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Properties

Label 4232.47
Modulus $4232$
Conductor $2116$
Order $46$
Real no
Primitive no
Minimal no
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(4232, base_ring=CyclotomicField(46)) M = H._module chi = DirichletCharacter(H, M([23,0,20]))
 
Copy content gp:[g,chi] = znchar(Mod(47, 4232))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4232.47");
 

Basic properties

Modulus: \(4232\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2116\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(46\)
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 \(\chi_{2116}(47,\cdot)\)
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: no
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 4232.v

\(\chi_{4232}(47,\cdot)\) \(\chi_{4232}(231,\cdot)\) \(\chi_{4232}(415,\cdot)\) \(\chi_{4232}(599,\cdot)\) \(\chi_{4232}(783,\cdot)\) \(\chi_{4232}(967,\cdot)\) \(\chi_{4232}(1151,\cdot)\) \(\chi_{4232}(1335,\cdot)\) \(\chi_{4232}(1519,\cdot)\) \(\chi_{4232}(1703,\cdot)\) \(\chi_{4232}(1887,\cdot)\) \(\chi_{4232}(2071,\cdot)\) \(\chi_{4232}(2255,\cdot)\) \(\chi_{4232}(2439,\cdot)\) \(\chi_{4232}(2623,\cdot)\) \(\chi_{4232}(2807,\cdot)\) \(\chi_{4232}(2991,\cdot)\) \(\chi_{4232}(3359,\cdot)\) \(\chi_{4232}(3543,\cdot)\) \(\chi_{4232}(3727,\cdot)\) \(\chi_{4232}(3911,\cdot)\) \(\chi_{4232}(4095,\cdot)\)

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: \(\Q(\zeta_{23})\)
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: 46.0.47800162176651263126448296111129436663616930416991022357727577384588531721814225992302333855612187520570978951002576428231439662710784.1
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((3175,2117,2121)\) → \((-1,1,e\left(\frac{10}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 4232 }(47, a) \) \(-1\)\(1\)\(e\left(\frac{21}{46}\right)\)\(e\left(\frac{10}{23}\right)\)\(e\left(\frac{27}{46}\right)\)\(e\left(\frac{21}{23}\right)\)\(e\left(\frac{33}{46}\right)\)\(e\left(\frac{14}{23}\right)\)\(e\left(\frac{41}{46}\right)\)\(e\left(\frac{14}{23}\right)\)\(e\left(\frac{21}{46}\right)\)\(e\left(\frac{1}{23}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 4232 }(47,a) \;\) at \(\;a = \) e.g. 2