Properties

Label 3150.1067
Modulus $3150$
Conductor $1575$
Order $60$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3150, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([50,39,10]))
 
Copy content pari:[g,chi] = znchar(Mod(1067,3150))
 

Basic properties

Modulus: \(3150\)
Conductor: \(1575\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(60\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1575}(1067,\cdot)\)
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 3150.ep

\(\chi_{3150}(47,\cdot)\) \(\chi_{3150}(173,\cdot)\) \(\chi_{3150}(437,\cdot)\) \(\chi_{3150}(563,\cdot)\) \(\chi_{3150}(677,\cdot)\) \(\chi_{3150}(803,\cdot)\) \(\chi_{3150}(1067,\cdot)\) \(\chi_{3150}(1433,\cdot)\) \(\chi_{3150}(1697,\cdot)\) \(\chi_{3150}(1823,\cdot)\) \(\chi_{3150}(1937,\cdot)\) \(\chi_{3150}(2063,\cdot)\) \(\chi_{3150}(2327,\cdot)\) \(\chi_{3150}(2453,\cdot)\) \(\chi_{3150}(2567,\cdot)\) \(\chi_{3150}(3083,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((2801,127,451)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{13}{20}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 3150 }(1067, a) \) \(-1\)\(1\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{1}{12}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 3150 }(1067,a) \;\) at \(\;a = \) e.g. 2