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Properties

Label 449.d
Modulus $449$
Conductor $449$
Order $7$
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(449, base_ring=CyclotomicField(14)) M = H._module chi = DirichletCharacter(H, M([8])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(18,449)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(449\)
Conductor: \(449\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(7\)
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: \(\Q(\zeta_{7})\)
Fixed field: 7.7.8193662024284801.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{449}(18,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(1\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{449}(25,\cdot)\) \(1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(1\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{449}(176,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(1\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{449}(324,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(1\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{449}(359,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(1\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{449}(444,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(1\) \(e\left(\frac{4}{7}\right)\)