Properties

Label 3825.1444
Modulus $3825$
Conductor $3825$
Order $30$
Real no
Primitive yes
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(3825, base_ring=CyclotomicField(30)) M = H._module chi = DirichletCharacter(H, M([10,27,15]))
 
Copy content pari:[g,chi] = znchar(Mod(1444,3825))
 

Basic properties

Modulus: \(3825\)
Conductor: \(3825\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(30\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
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 3825.dh

\(\chi_{3825}(169,\cdot)\) \(\chi_{3825}(679,\cdot)\) \(\chi_{3825}(934,\cdot)\) \(\chi_{3825}(1444,\cdot)\) \(\chi_{3825}(2209,\cdot)\) \(\chi_{3825}(2464,\cdot)\) \(\chi_{3825}(3229,\cdot)\) \(\chi_{3825}(3739,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Values on generators

\((2126,2602,2026)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{9}{10}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(19\)\(22\)
\( \chi_{ 3825 }(1444, a) \) \(1\)\(1\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{7}{15}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 3825 }(1444,a) \;\) at \(\;a = \) e.g. 2