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Properties

Label 97693.4071
Modulus $97693$
Conductor $97693$
Order $1155$
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(97693, base_ring=CyclotomicField(2310)) M = H._module chi = DirichletCharacter(H, M([1276,2010]))
 
Copy content gp:[g,chi] = znchar(Mod(4071, 97693))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("97693.4071");
 

Basic properties

Modulus: \(97693\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(97693\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1155\)
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 97693.xk

\(\chi_{97693}(84,\cdot)\) \(\chi_{97693}(120,\cdot)\) \(\chi_{97693}(381,\cdot)\) \(\chi_{97693}(383,\cdot)\) \(\chi_{97693}(419,\cdot)\) \(\chi_{97693}(527,\cdot)\) \(\chi_{97693}(903,\cdot)\) \(\chi_{97693}(983,\cdot)\) \(\chi_{97693}(1026,\cdot)\) \(\chi_{97693}(1209,\cdot)\) \(\chi_{97693}(1290,\cdot)\) \(\chi_{97693}(1533,\cdot)\) \(\chi_{97693}(1998,\cdot)\) \(\chi_{97693}(2190,\cdot)\) \(\chi_{97693}(2726,\cdot)\) \(\chi_{97693}(2827,\cdot)\) \(\chi_{97693}(2836,\cdot)\) \(\chi_{97693}(2862,\cdot)\) \(\chi_{97693}(2937,\cdot)\) \(\chi_{97693}(3234,\cdot)\) \(\chi_{97693}(3319,\cdot)\) \(\chi_{97693}(3385,\cdot)\) \(\chi_{97693}(3980,\cdot)\) \(\chi_{97693}(4071,\cdot)\) \(\chi_{97693}(4620,\cdot)\) \(\chi_{97693}(4694,\cdot)\) \(\chi_{97693}(5159,\cdot)\) \(\chi_{97693}(5163,\cdot)\) \(\chi_{97693}(5457,\cdot)\) \(\chi_{97693}(5605,\cdot)\) ...

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: $\Q(\zeta_{1155})$
Fixed field: Number field defined by a degree 1155 polynomial (not computed)

Values on generators

\((81026,33339)\) → \((e\left(\frac{58}{105}\right),e\left(\frac{67}{77}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 97693 }(4071, a) \) \(1\)\(1\)\(e\left(\frac{158}{1155}\right)\)\(e\left(\frac{719}{1155}\right)\)\(e\left(\frac{316}{1155}\right)\)\(e\left(\frac{262}{385}\right)\)\(e\left(\frac{877}{1155}\right)\)\(e\left(\frac{1052}{1155}\right)\)\(e\left(\frac{158}{385}\right)\)\(e\left(\frac{283}{1155}\right)\)\(e\left(\frac{944}{1155}\right)\)\(e\left(\frac{347}{385}\right)\)
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)
 
\( \chi_{ 97693 }(4071,a) \;\) at \(\;a = \) e.g. 2