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Properties

Label 4715.904
Modulus $4715$
Conductor $4715$
Order $220$
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(4715, base_ring=CyclotomicField(220)) M = H._module chi = DirichletCharacter(H, M([110,190,143]))
 
Copy content gp:[g,chi] = znchar(Mod(904, 4715))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4715.904");
 

Basic properties

Modulus: \(4715\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4715\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(220\)
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 4715.dz

\(\chi_{4715}(74,\cdot)\) \(\chi_{4715}(84,\cdot)\) \(\chi_{4715}(159,\cdot)\) \(\chi_{4715}(244,\cdot)\) \(\chi_{4715}(364,\cdot)\) \(\chi_{4715}(389,\cdot)\) \(\chi_{4715}(494,\cdot)\) \(\chi_{4715}(569,\cdot)\) \(\chi_{4715}(594,\cdot)\) \(\chi_{4715}(654,\cdot)\) \(\chi_{4715}(664,\cdot)\) \(\chi_{4715}(774,\cdot)\) \(\chi_{4715}(799,\cdot)\) \(\chi_{4715}(894,\cdot)\) \(\chi_{4715}(904,\cdot)\) \(\chi_{4715}(964,\cdot)\) \(\chi_{4715}(1004,\cdot)\) \(\chi_{4715}(1109,\cdot)\) \(\chi_{4715}(1169,\cdot)\) \(\chi_{4715}(1184,\cdot)\) \(\chi_{4715}(1194,\cdot)\) \(\chi_{4715}(1279,\cdot)\) \(\chi_{4715}(1374,\cdot)\) \(\chi_{4715}(1399,\cdot)\) \(\chi_{4715}(1414,\cdot)\) \(\chi_{4715}(1509,\cdot)\) \(\chi_{4715}(1579,\cdot)\) \(\chi_{4715}(1594,\cdot)\) \(\chi_{4715}(1604,\cdot)\) \(\chi_{4715}(1689,\cdot)\) ...

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: $\Q(\zeta_{220})$
Fixed field: Number field defined by a degree 220 polynomial (not computed)

Values on generators

\((1887,4306,2876)\) → \((-1,e\left(\frac{19}{22}\right),e\left(\frac{13}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 4715 }(904, a) \) \(-1\)\(1\)\(e\left(\frac{7}{55}\right)\)\(e\left(\frac{3}{44}\right)\)\(e\left(\frac{14}{55}\right)\)\(e\left(\frac{43}{220}\right)\)\(e\left(\frac{57}{220}\right)\)\(e\left(\frac{21}{55}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{159}{220}\right)\)\(e\left(\frac{71}{220}\right)\)\(e\left(\frac{163}{220}\right)\)
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)
 
\( \chi_{ 4715 }(904,a) \;\) at \(\;a = \) e.g. 2