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Properties

Label 1227.17
Modulus $1227$
Conductor $1227$
Order $102$
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(1227, base_ring=CyclotomicField(102)) M = H._module chi = DirichletCharacter(H, M([51,70]))
 
Copy content gp:[g,chi] = znchar(Mod(17, 1227))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1227.17");
 

Basic properties

Modulus: \(1227\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1227\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(102\)
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 1227.x

\(\chi_{1227}(17,\cdot)\) \(\chi_{1227}(71,\cdot)\) \(\chi_{1227}(77,\cdot)\) \(\chi_{1227}(80,\cdot)\) \(\chi_{1227}(98,\cdot)\) \(\chi_{1227}(101,\cdot)\) \(\chi_{1227}(167,\cdot)\) \(\chi_{1227}(179,\cdot)\) \(\chi_{1227}(197,\cdot)\) \(\chi_{1227}(203,\cdot)\) \(\chi_{1227}(218,\cdot)\) \(\chi_{1227}(272,\cdot)\) \(\chi_{1227}(389,\cdot)\) \(\chi_{1227}(425,\cdot)\) \(\chi_{1227}(494,\cdot)\) \(\chi_{1227}(542,\cdot)\) \(\chi_{1227}(548,\cdot)\) \(\chi_{1227}(593,\cdot)\) \(\chi_{1227}(665,\cdot)\) \(\chi_{1227}(674,\cdot)\) \(\chi_{1227}(695,\cdot)\) \(\chi_{1227}(698,\cdot)\) \(\chi_{1227}(773,\cdot)\) \(\chi_{1227}(794,\cdot)\) \(\chi_{1227}(809,\cdot)\) \(\chi_{1227}(899,\cdot)\) \(\chi_{1227}(914,\cdot)\) \(\chi_{1227}(920,\cdot)\) \(\chi_{1227}(1127,\cdot)\) \(\chi_{1227}(1136,\cdot)\) ...

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: $\Q(\zeta_{51})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 102 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((410,430)\) → \((-1,e\left(\frac{35}{51}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1227 }(17, a) \) \(-1\)\(1\)\(e\left(\frac{79}{102}\right)\)\(e\left(\frac{28}{51}\right)\)\(e\left(\frac{33}{34}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{34}\right)\)\(e\left(\frac{38}{51}\right)\)\(e\left(\frac{7}{34}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{15}{34}\right)\)\(e\left(\frac{5}{51}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 1227 }(17,a) \;\) at \(\;a = \) e.g. 2