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Properties

Label 27209.1557
Modulus $27209$
Conductor $27209$
Order $858$
Real no
Primitive yes
Minimal yes
Parity odd

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(27209, base_ring=CyclotomicField(858)) M = H._module chi = DirichletCharacter(H, M([143,517,312]))
 
Copy content gp:[g,chi] = znchar(Mod(1557, 27209))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("27209.1557");
 

Basic properties

Modulus: \(27209\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(27209\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(858\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 27209.ib

\(\chi_{27209}(82,\cdot)\) \(\chi_{27209}(101,\cdot)\) \(\chi_{27209}(173,\cdot)\) \(\chi_{27209}(374,\cdot)\) \(\chi_{27209}(446,\cdot)\) \(\chi_{27209}(537,\cdot)\) \(\chi_{27209}(556,\cdot)\) \(\chi_{27209}(647,\cdot)\) \(\chi_{27209}(719,\cdot)\) \(\chi_{27209}(738,\cdot)\) \(\chi_{27209}(901,\cdot)\) \(\chi_{27209}(1083,\cdot)\) \(\chi_{27209}(1557,\cdot)\) \(\chi_{27209}(1720,\cdot)\) \(\chi_{27209}(1830,\cdot)\) \(\chi_{27209}(1902,\cdot)\) \(\chi_{27209}(1921,\cdot)\) \(\chi_{27209}(2194,\cdot)\) \(\chi_{27209}(2266,\cdot)\) \(\chi_{27209}(2285,\cdot)\) \(\chi_{27209}(2467,\cdot)\) \(\chi_{27209}(2539,\cdot)\) \(\chi_{27209}(2630,\cdot)\) \(\chi_{27209}(2649,\cdot)\) \(\chi_{27209}(2740,\cdot)\) \(\chi_{27209}(2812,\cdot)\) \(\chi_{27209}(2831,\cdot)\) \(\chi_{27209}(2994,\cdot)\) \(\chi_{27209}(3085,\cdot)\) \(\chi_{27209}(3176,\cdot)\) ...

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: $\Q(\zeta_{429})$
Fixed field: Number field defined by a degree 858 polynomial (not computed)

Values on generators

\((3888,3382,5916)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{47}{78}\right),e\left(\frac{4}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 27209 }(1557, a) \) \(-1\)\(1\)\(e\left(\frac{569}{858}\right)\)\(e\left(\frac{201}{286}\right)\)\(e\left(\frac{140}{429}\right)\)\(e\left(\frac{266}{429}\right)\)\(e\left(\frac{157}{429}\right)\)\(e\left(\frac{283}{286}\right)\)\(e\left(\frac{58}{143}\right)\)\(e\left(\frac{81}{286}\right)\)\(e\left(\frac{1}{286}\right)\)\(e\left(\frac{25}{858}\right)\)
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)
 
\( \chi_{ 27209 }(1557,a) \;\) at \(\;a = \) e.g. 2