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Properties

Label 2620.1559
Modulus $2620$
Conductor $2620$
Order $130$
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(2620, base_ring=CyclotomicField(130)) M = H._module chi = DirichletCharacter(H, M([65,65,83]))
 
Copy content gp:[g,chi] = znchar(Mod(1559, 2620))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2620.1559");
 

Basic properties

Modulus: \(2620\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2620\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(130\)
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 2620.bq

\(\chi_{2620}(119,\cdot)\) \(\chi_{2620}(139,\cdot)\) \(\chi_{2620}(219,\cdot)\) \(\chi_{2620}(259,\cdot)\) \(\chi_{2620}(279,\cdot)\) \(\chi_{2620}(299,\cdot)\) \(\chi_{2620}(319,\cdot)\) \(\chi_{2620}(359,\cdot)\) \(\chi_{2620}(399,\cdot)\) \(\chi_{2620}(419,\cdot)\) \(\chi_{2620}(459,\cdot)\) \(\chi_{2620}(499,\cdot)\) \(\chi_{2620}(519,\cdot)\) \(\chi_{2620}(619,\cdot)\) \(\chi_{2620}(639,\cdot)\) \(\chi_{2620}(759,\cdot)\) \(\chi_{2620}(779,\cdot)\) \(\chi_{2620}(879,\cdot)\) \(\chi_{2620}(919,\cdot)\) \(\chi_{2620}(939,\cdot)\) \(\chi_{2620}(999,\cdot)\) \(\chi_{2620}(1039,\cdot)\) \(\chi_{2620}(1079,\cdot)\) \(\chi_{2620}(1159,\cdot)\) \(\chi_{2620}(1219,\cdot)\) \(\chi_{2620}(1299,\cdot)\) \(\chi_{2620}(1339,\cdot)\) \(\chi_{2620}(1539,\cdot)\) \(\chi_{2620}(1559,\cdot)\) \(\chi_{2620}(1639,\cdot)\) ...

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: $\Q(\zeta_{65})$
Fixed field: Number field defined by a degree 130 polynomial (not computed)

Values on generators

\((1311,2097,1181)\) → \((-1,-1,e\left(\frac{83}{130}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 2620 }(1559, a) \) \(1\)\(1\)\(e\left(\frac{63}{65}\right)\)\(e\left(\frac{19}{65}\right)\)\(e\left(\frac{61}{65}\right)\)\(e\left(\frac{33}{130}\right)\)\(e\left(\frac{129}{130}\right)\)\(e\left(\frac{62}{65}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{17}{65}\right)\)\(e\left(\frac{89}{130}\right)\)\(e\left(\frac{59}{65}\right)\)
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)
 
\( \chi_{ 2620 }(1559,a) \;\) at \(\;a = \) e.g. 2