Properties

Label 576.205
Modulus $576$
Conductor $576$
Order $48$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

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([0,45,32]))
 
pari: [g,chi] = znchar(Mod(205,576))
 

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.bm

\(\chi_{576}(13,\cdot)\) \(\chi_{576}(61,\cdot)\) \(\chi_{576}(85,\cdot)\) \(\chi_{576}(133,\cdot)\) \(\chi_{576}(157,\cdot)\) \(\chi_{576}(205,\cdot)\) \(\chi_{576}(229,\cdot)\) \(\chi_{576}(277,\cdot)\) \(\chi_{576}(301,\cdot)\) \(\chi_{576}(349,\cdot)\) \(\chi_{576}(373,\cdot)\) \(\chi_{576}(421,\cdot)\) \(\chi_{576}(445,\cdot)\) \(\chi_{576}(493,\cdot)\) \(\chi_{576}(517,\cdot)\) \(\chi_{576}(565,\cdot)\)

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

\((127,325,65)\) → \((1,e\left(\frac{15}{16}\right),e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{13}{48}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{17}{48}\right)\)\(e\left(\frac{19}{48}\right)\)\(i\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{5}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 576 }(205,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 576 }(205,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 576 }(205,·),\chi_{ 576 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 576 }(205,·)) \;\) at \(\; a,b = \) e.g. 1,2