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Properties

Label 212.i
Modulus $212$
Conductor $212$
Order $26$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(212, base_ring=CyclotomicField(26)) M = H._module chi = DirichletCharacter(H, M([13,6])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(15,212)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(212\)
Conductor: \(212\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(26\)
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: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{13})\)
Fixed field: 26.0.16195738565069746760238624235485184480969630941184.1

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(5\) \(7\) \(9\) \(11\) \(13\) \(15\) \(17\) \(19\) \(21\)
\(\chi_{212}(15,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{2}{13}\right)\)
\(\chi_{212}(47,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{3}{13}\right)\)
\(\chi_{212}(63,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{8}{13}\right)\)
\(\chi_{212}(95,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{1}{13}\right)\)
\(\chi_{212}(99,\cdot)\) \(-1\) \(1\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{11}{13}\right)\)
\(\chi_{212}(119,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{4}{13}\right)\)
\(\chi_{212}(155,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{9}{13}\right)\)
\(\chi_{212}(175,\cdot)\) \(-1\) \(1\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{5}{13}\right)\)
\(\chi_{212}(183,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{12}{13}\right)\)
\(\chi_{212}(187,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{7}{13}\right)\)
\(\chi_{212}(195,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{6}{13}\right)\)
\(\chi_{212}(203,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{10}{13}\right)\)