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Properties

Label 3125.351
Modulus $3125$
Conductor $625$
Order $125$
Real no
Primitive no
Minimal no
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(3125, base_ring=CyclotomicField(250)) M = H._module chi = DirichletCharacter(H, M([146]))
 
Copy content gp:[g,chi] = znchar(Mod(351, 3125))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3125.351");
 

Basic properties

Modulus: \(3125\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(625\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(125\)
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 \(\chi_{625}(146,\cdot)\)
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: no
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 3125.j

\(\chi_{3125}(26,\cdot)\) \(\chi_{3125}(51,\cdot)\) \(\chi_{3125}(76,\cdot)\) \(\chi_{3125}(101,\cdot)\) \(\chi_{3125}(151,\cdot)\) \(\chi_{3125}(176,\cdot)\) \(\chi_{3125}(201,\cdot)\) \(\chi_{3125}(226,\cdot)\) \(\chi_{3125}(276,\cdot)\) \(\chi_{3125}(301,\cdot)\) \(\chi_{3125}(326,\cdot)\) \(\chi_{3125}(351,\cdot)\) \(\chi_{3125}(401,\cdot)\) \(\chi_{3125}(426,\cdot)\) \(\chi_{3125}(451,\cdot)\) \(\chi_{3125}(476,\cdot)\) \(\chi_{3125}(526,\cdot)\) \(\chi_{3125}(551,\cdot)\) \(\chi_{3125}(576,\cdot)\) \(\chi_{3125}(601,\cdot)\) \(\chi_{3125}(651,\cdot)\) \(\chi_{3125}(676,\cdot)\) \(\chi_{3125}(701,\cdot)\) \(\chi_{3125}(726,\cdot)\) \(\chi_{3125}(776,\cdot)\) \(\chi_{3125}(801,\cdot)\) \(\chi_{3125}(826,\cdot)\) \(\chi_{3125}(851,\cdot)\) \(\chi_{3125}(901,\cdot)\) \(\chi_{3125}(926,\cdot)\) ...

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: $\Q(\zeta_{125})$
Fixed field: Number field defined by a degree 125 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{73}{125}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 3125 }(351, a) \) \(1\)\(1\)\(e\left(\frac{73}{125}\right)\)\(e\left(\frac{61}{125}\right)\)\(e\left(\frac{21}{125}\right)\)\(e\left(\frac{9}{125}\right)\)\(e\left(\frac{6}{25}\right)\)\(e\left(\frac{94}{125}\right)\)\(e\left(\frac{122}{125}\right)\)\(e\left(\frac{123}{125}\right)\)\(e\left(\frac{82}{125}\right)\)\(e\left(\frac{22}{125}\right)\)
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)
 
\( \chi_{ 3125 }(351,a) \;\) at \(\;a = \) e.g. 2