Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(338, base_ring=CyclotomicField(156)) M = H._module chi = DirichletCharacter(H, M([145]))
 
Copy content gp:[g,chi] = znchar(Mod(93, 338))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("338.93");
 

Basic properties

Modulus: \(338\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(169\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(156\)
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 \(\chi_{169}(93,\cdot)\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 338.l

\(\chi_{338}(7,\cdot)\) \(\chi_{338}(11,\cdot)\) \(\chi_{338}(15,\cdot)\) \(\chi_{338}(33,\cdot)\) \(\chi_{338}(37,\cdot)\) \(\chi_{338}(41,\cdot)\) \(\chi_{338}(45,\cdot)\) \(\chi_{338}(59,\cdot)\) \(\chi_{338}(63,\cdot)\) \(\chi_{338}(67,\cdot)\) \(\chi_{338}(71,\cdot)\) \(\chi_{338}(85,\cdot)\) \(\chi_{338}(93,\cdot)\) \(\chi_{338}(97,\cdot)\) \(\chi_{338}(111,\cdot)\) \(\chi_{338}(115,\cdot)\) \(\chi_{338}(119,\cdot)\) \(\chi_{338}(123,\cdot)\) \(\chi_{338}(137,\cdot)\) \(\chi_{338}(141,\cdot)\) \(\chi_{338}(145,\cdot)\) \(\chi_{338}(149,\cdot)\) \(\chi_{338}(163,\cdot)\) \(\chi_{338}(167,\cdot)\) \(\chi_{338}(171,\cdot)\) \(\chi_{338}(175,\cdot)\) \(\chi_{338}(189,\cdot)\) \(\chi_{338}(193,\cdot)\) \(\chi_{338}(197,\cdot)\) \(\chi_{338}(201,\cdot)\) ...

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: $\Q(\zeta_{156})$
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 156 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\(171\) → \(e\left(\frac{145}{156}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 338 }(93, a) \) \(-1\)\(1\)\(e\left(\frac{10}{39}\right)\)\(e\left(\frac{19}{52}\right)\)\(e\left(\frac{71}{156}\right)\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{115}{156}\right)\)\(e\left(\frac{97}{156}\right)\)\(e\left(\frac{55}{78}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{37}{52}\right)\)\(e\left(\frac{5}{6}\right)\)
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)
 
\( \chi_{ 338 }(93,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content comment:Gauss sum
 
Copy content sage:chi.gauss_sum(a)
 
Copy content gp:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 338 }(93,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 338 }(93,·),\chi_{ 338 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 338 }(93,·)) \;\) at \(\; a,b = \) e.g. 1,2