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Properties

Label 259200.517
Modulus $259200$
Conductor $259200$
Order $4320$
Real no
Primitive yes
Minimal yes
Parity odd

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(259200, base_ring=CyclotomicField(4320)) M = H._module chi = DirichletCharacter(H, M([0,135,1600,2808]))
 
Copy content gp:[g,chi] = znchar(Mod(517, 259200))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("259200.517");
 

Basic properties

Modulus: \(259200\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(259200\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(4320\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 259200.baq

\(\chi_{259200}(13,\cdot)\) \(\chi_{259200}(277,\cdot)\) \(\chi_{259200}(517,\cdot)\) \(\chi_{259200}(733,\cdot)\) \(\chi_{259200}(997,\cdot)\) \(\chi_{259200}(1213,\cdot)\) \(\chi_{259200}(1237,\cdot)\) \(\chi_{259200}(1453,\cdot)\)

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: $\Q(\zeta_{4320})$
Fixed field: Number field defined by a degree 4320 polynomial (not computed)

Values on generators

\((157951,202501,6401,72577)\) → \((1,e\left(\frac{1}{32}\right),e\left(\frac{10}{27}\right),e\left(\frac{13}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 259200 }(517, a) \) \(-1\)\(1\)\(e\left(\frac{211}{432}\right)\)\(e\left(\frac{3763}{4320}\right)\)\(e\left(\frac{3377}{4320}\right)\)\(e\left(\frac{197}{360}\right)\)\(e\left(\frac{283}{1440}\right)\)\(e\left(\frac{1429}{2160}\right)\)\(e\left(\frac{3661}{4320}\right)\)\(e\left(\frac{463}{540}\right)\)\(e\left(\frac{269}{1440}\right)\)\(e\left(\frac{361}{2160}\right)\)
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)
 
\( \chi_{ 259200 }(517,a) \;\) at \(\;a = \) e.g. 2