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Properties

Label 1215.809
Modulus $1215$
Conductor $45$
Order $6$
Real no
Primitive no
Minimal no
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1215, base_ring=CyclotomicField(6)) M = H._module chi = DirichletCharacter(H, M([1,3]))
 
Copy content pari:[g,chi] = znchar(Mod(809,1215))
 

Basic properties

Modulus: \(1215\)
Conductor: \(45\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(6\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{45}(29,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 1215.h

\(\chi_{1215}(404,\cdot)\) \(\chi_{1215}(809,\cdot)\)

Copy content sage:chi.galois_orbit()
 
Copy content pari:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\mathbb{Q}(\zeta_3)\)
Fixed field: 6.0.2460375.1

Values on generators

\((731,487)\) → \((e\left(\frac{1}{6}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 1215 }(809, a) \) \(-1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(1\)\(1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1215 }(809,a) \;\) at \(\;a = \) e.g. 2