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Properties

Label 1976.1527
Modulus $1976$
Conductor $988$
Order $12$
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(1976, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([6,0,5,4]))
 
Copy content pari:[g,chi] = znchar(Mod(1527,1976))
 

Basic properties

Modulus: \(1976\)
Conductor: \(988\)
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 \(\chi_{988}(539,\cdot)\)
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)
 

Galois orbit 1976.do

\(\chi_{1976}(7,\cdot)\) \(\chi_{1976}(847,\cdot)\) \(\chi_{1976}(999,\cdot)\) \(\chi_{1976}(1527,\cdot)\)

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: 12.12.124671053953887395784773632.1

Values on generators

\((495,989,457,1769)\) → \((-1,1,e\left(\frac{5}{12}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(21\)\(23\)\(25\)
\( \chi_{ 1976 }(1527, a) \) \(1\)\(1\)\(-1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1976 }(1527,a) \;\) at \(\;a = \) e.g. 2