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Properties

Label 8112.dm
Modulus $8112$
Conductor $1352$
Order $26$
Real no
Primitive no
Minimal no
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8112, base_ring=CyclotomicField(26)) M = H._module chi = DirichletCharacter(H, M([0,13,0,16])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(313,8112)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(8112\)
Conductor: \(1352\)
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: no, induced from 1352.bf
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
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 26 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(7\) \(11\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\) \(35\)
\(\chi_{8112}(313,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{11}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{23}{26}\right)\)
\(\chi_{8112}(937,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{17}{26}\right)\)
\(\chi_{8112}(1561,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{11}{26}\right)\)
\(\chi_{8112}(2185,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{5}{26}\right)\)
\(\chi_{8112}(2809,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{25}{26}\right)\)
\(\chi_{8112}(3433,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{19}{26}\right)\)
\(\chi_{8112}(4681,\cdot)\) \(1\) \(1\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{7}{26}\right)\)
\(\chi_{8112}(5305,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{5}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{1}{26}\right)\)
\(\chi_{8112}(5929,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{21}{26}\right)\)
\(\chi_{8112}(6553,\cdot)\) \(1\) \(1\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{15}{26}\right)\)
\(\chi_{8112}(7177,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{9}{26}\right)\)
\(\chi_{8112}(7801,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{2}{13}\right)\) \(-1\) \(1\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{3}{26}\right)\)