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Properties

Label 1275.1036
Modulus $1275$
Conductor $425$
Order $10$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1275, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([0,8,5]))
 
Copy content pari:[g,chi] = znchar(Mod(1036,1275))
 

Basic properties

Modulus: \(1275\)
Conductor: \(425\)
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 \(\chi_{425}(186,\cdot)\)
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)
 

Galois orbit 1275.bd

\(\chi_{1275}(16,\cdot)\) \(\chi_{1275}(271,\cdot)\) \(\chi_{1275}(781,\cdot)\) \(\chi_{1275}(1036,\cdot)\)

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: 10.10.216652984619140625.1

Values on generators

\((851,52,751)\) → \((1,e\left(\frac{4}{5}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(19\)\(22\)
\( \chi_{ 1275 }(1036, a) \) \(1\)\(1\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(-1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{10}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1275 }(1036,a) \;\) at \(\;a = \) e.g. 2