Properties

Label 576.101
Modulus $576$
Conductor $576$
Order $48$
Real no
Primitive yes
Minimal yes
Parity odd

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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([0,27,8]))
 
pari: [g,chi] = znchar(Mod(101,576))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 576.bn

\(\chi_{576}(5,\cdot)\) \(\chi_{576}(29,\cdot)\) \(\chi_{576}(77,\cdot)\) \(\chi_{576}(101,\cdot)\) \(\chi_{576}(149,\cdot)\) \(\chi_{576}(173,\cdot)\) \(\chi_{576}(221,\cdot)\) \(\chi_{576}(245,\cdot)\) \(\chi_{576}(293,\cdot)\) \(\chi_{576}(317,\cdot)\) \(\chi_{576}(365,\cdot)\) \(\chi_{576}(389,\cdot)\) \(\chi_{576}(437,\cdot)\) \(\chi_{576}(461,\cdot)\) \(\chi_{576}(509,\cdot)\) \(\chi_{576}(533,\cdot)\)

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

\((127,325,65)\) → \((1,e\left(\frac{9}{16}\right),e\left(\frac{1}{6}\right))\)

Values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 576 }(101, a) \) \(-1\)\(1\)\(e\left(\frac{19}{48}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{37}{48}\right)\)\(i\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{17}{48}\right)\)\(e\left(\frac{5}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 576 }(101,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 576 }(101,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 576 }(101,·),\chi_{ 576 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 576 }(101,·)) \;\) at \(\; a,b = \) e.g. 1,2