Properties

Label 5184.851
Modulus $5184$
Conductor $5184$
Order $432$
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(5184, base_ring=CyclotomicField(432))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([216,189,424]))
 
pari: [g,chi] = znchar(Mod(851,5184))
 

Basic properties

Modulus: \(5184\)
Conductor: \(5184\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(432\)
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 5184.db

\(\chi_{5184}(11,\cdot)\) \(\chi_{5184}(59,\cdot)\) \(\chi_{5184}(83,\cdot)\) \(\chi_{5184}(131,\cdot)\) \(\chi_{5184}(155,\cdot)\) \(\chi_{5184}(203,\cdot)\) \(\chi_{5184}(227,\cdot)\) \(\chi_{5184}(275,\cdot)\) \(\chi_{5184}(299,\cdot)\) \(\chi_{5184}(347,\cdot)\) \(\chi_{5184}(371,\cdot)\) \(\chi_{5184}(419,\cdot)\) \(\chi_{5184}(443,\cdot)\) \(\chi_{5184}(491,\cdot)\) \(\chi_{5184}(515,\cdot)\) \(\chi_{5184}(563,\cdot)\) \(\chi_{5184}(587,\cdot)\) \(\chi_{5184}(635,\cdot)\) \(\chi_{5184}(659,\cdot)\) \(\chi_{5184}(707,\cdot)\) \(\chi_{5184}(731,\cdot)\) \(\chi_{5184}(779,\cdot)\) \(\chi_{5184}(803,\cdot)\) \(\chi_{5184}(851,\cdot)\) \(\chi_{5184}(875,\cdot)\) \(\chi_{5184}(923,\cdot)\) \(\chi_{5184}(947,\cdot)\) \(\chi_{5184}(995,\cdot)\) \(\chi_{5184}(1019,\cdot)\) \(\chi_{5184}(1067,\cdot)\) ...

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: $\Q(\zeta_{432})$
Fixed field: Number field defined by a degree 432 polynomial (not computed)

Values on generators

\((2431,325,1217)\) → \((-1,e\left(\frac{7}{16}\right),e\left(\frac{53}{54}\right))\)

Values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 5184 }(851, a) \) \(1\)\(1\)\(e\left(\frac{5}{432}\right)\)\(e\left(\frac{125}{216}\right)\)\(e\left(\frac{193}{432}\right)\)\(e\left(\frac{179}{432}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{97}{144}\right)\)\(e\left(\frac{91}{216}\right)\)\(e\left(\frac{5}{216}\right)\)\(e\left(\frac{55}{432}\right)\)\(e\left(\frac{17}{27}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5184 }(851,a) \;\) at \(\;a = \) e.g. 2