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Properties

Label 77.m
Modulus $77$
Conductor $77$
Order $15$
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(77, base_ring=CyclotomicField(30)) M = H._module chi = DirichletCharacter(H, M([20,6])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(4,77)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(77\)
Conductor: \(77\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(15\)
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: \(\Q(\zeta_{15})\)
Fixed field: 15.15.886528337182930278529.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(8\) \(9\) \(10\) \(12\) \(13\)
\(\chi_{77}(4,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{5}\right)\)
\(\chi_{77}(9,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{5}\right)\)
\(\chi_{77}(16,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{5}\right)\)
\(\chi_{77}(25,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{4}{5}\right)\)
\(\chi_{77}(37,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{5}\right)\)
\(\chi_{77}(53,\cdot)\) \(1\) \(1\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{3}{5}\right)\)
\(\chi_{77}(58,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{4}{5}\right)\)
\(\chi_{77}(60,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{5}\right)\)