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Properties

Label 11137.1531
Modulus $11137$
Conductor $11137$
Order $84$
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(11137, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([70,77,34]))
 
Copy content gp:[g,chi] = znchar(Mod(1531, 11137))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("11137.1531");
 

Basic properties

Modulus: \(11137\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(11137\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(84\)
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 11137.re

\(\chi_{11137}(304,\cdot)\) \(\chi_{11137}(822,\cdot)\) \(\chi_{11137}(1531,\cdot)\) \(\chi_{11137}(2049,\cdot)\) \(\chi_{11137}(2308,\cdot)\) \(\chi_{11137}(2567,\cdot)\) \(\chi_{11137}(2635,\cdot)\) \(\chi_{11137}(2656,\cdot)\) \(\chi_{11137}(3603,\cdot)\) \(\chi_{11137}(4448,\cdot)\) \(\chi_{11137}(4707,\cdot)\) \(\chi_{11137}(6023,\cdot)\) \(\chi_{11137}(6541,\cdot)\) \(\chi_{11137}(6683,\cdot)\) \(\chi_{11137}(8074,\cdot)\) \(\chi_{11137}(8354,\cdot)\) \(\chi_{11137}(8755,\cdot)\) \(\chi_{11137}(9273,\cdot)\) \(\chi_{11137}(9532,\cdot)\) \(\chi_{11137}(9791,\cdot)\) \(\chi_{11137}(10167,\cdot)\) \(\chi_{11137}(10426,\cdot)\) \(\chi_{11137}(10596,\cdot)\) \(\chi_{11137}(10827,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((1592,4516,519)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{11}{12}\right),e\left(\frac{17}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 11137 }(1531, a) \) \(-1\)\(1\)\(e\left(\frac{43}{84}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{31}{84}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{2}{21}\right)\)
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)
 
\( \chi_{ 11137 }(1531,a) \;\) at \(\;a = \) e.g. 2