↑

Properties

Label 1225.244
Modulus $1225$
Conductor $175$
Order $10$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1225, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([9,5]))
 
Copy content pari:[g,chi] = znchar(Mod(244,1225))
 

Basic properties

Modulus: \(1225\)
Conductor: \(175\)
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 \(\chi_{175}(69,\cdot)\)
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 1225.n

\(\chi_{1225}(244,\cdot)\) \(\chi_{1225}(489,\cdot)\) \(\chi_{1225}(734,\cdot)\) \(\chi_{1225}(979,\cdot)\)

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: 10.0.12822723388671875.1

Values on generators

\((1177,101)\) → \((e\left(\frac{9}{10}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 1225 }(244, a) \) \(-1\)\(1\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{3}{5}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1225 }(244,a) \;\) at \(\;a = \) e.g. 2