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Properties

Label 8820.487
Modulus $8820$
Conductor $980$
Order $84$
Real no
Primitive no
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(8820, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([42,0,21,44]))
 
Copy content gp:[g,chi] = znchar(Mod(487, 8820))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8820.487");
 

Basic properties

Modulus: \(8820\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(980\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(84\)
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: no, induced from \(\chi_{980}(487,\cdot)\)
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 8820.ij

\(\chi_{8820}(163,\cdot)\) \(\chi_{8820}(487,\cdot)\) \(\chi_{8820}(1423,\cdot)\) \(\chi_{8820}(1747,\cdot)\) \(\chi_{8820}(1927,\cdot)\) \(\chi_{8820}(2503,\cdot)\) \(\chi_{8820}(2683,\cdot)\) \(\chi_{8820}(3187,\cdot)\) \(\chi_{8820}(3763,\cdot)\) \(\chi_{8820}(3943,\cdot)\) \(\chi_{8820}(4267,\cdot)\) \(\chi_{8820}(4447,\cdot)\) \(\chi_{8820}(5023,\cdot)\) \(\chi_{8820}(5203,\cdot)\) \(\chi_{8820}(5527,\cdot)\) \(\chi_{8820}(5707,\cdot)\) \(\chi_{8820}(6283,\cdot)\) \(\chi_{8820}(6463,\cdot)\) \(\chi_{8820}(6787,\cdot)\) \(\chi_{8820}(6967,\cdot)\) \(\chi_{8820}(7543,\cdot)\) \(\chi_{8820}(8047,\cdot)\) \(\chi_{8820}(8227,\cdot)\) \(\chi_{8820}(8803,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((4411,7841,7057,1081)\) → \((-1,1,i,e\left(\frac{11}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 8820 }(487, a) \) \(1\)\(1\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{29}{84}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{13}{84}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{84}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{11}{28}\right)\)
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)
 
\( \chi_{ 8820 }(487,a) \;\) at \(\;a = \) e.g. 2