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Properties

Label 4221.3713
Modulus $4221$
Conductor $4221$
Order $66$
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(4221, base_ring=CyclotomicField(66)) M = H._module chi = DirichletCharacter(H, M([55,11,25]))
 
Copy content gp:[g,chi] = znchar(Mod(3713, 4221))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4221.3713");
 

Basic properties

Modulus: \(4221\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4221\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(66\)
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 4221.fn

\(\chi_{4221}(185,\cdot)\) \(\chi_{4221}(299,\cdot)\) \(\chi_{4221}(614,\cdot)\) \(\chi_{4221}(677,\cdot)\) \(\chi_{4221}(1118,\cdot)\) \(\chi_{4221}(1256,\cdot)\) \(\chi_{4221}(1307,\cdot)\) \(\chi_{4221}(1760,\cdot)\) \(\chi_{4221}(2012,\cdot)\) \(\chi_{4221}(2138,\cdot)\) \(\chi_{4221}(2201,\cdot)\) \(\chi_{4221}(2252,\cdot)\) \(\chi_{4221}(2894,\cdot)\) \(\chi_{4221}(3398,\cdot)\) \(\chi_{4221}(3449,\cdot)\) \(\chi_{4221}(3461,\cdot)\) \(\chi_{4221}(3638,\cdot)\) \(\chi_{4221}(3713,\cdot)\) \(\chi_{4221}(3764,\cdot)\) \(\chi_{4221}(4016,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((470,1207,3151)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{6}\right),e\left(\frac{25}{66}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 4221 }(3713, a) \) \(-1\)\(1\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{41}{66}\right)\)
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)
 
\( \chi_{ 4221 }(3713,a) \;\) at \(\;a = \) e.g. 2