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Properties

Label 3055.348
Modulus $3055$
Conductor $3055$
Order $276$
Real no
Primitive yes
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(3055, base_ring=CyclotomicField(276)) M = H._module chi = DirichletCharacter(H, M([207,230,270]))
 
Copy content gp:[g,chi] = znchar(Mod(348, 3055))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3055.348");
 

Basic properties

Modulus: \(3055\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3055\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(276\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 3055.cv

\(\chi_{3055}(23,\cdot)\) \(\chi_{3055}(43,\cdot)\) \(\chi_{3055}(62,\cdot)\) \(\chi_{3055}(82,\cdot)\) \(\chi_{3055}(88,\cdot)\) \(\chi_{3055}(127,\cdot)\) \(\chi_{3055}(218,\cdot)\) \(\chi_{3055}(257,\cdot)\) \(\chi_{3055}(322,\cdot)\) \(\chi_{3055}(342,\cdot)\) \(\chi_{3055}(348,\cdot)\) \(\chi_{3055}(368,\cdot)\) \(\chi_{3055}(387,\cdot)\) \(\chi_{3055}(407,\cdot)\) \(\chi_{3055}(433,\cdot)\) \(\chi_{3055}(452,\cdot)\) \(\chi_{3055}(537,\cdot)\) \(\chi_{3055}(543,\cdot)\) \(\chi_{3055}(602,\cdot)\) \(\chi_{3055}(608,\cdot)\) \(\chi_{3055}(673,\cdot)\) \(\chi_{3055}(693,\cdot)\) \(\chi_{3055}(738,\cdot)\) \(\chi_{3055}(797,\cdot)\) \(\chi_{3055}(842,\cdot)\) \(\chi_{3055}(868,\cdot)\) \(\chi_{3055}(933,\cdot)\) \(\chi_{3055}(953,\cdot)\) \(\chi_{3055}(992,\cdot)\) \(\chi_{3055}(998,\cdot)\) ...

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: $\Q(\zeta_{276})$
Fixed field: Number field defined by a degree 276 polynomial (not computed)

Values on generators

\((612,236,2731)\) → \((-i,e\left(\frac{5}{6}\right),e\left(\frac{45}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\( \chi_{ 3055 }(348, a) \) \(1\)\(1\)\(e\left(\frac{53}{276}\right)\)\(e\left(\frac{41}{276}\right)\)\(e\left(\frac{53}{138}\right)\)\(e\left(\frac{47}{138}\right)\)\(e\left(\frac{61}{276}\right)\)\(e\left(\frac{53}{92}\right)\)\(e\left(\frac{41}{138}\right)\)\(e\left(\frac{47}{69}\right)\)\(e\left(\frac{49}{92}\right)\)\(e\left(\frac{19}{46}\right)\)
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)
 
\( \chi_{ 3055 }(348,a) \;\) at \(\;a = \) e.g. 2