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Properties

Label 305760.211681
Modulus $305760$
Conductor $13$
Order $12$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(305760, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([0,0,0,0,0,1]))
 
Copy content pari:[g,chi] = znchar(Mod(211681,305760))
 

Basic properties

Modulus: \(305760\)
Conductor: \(13\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(12\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{13}(2,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 305760.byi

\(\chi_{305760}(47041,\cdot)\) \(\chi_{305760}(141121,\cdot)\) \(\chi_{305760}(211681,\cdot)\) \(\chi_{305760}(282241,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: \(\Q(\zeta_{13})\)

Values on generators

\((95551,114661,101921,183457,18721,211681)\) → \((1,1,1,1,1,e\left(\frac{1}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 305760 }(211681, a) \) \(-1\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(-i\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 305760 }(211681,a) \;\) at \(\;a = \) e.g. 2