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Properties

Label 48139.146
Modulus $48139$
Conductor $48139$
Order $1518$
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(48139, base_ring=CyclotomicField(1518)) M = H._module chi = DirichletCharacter(H, M([759,506,150]))
 
Copy content gp:[g,chi] = znchar(Mod(146, 48139))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("48139.146");
 

Basic properties

Modulus: \(48139\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(48139\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1518\)
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 48139.hw

\(\chi_{48139}(48,\cdot)\) \(\chi_{48139}(55,\cdot)\) \(\chi_{48139}(146,\cdot)\) \(\chi_{48139}(328,\cdot)\) \(\chi_{48139}(510,\cdot)\) \(\chi_{48139}(601,\cdot)\) \(\chi_{48139}(685,\cdot)\) \(\chi_{48139}(692,\cdot)\) \(\chi_{48139}(867,\cdot)\) \(\chi_{48139}(1140,\cdot)\) \(\chi_{48139}(1231,\cdot)\) \(\chi_{48139}(1329,\cdot)\) \(\chi_{48139}(1504,\cdot)\) \(\chi_{48139}(1511,\cdot)\)

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: $\Q(\zeta_{759})$
Fixed field: Number field defined by a degree 1518 polynomial (not computed)

Values on generators

\((20632,22219,14288)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{25}{253}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 48139 }(146, a) \) \(-1\)\(1\)\(e\left(\frac{73}{759}\right)\)\(e\left(\frac{629}{1518}\right)\)\(e\left(\frac{146}{759}\right)\)\(e\left(\frac{303}{506}\right)\)\(e\left(\frac{775}{1518}\right)\)\(e\left(\frac{73}{253}\right)\)\(e\left(\frac{629}{759}\right)\)\(e\left(\frac{1055}{1518}\right)\)\(e\left(\frac{532}{759}\right)\)\(e\left(\frac{307}{506}\right)\)
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)
 
\( \chi_{ 48139 }(146,a) \;\) at \(\;a = \) e.g. 2