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Properties

Label 5220.1487
Modulus $5220$
Conductor $5220$
Order $84$
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(5220, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([42,14,21,9]))
 
Copy content gp:[g,chi] = znchar(Mod(1487, 5220))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5220.1487");
 

Basic properties

Modulus: \(5220\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5220\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(84\)
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 5220.fs

\(\chi_{5220}(47,\cdot)\) \(\chi_{5220}(263,\cdot)\) \(\chi_{5220}(623,\cdot)\) \(\chi_{5220}(1163,\cdot)\) \(\chi_{5220}(1487,\cdot)\) \(\chi_{5220}(1643,\cdot)\) \(\chi_{5220}(1703,\cdot)\) \(\chi_{5220}(1787,\cdot)\) \(\chi_{5220}(2003,\cdot)\) \(\chi_{5220}(2027,\cdot)\) \(\chi_{5220}(2363,\cdot)\) \(\chi_{5220}(2567,\cdot)\) \(\chi_{5220}(2903,\cdot)\) \(\chi_{5220}(2927,\cdot)\) \(\chi_{5220}(3143,\cdot)\) \(\chi_{5220}(3227,\cdot)\) \(\chi_{5220}(3287,\cdot)\) \(\chi_{5220}(3443,\cdot)\) \(\chi_{5220}(3767,\cdot)\) \(\chi_{5220}(4307,\cdot)\) \(\chi_{5220}(4667,\cdot)\) \(\chi_{5220}(4883,\cdot)\) \(\chi_{5220}(5027,\cdot)\) \(\chi_{5220}(5123,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((2611,4061,4177,901)\) → \((-1,e\left(\frac{1}{6}\right),i,e\left(\frac{3}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 5220 }(1487, a) \) \(1\)\(1\)\(e\left(\frac{59}{84}\right)\)\(e\left(\frac{29}{84}\right)\)\(e\left(\frac{1}{84}\right)\)\(1\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{19}{84}\right)\)\(e\left(\frac{79}{84}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{13}{42}\right)\)
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)
 
\( \chi_{ 5220 }(1487,a) \;\) at \(\;a = \) e.g. 2