↑

Properties

Label 7813.4041
Modulus $7813$
Conductor $7813$
Order $120$
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(7813, base_ring=CyclotomicField(120)) M = H._module chi = DirichletCharacter(H, M([70,9]))
 
Copy content gp:[g,chi] = znchar(Mod(4041, 7813))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7813.4041");
 

Basic properties

Modulus: \(7813\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(7813\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(120\)
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 7813.fo

\(\chi_{7813}(193,\cdot)\) \(\chi_{7813}(496,\cdot)\) \(\chi_{7813}(700,\cdot)\) \(\chi_{7813}(847,\cdot)\) \(\chi_{7813}(886,\cdot)\) \(\chi_{7813}(1103,\cdot)\) \(\chi_{7813}(1610,\cdot)\) \(\chi_{7813}(1718,\cdot)\) \(\chi_{7813}(1996,\cdot)\) \(\chi_{7813}(2489,\cdot)\) \(\chi_{7813}(2503,\cdot)\) \(\chi_{7813}(2906,\cdot)\) \(\chi_{7813}(3321,\cdot)\) \(\chi_{7813}(3360,\cdot)\) \(\chi_{7813}(3413,\cdot)\) \(\chi_{7813}(3703,\cdot)\) \(\chi_{7813}(3711,\cdot)\) \(\chi_{7813}(4041,\cdot)\) \(\chi_{7813}(4292,\cdot)\) \(\chi_{7813}(5124,\cdot)\) \(\chi_{7813}(5163,\cdot)\) \(\chi_{7813}(5506,\cdot)\) \(\chi_{7813}(5514,\cdot)\) \(\chi_{7813}(5575,\cdot)\) \(\chi_{7813}(5844,\cdot)\) \(\chi_{7813}(5913,\cdot)\) \(\chi_{7813}(6506,\cdot)\) \(\chi_{7813}(6857,\cdot)\) \(\chi_{7813}(6896,\cdot)\) \(\chi_{7813}(7378,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((5410,6618)\) → \((e\left(\frac{7}{12}\right),e\left(\frac{3}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 7813 }(4041, a) \) \(1\)\(1\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{29}{30}\right)\)\(-1\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{59}{120}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{103}{120}\right)\)
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)
 
\( \chi_{ 7813 }(4041,a) \;\) at \(\;a = \) e.g. 2