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Properties

Label 2900.289
Modulus $2900$
Conductor $725$
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(2900, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([0,3,5]))
 
Copy content pari:[g,chi] = znchar(Mod(289,2900))
 

Basic properties

Modulus: \(2900\)
Conductor: \(725\)
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 \(\chi_{725}(289,\cdot)\)
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 2900.x

\(\chi_{2900}(289,\cdot)\) \(\chi_{2900}(869,\cdot)\) \(\chi_{2900}(2029,\cdot)\) \(\chi_{2900}(2609,\cdot)\)

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: 10.10.15648764801025390625.1

Values on generators

\((1451,1277,901)\) → \((1,e\left(\frac{3}{10}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 2900 }(289, a) \) \(1\)\(1\)\(e\left(\frac{3}{5}\right)\)\(-1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{4}{5}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 2900 }(289,a) \;\) at \(\;a = \) e.g. 2