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Properties

Label 53.d
Modulus $53$
Conductor $53$
Order $13$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(53, base_ring=CyclotomicField(26)) M = H._module chi = DirichletCharacter(H, M([24])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(10,53)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(53\)
Conductor: \(53\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(13\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{13})\)
Fixed field: Number field defined by a degree 13 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{53}(10,\cdot)\) \(1\) \(1\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{7}{13}\right)\)
\(\chi_{53}(13,\cdot)\) \(1\) \(1\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{10}{13}\right)\)
\(\chi_{53}(15,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{5}{13}\right)\)
\(\chi_{53}(16,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{6}{13}\right)\)
\(\chi_{53}(24,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{4}{13}\right)\)
\(\chi_{53}(28,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{11}{13}\right)\)
\(\chi_{53}(36,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{2}{13}\right)\)
\(\chi_{53}(42,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{9}{13}\right)\)
\(\chi_{53}(44,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{12}{13}\right)\)
\(\chi_{53}(46,\cdot)\) \(1\) \(1\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{8}{13}\right)\)
\(\chi_{53}(47,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{1}{13}\right)\)
\(\chi_{53}(49,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{3}{13}\right)\)