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Properties

Label 6820.237
Modulus $6820$
Conductor $1705$
Order $60$
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(6820, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([0,15,54,16]))
 
Copy content gp:[g,chi] = znchar(Mod(237, 6820))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6820.237");
 

Basic properties

Modulus: \(6820\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1705\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(60\)
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 \(\chi_{1705}(237,\cdot)\)
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 6820.lz

\(\chi_{6820}(173,\cdot)\) \(\chi_{6820}(193,\cdot)\) \(\chi_{6820}(237,\cdot)\) \(\chi_{6820}(1537,\cdot)\) \(\chi_{6820}(1557,\cdot)\) \(\chi_{6820}(1993,\cdot)\) \(\chi_{6820}(2273,\cdot)\) \(\chi_{6820}(2933,\cdot)\) \(\chi_{6820}(3357,\cdot)\) \(\chi_{6820}(3637,\cdot)\) \(\chi_{6820}(4297,\cdot)\) \(\chi_{6820}(4633,\cdot)\) \(\chi_{6820}(5453,\cdot)\) \(\chi_{6820}(5693,\cdot)\) \(\chi_{6820}(5997,\cdot)\) \(\chi_{6820}(6817,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((3411,5457,1861,2421)\) → \((1,i,e\left(\frac{9}{10}\right),e\left(\frac{4}{15}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 6820 }(237, a) \) \(1\)\(1\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{1}{5}\right)\)
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)
 
\( \chi_{ 6820 }(237,a) \;\) at \(\;a = \) e.g. 2