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Properties

Label 5704.477
Modulus $5704$
Conductor $5704$
Order $330$
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(5704, base_ring=CyclotomicField(330)) M = H._module chi = DirichletCharacter(H, M([0,165,105,209]))
 
Copy content gp:[g,chi] = znchar(Mod(477, 5704))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5704.477");
 

Basic properties

Modulus: \(5704\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5704\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(330\)
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 5704.ep

\(\chi_{5704}(21,\cdot)\) \(\chi_{5704}(53,\cdot)\) \(\chi_{5704}(189,\cdot)\) \(\chi_{5704}(365,\cdot)\) \(\chi_{5704}(389,\cdot)\) \(\chi_{5704}(477,\cdot)\) \(\chi_{5704}(517,\cdot)\) \(\chi_{5704}(549,\cdot)\) \(\chi_{5704}(613,\cdot)\) \(\chi_{5704}(757,\cdot)\) \(\chi_{5704}(797,\cdot)\) \(\chi_{5704}(861,\cdot)\) \(\chi_{5704}(885,\cdot)\) \(\chi_{5704}(941,\cdot)\) \(\chi_{5704}(973,\cdot)\) \(\chi_{5704}(1045,\cdot)\) \(\chi_{5704}(1109,\cdot)\) \(\chi_{5704}(1253,\cdot)\) \(\chi_{5704}(1261,\cdot)\) \(\chi_{5704}(1293,\cdot)\) \(\chi_{5704}(1437,\cdot)\) \(\chi_{5704}(1469,\cdot)\) \(\chi_{5704}(1509,\cdot)\) \(\chi_{5704}(1629,\cdot)\) \(\chi_{5704}(1677,\cdot)\) \(\chi_{5704}(1717,\cdot)\) \(\chi_{5704}(1877,\cdot)\) \(\chi_{5704}(1965,\cdot)\) \(\chi_{5704}(1997,\cdot)\) \(\chi_{5704}(2173,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

\((4279,2853,3225,5521)\) → \((1,-1,e\left(\frac{7}{22}\right),e\left(\frac{19}{30}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 5704 }(477, a) \) \(1\)\(1\)\(e\left(\frac{37}{165}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{257}{330}\right)\)\(e\left(\frac{74}{165}\right)\)\(e\left(\frac{307}{330}\right)\)\(e\left(\frac{152}{165}\right)\)\(e\left(\frac{39}{55}\right)\)\(e\left(\frac{109}{165}\right)\)\(e\left(\frac{133}{165}\right)\)\(e\left(\frac{1}{330}\right)\)
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)
 
\( \chi_{ 5704 }(477,a) \;\) at \(\;a = \) e.g. 2