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Properties

Label 20097.736
Modulus $20097$
Conductor $2871$
Order $84$
Real no
Primitive no
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(20097, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([56,0,42,75]))
 
Copy content gp:[g,chi] = znchar(Mod(736, 20097))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("20097.736");
 

Basic properties

Modulus: \(20097\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2871\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(84\)
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: no, induced from \(\chi_{2871}(736,\cdot)\)
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 20097.np

\(\chi_{20097}(43,\cdot)\) \(\chi_{20097}(736,\cdot)\) \(\chi_{20097}(967,\cdot)\) \(\chi_{20097}(1429,\cdot)\) \(\chi_{20097}(2815,\cdot)\) \(\chi_{20097}(3739,\cdot)\) \(\chi_{20097}(5125,\cdot)\) \(\chi_{20097}(5587,\cdot)\) \(\chi_{20097}(5818,\cdot)\) \(\chi_{20097}(6511,\cdot)\) \(\chi_{20097}(7666,\cdot)\) \(\chi_{20097}(9283,\cdot)\) \(\chi_{20097}(10438,\cdot)\) \(\chi_{20097}(10669,\cdot)\) \(\chi_{20097}(11824,\cdot)\) \(\chi_{20097}(12517,\cdot)\) \(\chi_{20097}(13210,\cdot)\) \(\chi_{20097}(13441,\cdot)\) \(\chi_{20097}(14134,\cdot)\) \(\chi_{20097}(14827,\cdot)\) \(\chi_{20097}(15982,\cdot)\) \(\chi_{20097}(16213,\cdot)\) \(\chi_{20097}(17368,\cdot)\) \(\chi_{20097}(18985,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((2234,5743,1828,13168)\) → \((e\left(\frac{2}{3}\right),1,-1,e\left(\frac{25}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 20097 }(736, a) \) \(1\)\(1\)\(e\left(\frac{5}{84}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{5}{21}\right)\)\(i\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{2}{21}\right)\)
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)
 
\( \chi_{ 20097 }(736,a) \;\) at \(\;a = \) e.g. 2