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Properties

Label 3300.1993
Modulus $3300$
Conductor $55$
Order $20$
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(3300, base_ring=CyclotomicField(20)) M = H._module chi = DirichletCharacter(H, M([0,0,15,2]))
 
Copy content pari:[g,chi] = znchar(Mod(1993,3300))
 

Basic properties

Modulus: \(3300\)
Conductor: \(55\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(20\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{55}(13,\cdot)\)
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 3300.ey

\(\chi_{3300}(193,\cdot)\) \(\chi_{3300}(457,\cdot)\) \(\chi_{3300}(1393,\cdot)\) \(\chi_{3300}(1657,\cdot)\) \(\chi_{3300}(1993,\cdot)\) \(\chi_{3300}(2257,\cdot)\) \(\chi_{3300}(2593,\cdot)\) \(\chi_{3300}(2857,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: \(\Q(\zeta_{55})^+\)

Values on generators

\((1651,2201,2377,1201)\) → \((1,1,-i,e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 3300 }(1993, a) \) \(1\)\(1\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(i\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(-i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 3300 }(1993,a) \;\) at \(\;a = \) e.g. 2