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Properties

Label 9373.178
Modulus $9373$
Conductor $9373$
Order $102$
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(9373, base_ring=CyclotomicField(102)) M = H._module chi = DirichletCharacter(H, M([17,68,41]))
 
Copy content gp:[g,chi] = znchar(Mod(178, 9373))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9373.178");
 

Basic properties

Modulus: \(9373\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(9373\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(102\)
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 9373.jy

\(\chi_{9373}(87,\cdot)\) \(\chi_{9373}(178,\cdot)\) \(\chi_{9373}(614,\cdot)\) \(\chi_{9373}(978,\cdot)\) \(\chi_{9373}(1543,\cdot)\) \(\chi_{9373}(1615,\cdot)\) \(\chi_{9373}(1725,\cdot)\) \(\chi_{9373}(2434,\cdot)\) \(\chi_{9373}(2453,\cdot)\) \(\chi_{9373}(2525,\cdot)\) \(\chi_{9373}(2726,\cdot)\) \(\chi_{9373}(2889,\cdot)\) \(\chi_{9373}(2980,\cdot)\) \(\chi_{9373}(3545,\cdot)\) \(\chi_{9373}(3617,\cdot)\) \(\chi_{9373}(3981,\cdot)\) \(\chi_{9373}(4091,\cdot)\) \(\chi_{9373}(4182,\cdot)\) \(\chi_{9373}(4618,\cdot)\) \(\chi_{9373}(5710,\cdot)\) \(\chi_{9373}(5892,\cdot)\) \(\chi_{9373}(6165,\cdot)\) \(\chi_{9373}(6529,\cdot)\) \(\chi_{9373}(6730,\cdot)\) \(\chi_{9373}(7075,\cdot)\) \(\chi_{9373}(7185,\cdot)\) \(\chi_{9373}(7367,\cdot)\) \(\chi_{9373}(7530,\cdot)\) \(\chi_{9373}(7731,\cdot)\) \(\chi_{9373}(7913,\cdot)\) ...

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: $\Q(\zeta_{51})$
Fixed field: Number field defined by a degree 102 polynomial (not computed)

Values on generators

\((1340,7932,3095)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{2}{3}\right),e\left(\frac{41}{102}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 9373 }(178, a) \) \(1\)\(1\)\(e\left(\frac{35}{51}\right)\)\(e\left(\frac{26}{51}\right)\)\(e\left(\frac{19}{51}\right)\)\(e\left(\frac{4}{17}\right)\)\(e\left(\frac{10}{51}\right)\)\(e\left(\frac{1}{17}\right)\)\(e\left(\frac{1}{51}\right)\)\(e\left(\frac{47}{51}\right)\)\(e\left(\frac{29}{34}\right)\)\(e\left(\frac{15}{17}\right)\)
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)
 
\( \chi_{ 9373 }(178,a) \;\) at \(\;a = \) e.g. 2