Properties

Label 1011.797
Modulus $1011$
Conductor $1011$
Order $56$
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(1011, base_ring=CyclotomicField(56)) M = H._module chi = DirichletCharacter(H, M([28,33]))
 
Copy content gp:[g,chi] = znchar(Mod(797, 1011))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1011.797");
 

Basic properties

Modulus: \(1011\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1011\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(56\)
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 1011.bf

\(\chi_{1011}(47,\cdot)\) \(\chi_{1011}(56,\cdot)\) \(\chi_{1011}(137,\cdot)\) \(\chi_{1011}(200,\cdot)\) \(\chi_{1011}(281,\cdot)\) \(\chi_{1011}(290,\cdot)\) \(\chi_{1011}(344,\cdot)\) \(\chi_{1011}(362,\cdot)\) \(\chi_{1011}(380,\cdot)\) \(\chi_{1011}(458,\cdot)\) \(\chi_{1011}(551,\cdot)\) \(\chi_{1011}(626,\cdot)\) \(\chi_{1011}(635,\cdot)\) \(\chi_{1011}(647,\cdot)\) \(\chi_{1011}(668,\cdot)\) \(\chi_{1011}(680,\cdot)\) \(\chi_{1011}(701,\cdot)\) \(\chi_{1011}(713,\cdot)\) \(\chi_{1011}(722,\cdot)\) \(\chi_{1011}(797,\cdot)\) \(\chi_{1011}(890,\cdot)\) \(\chi_{1011}(968,\cdot)\) \(\chi_{1011}(986,\cdot)\) \(\chi_{1011}(1004,\cdot)\)

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: $\Q(\zeta_{56})$
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 56 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((338,10)\) → \((-1,e\left(\frac{33}{56}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1011 }(797, a) \) \(-1\)\(1\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{53}{56}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{33}{56}\right)\)\(e\left(\frac{15}{56}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{4}{7}\right)\)
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)
 
\( \chi_{ 1011 }(797,a) \;\) at \(\;a = \) e.g. 2