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Properties

Label 7807.635
Modulus $7807$
Conductor $7807$
Order $420$
Real no
Primitive yes
Minimal yes
Parity even

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(7807, base_ring=CyclotomicField(420)) M = H._module chi = DirichletCharacter(H, M([315,2]))
 
Copy content gp:[g,chi] = znchar(Mod(635, 7807))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7807.635");
 

Basic properties

Modulus: \(7807\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(7807\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(420\)
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 7807.gp

\(\chi_{7807}(142,\cdot)\) \(\chi_{7807}(191,\cdot)\) \(\chi_{7807}(228,\cdot)\) \(\chi_{7807}(302,\cdot)\) \(\chi_{7807}(327,\cdot)\) \(\chi_{7807}(376,\cdot)\) \(\chi_{7807}(413,\cdot)\) \(\chi_{7807}(549,\cdot)\) \(\chi_{7807}(586,\cdot)\) \(\chi_{7807}(635,\cdot)\) \(\chi_{7807}(672,\cdot)\) \(\chi_{7807}(808,\cdot)\) \(\chi_{7807}(820,\cdot)\) \(\chi_{7807}(919,\cdot)\) \(\chi_{7807}(956,\cdot)\) \(\chi_{7807}(993,\cdot)\) \(\chi_{7807}(1215,\cdot)\) \(\chi_{7807}(1301,\cdot)\) \(\chi_{7807}(1338,\cdot)\) \(\chi_{7807}(1585,\cdot)\) \(\chi_{7807}(1622,\cdot)\) \(\chi_{7807}(1745,\cdot)\) \(\chi_{7807}(1819,\cdot)\) \(\chi_{7807}(1893,\cdot)\) \(\chi_{7807}(2029,\cdot)\) \(\chi_{7807}(2041,\cdot)\) \(\chi_{7807}(2066,\cdot)\) \(\chi_{7807}(2226,\cdot)\) \(\chi_{7807}(2251,\cdot)\) \(\chi_{7807}(2362,\cdot)\) ...

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: $\Q(\zeta_{420})$
Fixed field: Number field defined by a degree 420 polynomial (not computed)

Values on generators

\((2111,5699)\) → \((-i,e\left(\frac{1}{210}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 7807 }(635, a) \) \(1\)\(1\)\(e\left(\frac{317}{420}\right)\)\(e\left(\frac{74}{105}\right)\)\(e\left(\frac{107}{210}\right)\)\(e\left(\frac{123}{140}\right)\)\(e\left(\frac{193}{420}\right)\)\(e\left(\frac{139}{210}\right)\)\(e\left(\frac{37}{140}\right)\)\(e\left(\frac{43}{105}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{19}{70}\right)\)
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)
 
\( \chi_{ 7807 }(635,a) \;\) at \(\;a = \) e.g. 2