $576$
Order $48$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(576, base_ring=CyclotomicField(48))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([24,21,40]))
 
pari: [g,chi] = znchar(Mod(275,576))
 

Basic properties

Modulus: \(576\)
Conductor: \(576\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
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 576.bl

\(\chi_{576}(11,\cdot)\) \(\chi_{576}(59,\cdot)\) \(\chi_{576}(83,\cdot)\) \(\chi_{576}(131,\cdot)\) \(\chi_{576}(155,\cdot)\) \(\chi_{576}(203,\cdot)\) \(\chi_{576}(227,\cdot)\) \(\chi_{576}(275,\cdot)\) \(\chi_{576}(299,\cdot)\) \(\chi_{576}(347,\cdot)\) \(\chi_{576}(371,\cdot)\) \(\chi_{576}(419,\cdot)\) \(\chi_{576}(443,\cdot)\) \(\chi_{576}(491,\cdot)\) \(\chi_{576}(515,\cdot)\) \(\chi_{576}(563,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((127,325,65)\) → \((-1,e\left(\frac{7}{16}\right),e\left(\frac{5}{6}\right))\)

Values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 576 }(275, a) \) \(1\)\(1\)\(e\left(\frac{29}{48}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{25}{48}\right)\)\(e\left(\frac{11}{48}\right)\)\(-i\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{31}{48}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 576 }(275,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 576 }(275,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 576 }(275,·),\chi_{ 576 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 576 }(275,·)) \;\) at \(\; a,b = \) e.g. 1,2