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Properties

Label 3025.62
Modulus $3025$
Conductor $3025$
Order $220$
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(3025, base_ring=CyclotomicField(220)) M = H._module chi = DirichletCharacter(H, M([99,174]))
 
Copy content gp:[g,chi] = znchar(Mod(62, 3025))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3025.62");
 

Basic properties

Modulus: \(3025\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3025\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(220\)
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 3025.cv

\(\chi_{3025}(17,\cdot)\) \(\chi_{3025}(52,\cdot)\) \(\chi_{3025}(62,\cdot)\) \(\chi_{3025}(83,\cdot)\) \(\chi_{3025}(173,\cdot)\) \(\chi_{3025}(178,\cdot)\) \(\chi_{3025}(222,\cdot)\) \(\chi_{3025}(238,\cdot)\) \(\chi_{3025}(292,\cdot)\) \(\chi_{3025}(327,\cdot)\) \(\chi_{3025}(337,\cdot)\) \(\chi_{3025}(358,\cdot)\) \(\chi_{3025}(448,\cdot)\) \(\chi_{3025}(453,\cdot)\) \(\chi_{3025}(497,\cdot)\) \(\chi_{3025}(513,\cdot)\) \(\chi_{3025}(567,\cdot)\) \(\chi_{3025}(612,\cdot)\) \(\chi_{3025}(633,\cdot)\) \(\chi_{3025}(728,\cdot)\) \(\chi_{3025}(772,\cdot)\) \(\chi_{3025}(788,\cdot)\) \(\chi_{3025}(842,\cdot)\) \(\chi_{3025}(877,\cdot)\) \(\chi_{3025}(908,\cdot)\) \(\chi_{3025}(998,\cdot)\) \(\chi_{3025}(1003,\cdot)\) \(\chi_{3025}(1047,\cdot)\) \(\chi_{3025}(1063,\cdot)\) \(\chi_{3025}(1117,\cdot)\) ...

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: $\Q(\zeta_{220})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 220 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((727,2301)\) → \((e\left(\frac{9}{20}\right),e\left(\frac{87}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 3025 }(62, a) \) \(1\)\(1\)\(e\left(\frac{53}{220}\right)\)\(-i\)\(e\left(\frac{53}{110}\right)\)\(e\left(\frac{109}{110}\right)\)\(e\left(\frac{173}{220}\right)\)\(e\left(\frac{159}{220}\right)\)\(-1\)\(e\left(\frac{51}{220}\right)\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{3}{110}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 3025 }(62,a) \;\) at \(\;a = \) e.g. 2