Properties

Label 3840.3413
Modulus $3840$
Conductor $3840$
Order $64$
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(3840, base_ring=CyclotomicField(64)) M = H._module chi = DirichletCharacter(H, M([0,29,32,48]))
 
Copy content gp:[g,chi] = znchar(Mod(3413, 3840))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3840.3413");
 

Basic properties

Modulus: \(3840\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3840\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(64\)
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 3840.dr

\(\chi_{3840}(53,\cdot)\) \(\chi_{3840}(77,\cdot)\) \(\chi_{3840}(293,\cdot)\) \(\chi_{3840}(317,\cdot)\) \(\chi_{3840}(533,\cdot)\) \(\chi_{3840}(557,\cdot)\) \(\chi_{3840}(773,\cdot)\) \(\chi_{3840}(797,\cdot)\) \(\chi_{3840}(1013,\cdot)\) \(\chi_{3840}(1037,\cdot)\) \(\chi_{3840}(1253,\cdot)\) \(\chi_{3840}(1277,\cdot)\) \(\chi_{3840}(1493,\cdot)\) \(\chi_{3840}(1517,\cdot)\) \(\chi_{3840}(1733,\cdot)\) \(\chi_{3840}(1757,\cdot)\) \(\chi_{3840}(1973,\cdot)\) \(\chi_{3840}(1997,\cdot)\) \(\chi_{3840}(2213,\cdot)\) \(\chi_{3840}(2237,\cdot)\) \(\chi_{3840}(2453,\cdot)\) \(\chi_{3840}(2477,\cdot)\) \(\chi_{3840}(2693,\cdot)\) \(\chi_{3840}(2717,\cdot)\) \(\chi_{3840}(2933,\cdot)\) \(\chi_{3840}(2957,\cdot)\) \(\chi_{3840}(3173,\cdot)\) \(\chi_{3840}(3197,\cdot)\) \(\chi_{3840}(3413,\cdot)\) \(\chi_{3840}(3437,\cdot)\) ...

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: $\Q(\zeta_{64})$
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: Number field defined by a degree 64 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((511,2821,2561,1537)\) → \((1,e\left(\frac{29}{64}\right),-1,-i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3840 }(3413, a) \) \(1\)\(1\)\(e\left(\frac{9}{32}\right)\)\(e\left(\frac{1}{64}\right)\)\(e\left(\frac{35}{64}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{59}{64}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{47}{64}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{5}{64}\right)\)\(e\left(\frac{19}{32}\right)\)
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)
 
\( \chi_{ 3840 }(3413,a) \;\) at \(\;a = \) e.g. 2