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Properties

Label 3025.686
Modulus $3025$
Conductor $275$
Order $5$
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(3025, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([8,2]))
 
Copy content pari:[g,chi] = znchar(Mod(686,3025))
 

Basic properties

Modulus: \(3025\)
Conductor: \(275\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(5\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{275}(136,\cdot)\)
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 3025.l

\(\chi_{3025}(366,\cdot)\) \(\chi_{3025}(686,\cdot)\) \(\chi_{3025}(856,\cdot)\) \(\chi_{3025}(1721,\cdot)\)

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: 5.5.5719140625.4

Values on generators

\((727,2301)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{1}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 3025 }(686, a) \) \(1\)\(1\)\(1\)\(e\left(\frac{1}{5}\right)\)\(1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{2}{5}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 3025 }(686,a) \;\) at \(\;a = \) e.g. 2