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Properties

Label 11687.1100
Modulus $11687$
Conductor $11687$
Order $140$
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(11687, base_ring=CyclotomicField(140)) M = H._module chi = DirichletCharacter(H, M([35,75,98]))
 
Copy content gp:[g,chi] = znchar(Mod(1100, 11687))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("11687.1100");
 

Basic properties

Modulus: \(11687\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(11687\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(140\)
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 11687.ma

\(\chi_{11687}(525,\cdot)\) \(\chi_{11687}(1100,\cdot)\) \(\chi_{11687}(1139,\cdot)\) \(\chi_{11687}(1162,\cdot)\) \(\chi_{11687}(1360,\cdot)\) \(\chi_{11687}(2309,\cdot)\) \(\chi_{11687}(2410,\cdot)\) \(\chi_{11687}(2631,\cdot)\) \(\chi_{11687}(2712,\cdot)\) \(\chi_{11687}(2774,\cdot)\) \(\chi_{11687}(2881,\cdot)\) \(\chi_{11687}(2943,\cdot)\) \(\chi_{11687}(3346,\cdot)\) \(\chi_{11687}(3778,\cdot)\) \(\chi_{11687}(4493,\cdot)\) \(\chi_{11687}(4555,\cdot)\) \(\chi_{11687}(4766,\cdot)\) \(\chi_{11687}(4828,\cdot)\) \(\chi_{11687}(5049,\cdot)\) \(\chi_{11687}(5130,\cdot)\) \(\chi_{11687}(5231,\cdot)\) \(\chi_{11687}(5702,\cdot)\) \(\chi_{11687}(6105,\cdot)\) \(\chi_{11687}(6167,\cdot)\) \(\chi_{11687}(6378,\cdot)\) \(\chi_{11687}(6401,\cdot)\) \(\chi_{11687}(6440,\cdot)\) \(\chi_{11687}(7405,\cdot)\) \(\chi_{11687}(7467,\cdot)\) \(\chi_{11687}(7548,\cdot)\) ...

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: $\Q(\zeta_{140})$
Fixed field: Number field defined by a degree 140 polynomial (not computed)

Values on generators

\((6294,7658,7164)\) → \((i,e\left(\frac{15}{28}\right),e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 11687 }(1100, a) \) \(-1\)\(1\)\(e\left(\frac{41}{70}\right)\)\(e\left(\frac{53}{140}\right)\)\(e\left(\frac{6}{35}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{109}{140}\right)\)\(e\left(\frac{53}{70}\right)\)\(e\left(\frac{53}{70}\right)\)\(e\left(\frac{87}{140}\right)\)\(e\left(\frac{17}{70}\right)\)
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)
 
\( \chi_{ 11687 }(1100,a) \;\) at \(\;a = \) e.g. 2