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Properties

Label 2535.1193
Modulus $2535$
Conductor $2535$
Order $156$
Real no
Primitive yes
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(2535, base_ring=CyclotomicField(156)) M = H._module chi = DirichletCharacter(H, M([78,117,10]))
 
Copy content gp:[g,chi] = znchar(Mod(1193, 2535))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2535.1193");
 

Basic properties

Modulus: \(2535\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2535\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(156\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 2535.cy

\(\chi_{2535}(17,\cdot)\) \(\chi_{2535}(62,\cdot)\) \(\chi_{2535}(173,\cdot)\) \(\chi_{2535}(212,\cdot)\) \(\chi_{2535}(218,\cdot)\) \(\chi_{2535}(257,\cdot)\) \(\chi_{2535}(368,\cdot)\) \(\chi_{2535}(407,\cdot)\) \(\chi_{2535}(413,\cdot)\) \(\chi_{2535}(452,\cdot)\) \(\chi_{2535}(563,\cdot)\) \(\chi_{2535}(602,\cdot)\) \(\chi_{2535}(608,\cdot)\) \(\chi_{2535}(647,\cdot)\) \(\chi_{2535}(758,\cdot)\) \(\chi_{2535}(797,\cdot)\) \(\chi_{2535}(803,\cdot)\) \(\chi_{2535}(842,\cdot)\) \(\chi_{2535}(953,\cdot)\) \(\chi_{2535}(998,\cdot)\) \(\chi_{2535}(1148,\cdot)\) \(\chi_{2535}(1187,\cdot)\) \(\chi_{2535}(1193,\cdot)\) \(\chi_{2535}(1232,\cdot)\) \(\chi_{2535}(1343,\cdot)\) \(\chi_{2535}(1382,\cdot)\) \(\chi_{2535}(1388,\cdot)\) \(\chi_{2535}(1427,\cdot)\) \(\chi_{2535}(1538,\cdot)\) \(\chi_{2535}(1577,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((1691,1522,1861)\) → \((-1,-i,e\left(\frac{5}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(14\)\(16\)\(17\)\(19\)\(22\)
\( \chi_{ 2535 }(1193, a) \) \(1\)\(1\)\(e\left(\frac{49}{156}\right)\)\(e\left(\frac{49}{78}\right)\)\(e\left(\frac{95}{156}\right)\)\(e\left(\frac{49}{52}\right)\)\(e\left(\frac{4}{39}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{10}{39}\right)\)\(e\left(\frac{95}{156}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{12}\right)\)
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)
 
\( \chi_{ 2535 }(1193,a) \;\) at \(\;a = \) e.g. 2