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Properties

Label 10535.3692
Modulus $10535$
Conductor $10535$
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(10535, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([21,50,14]))
 
Copy content gp:[g,chi] = znchar(Mod(3692, 10535))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("10535.3692");
 

Basic properties

Modulus: \(10535\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(10535\)
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 10535.qw

\(\chi_{10535}(222,\cdot)\) \(\chi_{10535}(523,\cdot)\) \(\chi_{10535}(682,\cdot)\) \(\chi_{10535}(983,\cdot)\) \(\chi_{10535}(1727,\cdot)\) \(\chi_{10535}(2488,\cdot)\) \(\chi_{10535}(3232,\cdot)\) \(\chi_{10535}(3533,\cdot)\) \(\chi_{10535}(3692,\cdot)\) \(\chi_{10535}(3993,\cdot)\) \(\chi_{10535}(4737,\cdot)\) \(\chi_{10535}(5038,\cdot)\) \(\chi_{10535}(5197,\cdot)\) \(\chi_{10535}(5498,\cdot)\) \(\chi_{10535}(6543,\cdot)\) \(\chi_{10535}(6702,\cdot)\) \(\chi_{10535}(7003,\cdot)\) \(\chi_{10535}(7747,\cdot)\) \(\chi_{10535}(8048,\cdot)\) \(\chi_{10535}(8207,\cdot)\) \(\chi_{10535}(9252,\cdot)\) \(\chi_{10535}(9553,\cdot)\) \(\chi_{10535}(9712,\cdot)\) \(\chi_{10535}(10013,\cdot)\)

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

\((6322,2796,5636)\) → \((i,e\left(\frac{25}{42}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 10535 }(3692, a) \) \(-1\)\(1\)\(e\left(\frac{19}{84}\right)\)\(e\left(\frac{43}{84}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{61}{84}\right)\)\(e\left(\frac{19}{21}\right)\)
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)
 
\( \chi_{ 10535 }(3692,a) \;\) at \(\;a = \) e.g. 2