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Properties

Label 3981.3050
Modulus $3981$
Conductor $3981$
Order $34$
Real no
Primitive yes
Minimal yes
Parity odd

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(3981, base_ring=CyclotomicField(34)) M = H._module chi = DirichletCharacter(H, M([17,6]))
 
Copy content gp:[g,chi] = znchar(Mod(3050, 3981))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3981.3050");
 

Basic properties

Modulus: \(3981\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3981\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(34\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 3981.p

\(\chi_{3981}(230,\cdot)\) \(\chi_{3981}(317,\cdot)\) \(\chi_{3981}(794,\cdot)\) \(\chi_{3981}(821,\cdot)\) \(\chi_{3981}(1064,\cdot)\) \(\chi_{3981}(2012,\cdot)\) \(\chi_{3981}(2171,\cdot)\) \(\chi_{3981}(2222,\cdot)\) \(\chi_{3981}(2291,\cdot)\) \(\chi_{3981}(2474,\cdot)\) \(\chi_{3981}(2579,\cdot)\) \(\chi_{3981}(2765,\cdot)\) \(\chi_{3981}(2819,\cdot)\) \(\chi_{3981}(3032,\cdot)\) \(\chi_{3981}(3050,\cdot)\) \(\chi_{3981}(3206,\cdot)\)

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: \(\Q(\zeta_{17})\)
Fixed field: Number field defined by a degree 34 polynomial

Values on generators

\((1328,1330)\) → \((-1,e\left(\frac{3}{17}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 3981 }(3050, a) \) \(-1\)\(1\)\(e\left(\frac{29}{34}\right)\)\(e\left(\frac{12}{17}\right)\)\(e\left(\frac{21}{34}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{19}{34}\right)\)\(e\left(\frac{8}{17}\right)\)\(e\left(\frac{25}{34}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{15}{34}\right)\)\(e\left(\frac{7}{17}\right)\)
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)
 
\( \chi_{ 3981 }(3050,a) \;\) at \(\;a = \) e.g. 2