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Properties

Label 46225.729
Modulus $46225$
Conductor $46225$
Order $3010$
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(46225, base_ring=CyclotomicField(3010)) M = H._module chi = DirichletCharacter(H, M([301,10]))
 
Copy content gp:[g,chi] = znchar(Mod(729, 46225))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("46225.729");
 

Basic properties

Modulus: \(46225\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(46225\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(3010\)
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 46225.dh

\(\chi_{46225}(4,\cdot)\) \(\chi_{46225}(54,\cdot)\) \(\chi_{46225}(59,\cdot)\) \(\chi_{46225}(64,\cdot)\) \(\chi_{46225}(84,\cdot)\) \(\chi_{46225}(164,\cdot)\) \(\chi_{46225}(219,\cdot)\) \(\chi_{46225}(269,\cdot)\) \(\chi_{46225}(279,\cdot)\) \(\chi_{46225}(379,\cdot)\) \(\chi_{46225}(434,\cdot)\) \(\chi_{46225}(484,\cdot)\) \(\chi_{46225}(489,\cdot)\) \(\chi_{46225}(494,\cdot)\) \(\chi_{46225}(514,\cdot)\) \(\chi_{46225}(594,\cdot)\) \(\chi_{46225}(704,\cdot)\) \(\chi_{46225}(709,\cdot)\) \(\chi_{46225}(729,\cdot)\) \(\chi_{46225}(809,\cdot)\) \(\chi_{46225}(864,\cdot)\) \(\chi_{46225}(914,\cdot)\) \(\chi_{46225}(919,\cdot)\) \(\chi_{46225}(944,\cdot)\) \(\chi_{46225}(1079,\cdot)\) \(\chi_{46225}(1129,\cdot)\) \(\chi_{46225}(1134,\cdot)\) \(\chi_{46225}(1139,\cdot)\) \(\chi_{46225}(1159,\cdot)\) \(\chi_{46225}(1239,\cdot)\) ...

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: $\Q(\zeta_{1505})$
Fixed field: Number field defined by a degree 3010 polynomial (not computed)

Values on generators

\((44377,3701)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{1}{301}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 46225 }(729, a) \) \(1\)\(1\)\(e\left(\frac{641}{3010}\right)\)\(e\left(\frac{2117}{3010}\right)\)\(e\left(\frac{641}{1505}\right)\)\(e\left(\frac{197}{215}\right)\)\(e\left(\frac{27}{86}\right)\)\(e\left(\frac{1923}{3010}\right)\)\(e\left(\frac{612}{1505}\right)\)\(e\left(\frac{668}{1505}\right)\)\(e\left(\frac{389}{3010}\right)\)\(e\left(\frac{1069}{3010}\right)\)
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)
 
\( \chi_{ 46225 }(729,a) \;\) at \(\;a = \) e.g. 2