↑

Properties

Label 49619.1046
Modulus $49619$
Conductor $49619$
Order $812$
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(49619, base_ring=CyclotomicField(812)) M = H._module chi = DirichletCharacter(H, M([505,462]))
 
Copy content gp:[g,chi] = znchar(Mod(1046, 49619))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("49619.1046");
 

Basic properties

Modulus: \(49619\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(49619\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(812\)
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 49619.nr

\(\chi_{49619}(55,\cdot)\) \(\chi_{49619}(131,\cdot)\) \(\chi_{49619}(392,\cdot)\) \(\chi_{49619}(578,\cdot)\) \(\chi_{49619}(693,\cdot)\) \(\chi_{49619}(798,\cdot)\) \(\chi_{49619}(918,\cdot)\) \(\chi_{49619}(1036,\cdot)\) \(\chi_{49619}(1046,\cdot)\) \(\chi_{49619}(1055,\cdot)\) \(\chi_{49619}(1094,\cdot)\) \(\chi_{49619}(1105,\cdot)\) \(\chi_{49619}(1116,\cdot)\) \(\chi_{49619}(1245,\cdot)\) \(\chi_{49619}(1294,\cdot)\) \(\chi_{49619}(1458,\cdot)\) \(\chi_{49619}(1468,\cdot)\) \(\chi_{49619}(1643,\cdot)\) \(\chi_{49619}(1755,\cdot)\) \(\chi_{49619}(1808,\cdot)\) \(\chi_{49619}(2109,\cdot)\) \(\chi_{49619}(2502,\cdot)\) \(\chi_{49619}(3105,\cdot)\) \(\chi_{49619}(3122,\cdot)\) \(\chi_{49619}(3395,\cdot)\) \(\chi_{49619}(3767,\cdot)\) \(\chi_{49619}(4840,\cdot)\) \(\chi_{49619}(4893,\cdot)\) \(\chi_{49619}(4911,\cdot)\) \(\chi_{49619}(5318,\cdot)\) ...

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: $\Q(\zeta_{812})$
Fixed field: Number field defined by a degree 812 polynomial (not computed)

Values on generators

\((46257,3365)\) → \((e\left(\frac{505}{812}\right),e\left(\frac{33}{58}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 49619 }(1046, a) \) \(1\)\(1\)\(e\left(\frac{155}{812}\right)\)\(e\left(\frac{341}{812}\right)\)\(e\left(\frac{155}{406}\right)\)\(e\left(\frac{95}{406}\right)\)\(e\left(\frac{124}{203}\right)\)\(e\left(\frac{17}{203}\right)\)\(e\left(\frac{465}{812}\right)\)\(e\left(\frac{341}{406}\right)\)\(e\left(\frac{345}{812}\right)\)\(e\left(\frac{487}{812}\right)\)
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)
 
\( \chi_{ 49619 }(1046,a) \;\) at \(\;a = \) e.g. 2