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Properties

Label 1732.1207
Modulus $1732$
Conductor $1732$
Order $432$
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(1732, base_ring=CyclotomicField(432)) M = H._module chi = DirichletCharacter(H, M([216,119]))
 
Copy content gp:[g,chi] = znchar(Mod(1207, 1732))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1732.1207");
 

Basic properties

Modulus: \(1732\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1732\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(432\)
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 1732.bm

\(\chi_{1732}(7,\cdot)\) \(\chi_{1732}(15,\cdot)\) \(\chi_{1732}(19,\cdot)\) \(\chi_{1732}(23,\cdot)\) \(\chi_{1732}(31,\cdot)\) \(\chi_{1732}(47,\cdot)\) \(\chi_{1732}(55,\cdot)\) \(\chi_{1732}(63,\cdot)\) \(\chi_{1732}(71,\cdot)\) \(\chi_{1732}(83,\cdot)\) \(\chi_{1732}(87,\cdot)\) \(\chi_{1732}(107,\cdot)\) \(\chi_{1732}(135,\cdot)\) \(\chi_{1732}(163,\cdot)\) \(\chi_{1732}(175,\cdot)\) \(\chi_{1732}(207,\cdot)\) \(\chi_{1732}(219,\cdot)\) \(\chi_{1732}(231,\cdot)\) \(\chi_{1732}(247,\cdot)\) \(\chi_{1732}(259,\cdot)\) \(\chi_{1732}(263,\cdot)\) \(\chi_{1732}(267,\cdot)\) \(\chi_{1732}(291,\cdot)\) \(\chi_{1732}(303,\cdot)\) \(\chi_{1732}(307,\cdot)\) \(\chi_{1732}(311,\cdot)\) \(\chi_{1732}(319,\cdot)\) \(\chi_{1732}(323,\cdot)\) \(\chi_{1732}(339,\cdot)\) \(\chi_{1732}(371,\cdot)\) ...

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: $\Q(\zeta_{432})$
Fixed field: Number field defined by a degree 432 polynomial (not computed)

Values on generators

\((867,5)\) → \((-1,e\left(\frac{119}{432}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1732 }(1207, a) \) \(1\)\(1\)\(e\left(\frac{37}{54}\right)\)\(e\left(\frac{119}{432}\right)\)\(e\left(\frac{19}{432}\right)\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{67}{216}\right)\)\(e\left(\frac{103}{216}\right)\)\(e\left(\frac{415}{432}\right)\)\(e\left(\frac{11}{27}\right)\)\(e\left(\frac{71}{432}\right)\)\(e\left(\frac{35}{48}\right)\)
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)
 
\( \chi_{ 1732 }(1207,a) \;\) at \(\;a = \) e.g. 2