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Properties

Label 1985.1089
Modulus $1985$
Conductor $1985$
Order $198$
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(1985, base_ring=CyclotomicField(198)) M = H._module chi = DirichletCharacter(H, M([99,124]))
 
Copy content gp:[g,chi] = znchar(Mod(1089, 1985))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1985.1089");
 

Basic properties

Modulus: \(1985\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1985\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(198\)
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 1985.ca

\(\chi_{1985}(9,\cdot)\) \(\chi_{1985}(19,\cdot)\) \(\chi_{1985}(29,\cdot)\) \(\chi_{1985}(69,\cdot)\) \(\chi_{1985}(129,\cdot)\) \(\chi_{1985}(144,\cdot)\) \(\chi_{1985}(154,\cdot)\) \(\chi_{1985}(209,\cdot)\) \(\chi_{1985}(224,\cdot)\) \(\chi_{1985}(249,\cdot)\) \(\chi_{1985}(279,\cdot)\) \(\chi_{1985}(319,\cdot)\) \(\chi_{1985}(349,\cdot)\) \(\chi_{1985}(364,\cdot)\) \(\chi_{1985}(374,\cdot)\) \(\chi_{1985}(394,\cdot)\) \(\chi_{1985}(409,\cdot)\) \(\chi_{1985}(434,\cdot)\) \(\chi_{1985}(444,\cdot)\) \(\chi_{1985}(464,\cdot)\) \(\chi_{1985}(479,\cdot)\) \(\chi_{1985}(489,\cdot)\) \(\chi_{1985}(494,\cdot)\) \(\chi_{1985}(519,\cdot)\) \(\chi_{1985}(524,\cdot)\) \(\chi_{1985}(529,\cdot)\) \(\chi_{1985}(589,\cdot)\) \(\chi_{1985}(754,\cdot)\) \(\chi_{1985}(769,\cdot)\) \(\chi_{1985}(804,\cdot)\) ...

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: $\Q(\zeta_{99})$
Fixed field: Number field defined by a degree 198 polynomial (not computed)

Values on generators

\((1192,1196)\) → \((-1,e\left(\frac{62}{99}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1985 }(1089, a) \) \(1\)\(1\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{95}{198}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{16}{99}\right)\)\(e\left(\frac{107}{198}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{95}{99}\right)\)\(e\left(\frac{67}{99}\right)\)\(e\left(\frac{167}{198}\right)\)\(e\left(\frac{149}{198}\right)\)
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)
 
\( \chi_{ 1985 }(1089,a) \;\) at \(\;a = \) e.g. 2