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Properties

Label 3136.cb
Modulus $3136$
Conductor $196$
Order $42$
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(3136, base_ring=CyclotomicField(42)) M = H._module chi = DirichletCharacter(H, M([21,0,13])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(255,3136)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(3136\)
Conductor: \(196\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(42\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from 196.p
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: \(\Q(\zeta_{21})\)
Fixed field: \(\Q(\zeta_{196})^+\)

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(5\) \(9\) \(11\) \(13\) \(15\) \(17\) \(19\) \(23\) \(25\)
\(\chi_{3136}(255,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{20}{21}\right)\)
\(\chi_{3136}(383,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{16}{21}\right)\)
\(\chi_{3136}(703,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{11}{21}\right)\)
\(\chi_{3136}(831,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{19}{21}\right)\)
\(\chi_{3136}(1151,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{2}{21}\right)\)
\(\chi_{3136}(1279,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{1}{21}\right)\)
\(\chi_{3136}(1727,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{4}{21}\right)\)
\(\chi_{3136}(2047,\cdot)\) \(1\) \(1\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{5}{21}\right)\)
\(\chi_{3136}(2495,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{17}{21}\right)\)
\(\chi_{3136}(2623,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{10}{21}\right)\)
\(\chi_{3136}(2943,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{8}{21}\right)\)
\(\chi_{3136}(3071,\cdot)\) \(1\) \(1\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{13}{21}\right)\)