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Properties

Label 3121.335
Modulus $3121$
Conductor $3121$
Order $1560$
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(3121, base_ring=CyclotomicField(1560)) M = H._module chi = DirichletCharacter(H, M([593]))
 
Copy content gp:[g,chi] = znchar(Mod(335, 3121))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3121.335");
 

Basic properties

Modulus: \(3121\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3121\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1560\)
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 3121.bm

\(\chi_{3121}(3,\cdot)\) \(\chi_{3121}(6,\cdot)\) \(\chi_{3121}(15,\cdot)\) \(\chi_{3121}(24,\cdot)\) \(\chi_{3121}(26,\cdot)\) \(\chi_{3121}(46,\cdot)\) \(\chi_{3121}(48,\cdot)\) \(\chi_{3121}(49,\cdot)\) \(\chi_{3121}(52,\cdot)\) \(\chi_{3121}(54,\cdot)\) \(\chi_{3121}(60,\cdot)\) \(\chi_{3121}(65,\cdot)\) \(\chi_{3121}(108,\cdot)\) \(\chi_{3121}(115,\cdot)\) \(\chi_{3121}(117,\cdot)\) \(\chi_{3121}(120,\cdot)\) \(\chi_{3121}(123,\cdot)\) \(\chi_{3121}(127,\cdot)\) \(\chi_{3121}(129,\cdot)\) \(\chi_{3121}(130,\cdot)\) \(\chi_{3121}(134,\cdot)\) \(\chi_{3121}(135,\cdot)\) \(\chi_{3121}(150,\cdot)\) \(\chi_{3121}(177,\cdot)\) \(\chi_{3121}(192,\cdot)\) \(\chi_{3121}(203,\cdot)\) \(\chi_{3121}(208,\cdot)\) \(\chi_{3121}(230,\cdot)\) \(\chi_{3121}(237,\cdot)\) \(\chi_{3121}(242,\cdot)\) ...

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: $\Q(\zeta_{1560})$
Fixed field: Number field defined by a degree 1560 polynomial (not computed)

Values on generators

\(7\) → \(e\left(\frac{593}{1560}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 3121 }(335, a) \) \(1\)\(1\)\(e\left(\frac{49}{78}\right)\)\(e\left(\frac{373}{780}\right)\)\(e\left(\frac{10}{39}\right)\)\(e\left(\frac{2}{39}\right)\)\(e\left(\frac{83}{780}\right)\)\(e\left(\frac{593}{1560}\right)\)\(e\left(\frac{23}{26}\right)\)\(e\left(\frac{373}{390}\right)\)\(e\left(\frac{53}{78}\right)\)\(e\left(\frac{91}{120}\right)\)
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)
 
\( \chi_{ 3121 }(335,a) \;\) at \(\;a = \) e.g. 2