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Properties

Label 4300.427
Modulus $4300$
Conductor $4300$
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(4300, base_ring=CyclotomicField(420)) M = H._module chi = DirichletCharacter(H, M([210,21,220]))
 
Copy content gp:[g,chi] = znchar(Mod(427, 4300))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4300.427");
 

Basic properties

Modulus: \(4300\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4300\)
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 4300.do

\(\chi_{4300}(23,\cdot)\) \(\chi_{4300}(67,\cdot)\) \(\chi_{4300}(83,\cdot)\) \(\chi_{4300}(103,\cdot)\) \(\chi_{4300}(167,\cdot)\) \(\chi_{4300}(187,\cdot)\) \(\chi_{4300}(203,\cdot)\) \(\chi_{4300}(267,\cdot)\) \(\chi_{4300}(283,\cdot)\) \(\chi_{4300}(367,\cdot)\) \(\chi_{4300}(427,\cdot)\) \(\chi_{4300}(447,\cdot)\) \(\chi_{4300}(483,\cdot)\) \(\chi_{4300}(487,\cdot)\) \(\chi_{4300}(547,\cdot)\) \(\chi_{4300}(583,\cdot)\) \(\chi_{4300}(627,\cdot)\) \(\chi_{4300}(683,\cdot)\) \(\chi_{4300}(703,\cdot)\) \(\chi_{4300}(783,\cdot)\) \(\chi_{4300}(787,\cdot)\) \(\chi_{4300}(827,\cdot)\) \(\chi_{4300}(883,\cdot)\) \(\chi_{4300}(927,\cdot)\) \(\chi_{4300}(963,\cdot)\) \(\chi_{4300}(1003,\cdot)\) \(\chi_{4300}(1027,\cdot)\) \(\chi_{4300}(1047,\cdot)\) \(\chi_{4300}(1063,\cdot)\) \(\chi_{4300}(1127,\cdot)\) ...

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

\((2151,1377,3701)\) → \((-1,e\left(\frac{1}{20}\right),e\left(\frac{11}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4300 }(427, a) \) \(1\)\(1\)\(e\left(\frac{157}{420}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{157}{210}\right)\)\(e\left(\frac{1}{70}\right)\)\(e\left(\frac{299}{420}\right)\)\(e\left(\frac{233}{420}\right)\)\(e\left(\frac{37}{105}\right)\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{181}{420}\right)\)\(e\left(\frac{17}{140}\right)\)
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)
 
\( \chi_{ 4300 }(427,a) \;\) at \(\;a = \) e.g. 2