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Properties

Label 48552.4189
Modulus $48552$
Conductor $16184$
Order $816$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(48552, base_ring=CyclotomicField(816)) M = H._module chi = DirichletCharacter(H, M([0,408,0,136,657]))
 
Copy content gp:[g,chi] = znchar(Mod(4189, 48552))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("48552.4189");
 

Basic properties

Modulus: \(48552\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(16184\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(816\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: no, induced from \(\chi_{16184}(4189,\cdot)\)
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 48552.ma

\(\chi_{48552}(61,\cdot)\) \(\chi_{48552}(397,\cdot)\) \(\chi_{48552}(997,\cdot)\) \(\chi_{48552}(1333,\cdot)\) \(\chi_{48552}(1501,\cdot)\) \(\chi_{48552}(1741,\cdot)\) \(\chi_{48552}(1909,\cdot)\) \(\chi_{48552}(2077,\cdot)\) \(\chi_{48552}(2173,\cdot)\) \(\chi_{48552}(2341,\cdot)\) \(\chi_{48552}(2509,\cdot)\) \(\chi_{48552}(2581,\cdot)\) \(\chi_{48552}(2749,\cdot)\) \(\chi_{48552}(2845,\cdot)\) \(\chi_{48552}(2917,\cdot)\) \(\chi_{48552}(3253,\cdot)\) \(\chi_{48552}(3853,\cdot)\) \(\chi_{48552}(4189,\cdot)\) \(\chi_{48552}(4261,\cdot)\) \(\chi_{48552}(4357,\cdot)\) \(\chi_{48552}(4525,\cdot)\) \(\chi_{48552}(4597,\cdot)\) \(\chi_{48552}(4765,\cdot)\) \(\chi_{48552}(4933,\cdot)\) \(\chi_{48552}(5029,\cdot)\) \(\chi_{48552}(5197,\cdot)\) \(\chi_{48552}(5365,\cdot)\) \(\chi_{48552}(5437,\cdot)\) \(\chi_{48552}(5605,\cdot)\) \(\chi_{48552}(5701,\cdot)\) ...

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{816})$
Fixed field: Number field defined by a degree 816 polynomial (not computed)

Values on generators

\((36415,24277,32369,34681,34105)\) → \((1,-1,1,e\left(\frac{1}{6}\right),e\left(\frac{219}{272}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 48552 }(4189, a) \) \(1\)\(1\)\(e\left(\frac{581}{816}\right)\)\(e\left(\frac{559}{816}\right)\)\(e\left(\frac{55}{68}\right)\)\(e\left(\frac{247}{408}\right)\)\(e\left(\frac{719}{816}\right)\)\(e\left(\frac{173}{408}\right)\)\(e\left(\frac{39}{272}\right)\)\(e\left(\frac{337}{816}\right)\)\(e\left(\frac{569}{816}\right)\)\(e\left(\frac{49}{272}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 48552 }(4189,a) \;\) at \(\;a = \) e.g. 2