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Properties

Label 1452.403
Modulus $1452$
Conductor $44$
Order $10$
Real no
Primitive no
Minimal no
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1452, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([5,0,7]))
 
Copy content pari:[g,chi] = znchar(Mod(403,1452))
 

Basic properties

Modulus: \(1452\)
Conductor: \(44\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(10\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{44}(7,\cdot)\)
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 1452.j

\(\chi_{1452}(403,\cdot)\) \(\chi_{1452}(475,\cdot)\) \(\chi_{1452}(1183,\cdot)\) \(\chi_{1452}(1207,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: \(\Q(\zeta_{44})^+\)

Values on generators

\((727,485,1333)\) → \((-1,1,e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 1452 }(403, a) \) \(1\)\(1\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(-1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{5}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1452 }(403,a) \;\) at \(\;a = \) e.g. 2