Properties

Label 53.e
Modulus $53$
Conductor $53$
Order $26$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(53, base_ring=CyclotomicField(26))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(4,53))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(53\)
Conductor: \(53\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(26\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{13})\)
Fixed field: \(\Q(\zeta_{53})^+\)

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{53}(4,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{3}{13}\right)\)
\(\chi_{53}(6,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{1}{13}\right)\)
\(\chi_{53}(7,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{8}{13}\right)\)
\(\chi_{53}(9,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{12}{13}\right)\)
\(\chi_{53}(11,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{9}{13}\right)\)
\(\chi_{53}(17,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{2}{13}\right)\)
\(\chi_{53}(25,\cdot)\) \(1\) \(1\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{11}{13}\right)\)
\(\chi_{53}(29,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{4}{13}\right)\)
\(\chi_{53}(37,\cdot)\) \(1\) \(1\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{6}{13}\right)\)
\(\chi_{53}(38,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{5}{13}\right)\)
\(\chi_{53}(40,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{10}{13}\right)\)
\(\chi_{53}(43,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{7}{13}\right)\)