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Properties

Label 7525.2046
Modulus $7525$
Conductor $7525$
Order $105$
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(7525, base_ring=CyclotomicField(210)) M = H._module chi = DirichletCharacter(H, M([126,70,40]))
 
Copy content gp:[g,chi] = znchar(Mod(2046, 7525))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7525.2046");
 

Basic properties

Modulus: \(7525\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(7525\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(105\)
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 7525.hh

\(\chi_{7525}(81,\cdot)\) \(\chi_{7525}(541,\cdot)\) \(\chi_{7525}(611,\cdot)\) \(\chi_{7525}(711,\cdot)\) \(\chi_{7525}(956,\cdot)\) \(\chi_{7525}(961,\cdot)\) \(\chi_{7525}(1271,\cdot)\) \(\chi_{7525}(1346,\cdot)\) \(\chi_{7525}(1586,\cdot)\) \(\chi_{7525}(1906,\cdot)\) \(\chi_{7525}(2046,\cdot)\) \(\chi_{7525}(2081,\cdot)\) \(\chi_{7525}(2116,\cdot)\) \(\chi_{7525}(2181,\cdot)\) \(\chi_{7525}(2216,\cdot)\) \(\chi_{7525}(2461,\cdot)\) \(\chi_{7525}(2466,\cdot)\) \(\chi_{7525}(2706,\cdot)\) \(\chi_{7525}(3091,\cdot)\) \(\chi_{7525}(3411,\cdot)\) \(\chi_{7525}(3586,\cdot)\) \(\chi_{7525}(3621,\cdot)\) \(\chi_{7525}(3686,\cdot)\) \(\chi_{7525}(3721,\cdot)\) \(\chi_{7525}(3966,\cdot)\) \(\chi_{7525}(3971,\cdot)\) \(\chi_{7525}(4211,\cdot)\) \(\chi_{7525}(4281,\cdot)\) \(\chi_{7525}(4356,\cdot)\) \(\chi_{7525}(4596,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 105 polynomial (not computed)

Values on generators

\((302,4301,2626)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{1}{3}\right),e\left(\frac{4}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 7525 }(2046, a) \) \(1\)\(1\)\(e\left(\frac{43}{105}\right)\)\(e\left(\frac{76}{105}\right)\)\(e\left(\frac{86}{105}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{8}{35}\right)\)\(e\left(\frac{47}{105}\right)\)\(e\left(\frac{68}{105}\right)\)\(e\left(\frac{19}{35}\right)\)\(e\left(\frac{52}{105}\right)\)\(e\left(\frac{67}{105}\right)\)
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)
 
\( \chi_{ 7525 }(2046,a) \;\) at \(\;a = \) e.g. 2