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Properties

Label 7569.457
Modulus $7569$
Conductor $7569$
Order $1218$
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(7569, base_ring=CyclotomicField(1218)) M = H._module chi = DirichletCharacter(H, M([812,207]))
 
Copy content gp:[g,chi] = znchar(Mod(457, 7569))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7569.457");
 

Basic properties

Modulus: \(7569\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(7569\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1218\)
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 7569.bs

\(\chi_{7569}(4,\cdot)\) \(\chi_{7569}(13,\cdot)\) \(\chi_{7569}(22,\cdot)\) \(\chi_{7569}(34,\cdot)\) \(\chi_{7569}(67,\cdot)\) \(\chi_{7569}(121,\cdot)\) \(\chi_{7569}(151,\cdot)\) \(\chi_{7569}(178,\cdot)\) \(\chi_{7569}(187,\cdot)\) \(\chi_{7569}(238,\cdot)\) \(\chi_{7569}(241,\cdot)\) \(\chi_{7569}(265,\cdot)\) \(\chi_{7569}(274,\cdot)\) \(\chi_{7569}(283,\cdot)\) \(\chi_{7569}(295,\cdot)\) \(\chi_{7569}(328,\cdot)\) \(\chi_{7569}(382,\cdot)\) \(\chi_{7569}(412,\cdot)\) \(\chi_{7569}(439,\cdot)\) \(\chi_{7569}(448,\cdot)\) \(\chi_{7569}(457,\cdot)\) \(\chi_{7569}(499,\cdot)\) \(\chi_{7569}(502,\cdot)\) \(\chi_{7569}(526,\cdot)\) \(\chi_{7569}(535,\cdot)\) \(\chi_{7569}(544,\cdot)\) \(\chi_{7569}(556,\cdot)\) \(\chi_{7569}(589,\cdot)\) \(\chi_{7569}(643,\cdot)\) \(\chi_{7569}(673,\cdot)\) ...

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: $\Q(\zeta_{609})$
Fixed field: Number field defined by a degree 1218 polynomial (not computed)

Values on generators

\((5888,1684)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{69}{406}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 7569 }(457, a) \) \(1\)\(1\)\(e\left(\frac{1019}{1218}\right)\)\(e\left(\frac{410}{609}\right)\)\(e\left(\frac{401}{609}\right)\)\(e\left(\frac{346}{609}\right)\)\(e\left(\frac{207}{406}\right)\)\(e\left(\frac{201}{406}\right)\)\(e\left(\frac{527}{1218}\right)\)\(e\left(\frac{512}{609}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{211}{609}\right)\)
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)
 
\( \chi_{ 7569 }(457,a) \;\) at \(\;a = \) e.g. 2