↑

Properties

Label 13552.12913
Modulus $13552$
Conductor $847$
Order $66$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(13552, base_ring=CyclotomicField(66)) M = H._module chi = DirichletCharacter(H, M([0,0,55,63]))
 
Copy content gp:[g,chi] = znchar(Mod(12913, 13552))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("13552.12913");
 

Basic properties

Modulus: \(13552\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(847\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(66\)
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 \(\chi_{847}(208,\cdot)\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 13552.fa

\(\chi_{13552}(593,\cdot)\) \(\chi_{13552}(1473,\cdot)\) \(\chi_{13552}(1825,\cdot)\) \(\chi_{13552}(2705,\cdot)\) \(\chi_{13552}(3057,\cdot)\) \(\chi_{13552}(3937,\cdot)\) \(\chi_{13552}(4289,\cdot)\) \(\chi_{13552}(5169,\cdot)\) \(\chi_{13552}(5521,\cdot)\) \(\chi_{13552}(6401,\cdot)\) \(\chi_{13552}(6753,\cdot)\) \(\chi_{13552}(7633,\cdot)\) \(\chi_{13552}(8865,\cdot)\) \(\chi_{13552}(9217,\cdot)\) \(\chi_{13552}(10097,\cdot)\) \(\chi_{13552}(10449,\cdot)\) \(\chi_{13552}(11329,\cdot)\) \(\chi_{13552}(11681,\cdot)\) \(\chi_{13552}(12561,\cdot)\) \(\chi_{13552}(12913,\cdot)\)

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: \(\Q(\zeta_{33})\)
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 66 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((1695,10165,7745,10529)\) → \((1,1,e\left(\frac{5}{6}\right),e\left(\frac{21}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 13552 }(12913, a) \) \(1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{53}{66}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{13}{33}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{20}{33}\right)\)\(-1\)
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)
 
\( \chi_{ 13552 }(12913,a) \;\) at \(\;a = \) e.g. 2