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Properties

Label 5851.4692
Modulus $5851$
Conductor $5851$
Order $150$
Real no
Primitive yes
Minimal yes
Parity odd

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(5851, base_ring=CyclotomicField(150)) M = H._module chi = DirichletCharacter(H, M([49]))
 
Copy content gp:[g,chi] = znchar(Mod(4692, 5851))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5851.4692");
 

Basic properties

Modulus: \(5851\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5851\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(150\)
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: yes
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 5851.w

\(\chi_{5851}(404,\cdot)\) \(\chi_{5851}(420,\cdot)\) \(\chi_{5851}(528,\cdot)\) \(\chi_{5851}(612,\cdot)\) \(\chi_{5851}(778,\cdot)\) \(\chi_{5851}(843,\cdot)\) \(\chi_{5851}(866,\cdot)\) \(\chi_{5851}(1147,\cdot)\) \(\chi_{5851}(1292,\cdot)\) \(\chi_{5851}(1337,\cdot)\) \(\chi_{5851}(1642,\cdot)\) \(\chi_{5851}(2064,\cdot)\) \(\chi_{5851}(2104,\cdot)\) \(\chi_{5851}(2247,\cdot)\) \(\chi_{5851}(2347,\cdot)\) \(\chi_{5851}(2391,\cdot)\) \(\chi_{5851}(2407,\cdot)\) \(\chi_{5851}(2449,\cdot)\) \(\chi_{5851}(2837,\cdot)\) \(\chi_{5851}(2851,\cdot)\) \(\chi_{5851}(3173,\cdot)\) \(\chi_{5851}(3220,\cdot)\) \(\chi_{5851}(3233,\cdot)\) \(\chi_{5851}(3448,\cdot)\) \(\chi_{5851}(3730,\cdot)\) \(\chi_{5851}(4048,\cdot)\) \(\chi_{5851}(4122,\cdot)\) \(\chi_{5851}(4520,\cdot)\) \(\chi_{5851}(4689,\cdot)\) \(\chi_{5851}(4692,\cdot)\) ...

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: $\Q(\zeta_{75})$
Fixed field: Number field defined by a degree 150 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{49}{150}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 5851 }(4692, a) \) \(-1\)\(1\)\(e\left(\frac{49}{150}\right)\)\(e\left(\frac{43}{50}\right)\)\(e\left(\frac{49}{75}\right)\)\(e\left(\frac{46}{75}\right)\)\(e\left(\frac{14}{75}\right)\)\(e\left(\frac{67}{75}\right)\)\(e\left(\frac{49}{50}\right)\)\(e\left(\frac{18}{25}\right)\)\(e\left(\frac{47}{50}\right)\)\(e\left(\frac{61}{75}\right)\)
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)
 
\( \chi_{ 5851 }(4692,a) \;\) at \(\;a = \) e.g. 2