Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9680, base_ring=CyclotomicField(110)) M = H._module chi = DirichletCharacter(H, M([0,55,0,13]))
 
Copy content gp:[g,chi] = znchar(Mod(4441, 9680))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9680.4441");
 

Basic properties

Modulus: \(9680\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(968\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(110\)
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 \(\chi_{968}(85,\cdot)\)
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: no
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 9680.fa

\(\chi_{9680}(41,\cdot)\) \(\chi_{9680}(281,\cdot)\) \(\chi_{9680}(601,\cdot)\) \(\chi_{9680}(761,\cdot)\) \(\chi_{9680}(921,\cdot)\) \(\chi_{9680}(1161,\cdot)\) \(\chi_{9680}(1481,\cdot)\) \(\chi_{9680}(1641,\cdot)\) \(\chi_{9680}(1801,\cdot)\) \(\chi_{9680}(2041,\cdot)\) \(\chi_{9680}(2361,\cdot)\) \(\chi_{9680}(2521,\cdot)\) \(\chi_{9680}(2681,\cdot)\) \(\chi_{9680}(2921,\cdot)\) \(\chi_{9680}(3241,\cdot)\) \(\chi_{9680}(3401,\cdot)\) \(\chi_{9680}(3561,\cdot)\) \(\chi_{9680}(3801,\cdot)\) \(\chi_{9680}(4121,\cdot)\) \(\chi_{9680}(4281,\cdot)\) \(\chi_{9680}(4441,\cdot)\) \(\chi_{9680}(4681,\cdot)\) \(\chi_{9680}(5161,\cdot)\) \(\chi_{9680}(5561,\cdot)\) \(\chi_{9680}(5881,\cdot)\) \(\chi_{9680}(6201,\cdot)\) \(\chi_{9680}(6441,\cdot)\) \(\chi_{9680}(6761,\cdot)\) \(\chi_{9680}(6921,\cdot)\) \(\chi_{9680}(7081,\cdot)\) ...

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: $\Q(\zeta_{55})$
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 110 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((3631,2421,1937,4721)\) → \((1,-1,1,e\left(\frac{13}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 9680 }(4441, a) \) \(-1\)\(1\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{91}{110}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{24}{55}\right)\)\(e\left(\frac{87}{110}\right)\)\(e\left(\frac{17}{55}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{28}{55}\right)\)
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)
 
\( \chi_{ 9680 }(4441,a) \;\) at \(\;a = \) e.g. 2