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Properties

Label 5336.331
Modulus $5336$
Conductor $5336$
Order $44$
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(5336, base_ring=CyclotomicField(44)) M = H._module chi = DirichletCharacter(H, M([22,22,20,11]))
 
Copy content gp:[g,chi] = znchar(Mod(331, 5336))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5336.331");
 

Basic properties

Modulus: \(5336\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5336\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(44\)
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 5336.co

\(\chi_{5336}(75,\cdot)\) \(\chi_{5336}(307,\cdot)\) \(\chi_{5336}(331,\cdot)\) \(\chi_{5336}(771,\cdot)\) \(\chi_{5336}(795,\cdot)\) \(\chi_{5336}(1235,\cdot)\) \(\chi_{5336}(1467,\cdot)\) \(\chi_{5336}(2187,\cdot)\) \(\chi_{5336}(2395,\cdot)\) \(\chi_{5336}(2419,\cdot)\) \(\chi_{5336}(2651,\cdot)\) \(\chi_{5336}(2883,\cdot)\) \(\chi_{5336}(3091,\cdot)\) \(\chi_{5336}(3347,\cdot)\) \(\chi_{5336}(3555,\cdot)\) \(\chi_{5336}(3811,\cdot)\) \(\chi_{5336}(4043,\cdot)\) \(\chi_{5336}(4947,\cdot)\) \(\chi_{5336}(4971,\cdot)\) \(\chi_{5336}(5179,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((1335,2669,465,553)\) → \((-1,-1,e\left(\frac{5}{11}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 5336 }(331, a) \) \(1\)\(1\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{3}{44}\right)\)\(e\left(\frac{29}{44}\right)\)
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)
 
\( \chi_{ 5336 }(331,a) \;\) at \(\;a = \) e.g. 2