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Properties

Label 5408.2369
Modulus $5408$
Conductor $169$
Order $39$
Real no
Primitive no
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(5408, base_ring=CyclotomicField(78)) M = H._module chi = DirichletCharacter(H, M([0,0,62]))
 
Copy content gp:[g,chi] = znchar(Mod(2369, 5408))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5408.2369");
 

Basic properties

Modulus: \(5408\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(169\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(39\)
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: no, induced from \(\chi_{169}(3,\cdot)\)
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 5408.cm

\(\chi_{5408}(289,\cdot)\) \(\chi_{5408}(321,\cdot)\) \(\chi_{5408}(705,\cdot)\) \(\chi_{5408}(737,\cdot)\) \(\chi_{5408}(1121,\cdot)\) \(\chi_{5408}(1153,\cdot)\) \(\chi_{5408}(1537,\cdot)\) \(\chi_{5408}(1569,\cdot)\) \(\chi_{5408}(1953,\cdot)\) \(\chi_{5408}(1985,\cdot)\) \(\chi_{5408}(2369,\cdot)\) \(\chi_{5408}(2401,\cdot)\) \(\chi_{5408}(2785,\cdot)\) \(\chi_{5408}(2817,\cdot)\) \(\chi_{5408}(3201,\cdot)\) \(\chi_{5408}(3617,\cdot)\) \(\chi_{5408}(3649,\cdot)\) \(\chi_{5408}(4065,\cdot)\) \(\chi_{5408}(4449,\cdot)\) \(\chi_{5408}(4481,\cdot)\) \(\chi_{5408}(4865,\cdot)\) \(\chi_{5408}(4897,\cdot)\) \(\chi_{5408}(5281,\cdot)\) \(\chi_{5408}(5313,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 39 polynomial

Values on generators

\((2367,677,1185)\) → \((1,1,e\left(\frac{31}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 5408 }(2369, a) \) \(1\)\(1\)\(e\left(\frac{22}{39}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{2}{39}\right)\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{34}{39}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{2}{39}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{1}{3}\right)\)
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)
 
\( \chi_{ 5408 }(2369,a) \;\) at \(\;a = \) e.g. 2