↑

Properties

Label 6175.1034
Modulus $6175$
Conductor $6175$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6175, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([42,55,10]))
 
Copy content gp:[g,chi] = znchar(Mod(1034, 6175))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6175.1034");
 

Basic properties

Modulus: \(6175\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6175\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(60\)
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 6175.jo

\(\chi_{6175}(84,\cdot)\) \(\chi_{6175}(544,\cdot)\) \(\chi_{6175}(639,\cdot)\) \(\chi_{6175}(1034,\cdot)\) \(\chi_{6175}(1319,\cdot)\) \(\chi_{6175}(1779,\cdot)\) \(\chi_{6175}(2269,\cdot)\) \(\chi_{6175}(2554,\cdot)\) \(\chi_{6175}(3014,\cdot)\) \(\chi_{6175}(3109,\cdot)\) \(\chi_{6175}(3504,\cdot)\) \(\chi_{6175}(3789,\cdot)\) \(\chi_{6175}(4344,\cdot)\) \(\chi_{6175}(4739,\cdot)\) \(\chi_{6175}(5484,\cdot)\) \(\chi_{6175}(5579,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((1977,951,3251)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{11}{12}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\( \chi_{ 6175 }(1034, a) \) \(1\)\(1\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{11}{30}\right)\)
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)
 
\( \chi_{ 6175 }(1034,a) \;\) at \(\;a = \) e.g. 2