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Properties

Label 34727.598
Modulus $34727$
Conductor $34727$
Order $1320$
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(34727, base_ring=CyclotomicField(1320)) M = H._module chi = DirichletCharacter(H, M([220,744,429]))
 
Copy content gp:[g,chi] = znchar(Mod(598, 34727))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("34727.598");
 

Basic properties

Modulus: \(34727\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(34727\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1320\)
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 34727.qz

\(\chi_{34727}(75,\cdot)\) \(\chi_{34727}(192,\cdot)\) \(\chi_{34727}(234,\cdot)\) \(\chi_{34727}(257,\cdot)\) \(\chi_{34727}(306,\cdot)\) \(\chi_{34727}(423,\cdot)\) \(\chi_{34727}(586,\cdot)\) \(\chi_{34727}(598,\cdot)\) \(\chi_{34727}(621,\cdot)\) \(\chi_{34727}(796,\cdot)\) \(\chi_{34727}(801,\cdot)\) \(\chi_{34727}(1032,\cdot)\) \(\chi_{34727}(1081,\cdot)\) \(\chi_{34727}(1137,\cdot)\) \(\chi_{34727}(1347,\cdot)\) \(\chi_{34727}(1543,\cdot)\) \(\chi_{34727}(1545,\cdot)\)

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: $\Q(\zeta_{1320})$
Fixed field: Number field defined by a degree 1320 polynomial (not computed)

Values on generators

\((29767,22387,22023)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{31}{55}\right),e\left(\frac{13}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 34727 }(598, a) \) \(1\)\(1\)\(e\left(\frac{229}{660}\right)\)\(e\left(\frac{77}{120}\right)\)\(e\left(\frac{229}{330}\right)\)\(e\left(\frac{457}{660}\right)\)\(e\left(\frac{87}{88}\right)\)\(e\left(\frac{9}{220}\right)\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{13}{330}\right)\)\(e\left(\frac{443}{1320}\right)\)\(e\left(\frac{221}{440}\right)\)
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)
 
\( \chi_{ 34727 }(598,a) \;\) at \(\;a = \) e.g. 2