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Properties

Label 10097.73
Modulus $10097$
Conductor $10097$
Order $803$
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(10097, base_ring=CyclotomicField(1606)) M = H._module chi = DirichletCharacter(H, M([292,968]))
 
Copy content gp:[g,chi] = znchar(Mod(73, 10097))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("10097.73");
 

Basic properties

Modulus: \(10097\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(10097\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(803\)
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 10097.y

\(\chi_{10097}(2,\cdot)\) \(\chi_{10097}(4,\cdot)\) \(\chi_{10097}(8,\cdot)\) \(\chi_{10097}(9,\cdot)\) \(\chi_{10097}(16,\cdot)\) \(\chi_{10097}(18,\cdot)\) \(\chi_{10097}(32,\cdot)\) \(\chi_{10097}(36,\cdot)\) \(\chi_{10097}(49,\cdot)\) \(\chi_{10097}(55,\cdot)\) \(\chi_{10097}(64,\cdot)\) \(\chi_{10097}(72,\cdot)\) \(\chi_{10097}(73,\cdot)\) \(\chi_{10097}(81,\cdot)\) \(\chi_{10097}(98,\cdot)\) \(\chi_{10097}(110,\cdot)\) \(\chi_{10097}(128,\cdot)\) \(\chi_{10097}(141,\cdot)\) \(\chi_{10097}(144,\cdot)\) \(\chi_{10097}(146,\cdot)\) \(\chi_{10097}(196,\cdot)\) \(\chi_{10097}(209,\cdot)\) \(\chi_{10097}(220,\cdot)\) \(\chi_{10097}(223,\cdot)\) \(\chi_{10097}(256,\cdot)\) \(\chi_{10097}(271,\cdot)\) \(\chi_{10097}(282,\cdot)\) \(\chi_{10097}(288,\cdot)\) \(\chi_{10097}(292,\cdot)\) \(\chi_{10097}(324,\cdot)\) ...

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: $\Q(\zeta_{803})$
Fixed field: Number field defined by a degree 803 polynomial (not computed)

Values on generators

\((879,7039)\) → \((e\left(\frac{2}{11}\right),e\left(\frac{44}{73}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 10097 }(73, a) \) \(1\)\(1\)\(e\left(\frac{303}{803}\right)\)\(e\left(\frac{620}{803}\right)\)\(e\left(\frac{606}{803}\right)\)\(e\left(\frac{740}{803}\right)\)\(e\left(\frac{120}{803}\right)\)\(e\left(\frac{79}{803}\right)\)\(e\left(\frac{106}{803}\right)\)\(e\left(\frac{437}{803}\right)\)\(e\left(\frac{240}{803}\right)\)\(e\left(\frac{687}{803}\right)\)
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)
 
\( \chi_{ 10097 }(73,a) \;\) at \(\;a = \) e.g. 2