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Properties

Label 1650.13
Modulus $1650$
Conductor $275$
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(1650, base_ring=CyclotomicField(20)) M = H._module chi = DirichletCharacter(H, M([0,19,2]))
 
Copy content pari:[g,chi] = znchar(Mod(13,1650))
 

Basic properties

Modulus: \(1650\)
Conductor: \(275\)
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 \(\chi_{275}(13,\cdot)\)
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 1650.cl

\(\chi_{1650}(13,\cdot)\) \(\chi_{1650}(127,\cdot)\) \(\chi_{1650}(523,\cdot)\) \(\chi_{1650}(547,\cdot)\) \(\chi_{1650}(667,\cdot)\) \(\chi_{1650}(733,\cdot)\) \(\chi_{1650}(937,\cdot)\) \(\chi_{1650}(1603,\cdot)\)

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: Number field defined by a degree 20 polynomial

Values on generators

\((551,727,1201)\) → \((1,e\left(\frac{19}{20}\right),e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 1650 }(13, a) \) \(1\)\(1\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(i\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(-i\)\(e\left(\frac{1}{10}\right)\)\(-i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1650 }(13,a) \;\) at \(\;a = \) e.g. 2