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Properties

Label 4276.473
Modulus $4276$
Conductor $1069$
Order $89$
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(4276, base_ring=CyclotomicField(178)) M = H._module chi = DirichletCharacter(H, M([0,136]))
 
Copy content gp:[g,chi] = znchar(Mod(473, 4276))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4276.473");
 

Basic properties

Modulus: \(4276\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1069\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(89\)
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 \(\chi_{1069}(473,\cdot)\)
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 4276.m

\(\chi_{4276}(29,\cdot)\) \(\chi_{4276}(45,\cdot)\) \(\chi_{4276}(57,\cdot)\) \(\chi_{4276}(85,\cdot)\) \(\chi_{4276}(125,\cdot)\) \(\chi_{4276}(149,\cdot)\) \(\chi_{4276}(253,\cdot)\) \(\chi_{4276}(473,\cdot)\) \(\chi_{4276}(477,\cdot)\) \(\chi_{4276}(569,\cdot)\) \(\chi_{4276}(637,\cdot)\) \(\chi_{4276}(729,\cdot)\) \(\chi_{4276}(821,\cdot)\) \(\chi_{4276}(841,\cdot)\) \(\chi_{4276}(889,\cdot)\) \(\chi_{4276}(901,\cdot)\) \(\chi_{4276}(913,\cdot)\) \(\chi_{4276}(953,\cdot)\) \(\chi_{4276}(1005,\cdot)\) \(\chi_{4276}(1065,\cdot)\) \(\chi_{4276}(1085,\cdot)\) \(\chi_{4276}(1177,\cdot)\) \(\chi_{4276}(1217,\cdot)\) \(\chi_{4276}(1273,\cdot)\) \(\chi_{4276}(1305,\cdot)\) \(\chi_{4276}(1325,\cdot)\) \(\chi_{4276}(1329,\cdot)\) \(\chi_{4276}(1349,\cdot)\) \(\chi_{4276}(1369,\cdot)\) \(\chi_{4276}(1377,\cdot)\) ...

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: $\Q(\zeta_{89})$
Fixed field: Number field defined by a degree 89 polynomial

Values on generators

\((2139,3213)\) → \((1,e\left(\frac{68}{89}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 4276 }(473, a) \) \(1\)\(1\)\(e\left(\frac{30}{89}\right)\)\(e\left(\frac{53}{89}\right)\)\(e\left(\frac{16}{89}\right)\)\(e\left(\frac{60}{89}\right)\)\(e\left(\frac{79}{89}\right)\)\(e\left(\frac{32}{89}\right)\)\(e\left(\frac{83}{89}\right)\)\(e\left(\frac{51}{89}\right)\)\(e\left(\frac{12}{89}\right)\)\(e\left(\frac{46}{89}\right)\)
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)
 
\( \chi_{ 4276 }(473,a) \;\) at \(\;a = \) e.g. 2