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Properties

Label 1017.262
Modulus $1017$
Conductor $113$
Order $56$
Real no
Primitive no
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(1017, base_ring=CyclotomicField(56)) M = H._module chi = DirichletCharacter(H, M([0,13]))
 
Copy content gp:[g,chi] = znchar(Mod(262, 1017))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1017.262");
 

Basic properties

Modulus: \(1017\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(113\)
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: no, induced from \(\chi_{113}(36,\cdot)\)
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: 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 1017.bf

\(\chi_{1017}(82,\cdot)\) \(\chi_{1017}(91,\cdot)\) \(\chi_{1017}(100,\cdot)\) \(\chi_{1017}(154,\cdot)\) \(\chi_{1017}(163,\cdot)\) \(\chi_{1017}(190,\cdot)\) \(\chi_{1017}(217,\cdot)\) \(\chi_{1017}(235,\cdot)\) \(\chi_{1017}(262,\cdot)\) \(\chi_{1017}(289,\cdot)\) \(\chi_{1017}(298,\cdot)\) \(\chi_{1017}(352,\cdot)\) \(\chi_{1017}(361,\cdot)\) \(\chi_{1017}(370,\cdot)\) \(\chi_{1017}(478,\cdot)\) \(\chi_{1017}(514,\cdot)\) \(\chi_{1017}(667,\cdot)\) \(\chi_{1017}(703,\cdot)\) \(\chi_{1017}(730,\cdot)\) \(\chi_{1017}(739,\cdot)\) \(\chi_{1017}(766,\cdot)\) \(\chi_{1017}(802,\cdot)\) \(\chi_{1017}(955,\cdot)\) \(\chi_{1017}(991,\cdot)\)

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: $\Q(\zeta_{56})$
Fixed field: Number field defined by a degree 56 polynomial

Values on generators

\((227,568)\) → \((1,e\left(\frac{13}{56}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1017 }(262, a) \) \(1\)\(1\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{15}{56}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{3}{56}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{1}{7}\right)\)
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)
 
\( \chi_{ 1017 }(262,a) \;\) at \(\;a = \) e.g. 2