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Properties

Label 4147.1059
Modulus $4147$
Conductor $4147$
Order $420$
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(4147, base_ring=CyclotomicField(420)) M = H._module chi = DirichletCharacter(H, M([336,175,405]))
 
Copy content gp:[g,chi] = znchar(Mod(1059, 4147))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4147.1059");
 

Basic properties

Modulus: \(4147\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4147\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(420\)
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 4147.ft

\(\chi_{4147}(15,\cdot)\) \(\chi_{4147}(97,\cdot)\) \(\chi_{4147}(137,\cdot)\) \(\chi_{4147}(201,\cdot)\) \(\chi_{4147}(240,\cdot)\) \(\chi_{4147}(279,\cdot)\) \(\chi_{4147}(280,\cdot)\) \(\chi_{4147}(366,\cdot)\) \(\chi_{4147}(388,\cdot)\) \(\chi_{4147}(427,\cdot)\) \(\chi_{4147}(449,\cdot)\) \(\chi_{4147}(466,\cdot)\) \(\chi_{4147}(548,\cdot)\) \(\chi_{4147}(553,\cdot)\) \(\chi_{4147}(570,\cdot)\) \(\chi_{4147}(665,\cdot)\) \(\chi_{4147}(851,\cdot)\) \(\chi_{4147}(852,\cdot)\) \(\chi_{4147}(938,\cdot)\) \(\chi_{4147}(955,\cdot)\) \(\chi_{4147}(960,\cdot)\) \(\chi_{4147}(994,\cdot)\) \(\chi_{4147}(1059,\cdot)\) \(\chi_{4147}(1081,\cdot)\) \(\chi_{4147}(1116,\cdot)\) \(\chi_{4147}(1120,\cdot)\) \(\chi_{4147}(1142,\cdot)\) \(\chi_{4147}(1181,\cdot)\) \(\chi_{4147}(1203,\cdot)\) \(\chi_{4147}(1215,\cdot)\) ...

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: $\Q(\zeta_{420})$
Fixed field: Number field defined by a degree 420 polynomial (not computed)

Values on generators

\((1509,639,2003)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{5}{12}\right),e\left(\frac{27}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 4147 }(1059, a) \) \(1\)\(1\)\(e\left(\frac{19}{105}\right)\)\(e\left(\frac{373}{420}\right)\)\(e\left(\frac{38}{105}\right)\)\(e\left(\frac{23}{140}\right)\)\(e\left(\frac{29}{420}\right)\)\(e\left(\frac{317}{420}\right)\)\(e\left(\frac{19}{35}\right)\)\(e\left(\frac{163}{210}\right)\)\(e\left(\frac{29}{84}\right)\)\(i\)
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)
 
\( \chi_{ 4147 }(1059,a) \;\) at \(\;a = \) e.g. 2