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Properties

Label 4187.3432
Modulus $4187$
Conductor $4187$
Order $78$
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(4187, base_ring=CyclotomicField(78)) M = H._module chi = DirichletCharacter(H, M([75,37]))
 
Copy content gp:[g,chi] = znchar(Mod(3432, 4187))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4187.3432");
 

Basic properties

Modulus: \(4187\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4187\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(78\)
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 4187.ey

\(\chi_{4187}(6,\cdot)\) \(\chi_{4187}(7,\cdot)\) \(\chi_{4187}(59,\cdot)\) \(\chi_{4187}(698,\cdot)\) \(\chi_{4187}(706,\cdot)\) \(\chi_{4187}(1786,\cdot)\) \(\chi_{4187}(1845,\cdot)\) \(\chi_{4187}(1880,\cdot)\) \(\chi_{4187}(2124,\cdot)\) \(\chi_{4187}(2129,\cdot)\) \(\chi_{4187}(2210,\cdot)\) \(\chi_{4187}(2423,\cdot)\) \(\chi_{4187}(2478,\cdot)\) \(\chi_{4187}(2675,\cdot)\) \(\chi_{4187}(2891,\cdot)\) \(\chi_{4187}(3432,\cdot)\) \(\chi_{4187}(3558,\cdot)\) \(\chi_{4187}(3589,\cdot)\) \(\chi_{4187}(3594,\cdot)\) \(\chi_{4187}(3615,\cdot)\) \(\chi_{4187}(4004,\cdot)\) \(\chi_{4187}(4138,\cdot)\) \(\chi_{4187}(4145,\cdot)\) \(\chi_{4187}(4151,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((4030,319)\) → \((e\left(\frac{25}{26}\right),e\left(\frac{37}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4187 }(3432, a) \) \(-1\)\(1\)\(e\left(\frac{67}{78}\right)\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{47}{78}\right)\)\(e\left(\frac{53}{78}\right)\)\(e\left(\frac{47}{78}\right)\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{25}{39}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{1}{39}\right)\)
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)
 
\( \chi_{ 4187 }(3432,a) \;\) at \(\;a = \) e.g. 2